Dijkstra's algorithm
You are encouraged to solve this task according to the task description, using any language you may know.
Dijkstra's algorithm, conceived by Dutch computer scientist Edsger Dijkstra in 1956 and published in 1959, is a graph search algorithm that solves the single-source shortest path problem for a graph with non-negative edge path costs, producing a shortest path tree.
This algorithm is often used in routing and as a subroutine in other graph algorithms.
For a given source vertex (node) in the graph, the algorithm finds the path with lowest cost (i.e. the shortest path) between that vertex and every other vertex.
- For instance
If the vertices of the graph represent cities and edge path costs represent driving distances between pairs of cities connected by a direct road, Dijkstra's algorithm can be used to find the shortest route between one city and all other cities.
As a result, the shortest path first is widely used in network routing protocols, most notably:
- Important note
The inputs to Dijkstra's algorithm are a directed and weighted graph consisting of 2 or more nodes, generally represented by:
- an adjacency matrix or list, and
- a start node.
A destination node is not specified.
The output is a set of edges depicting the shortest path to each destination node.
- An example, starting with
<lang> a──►b, cost=7, lastNode=a
a──►c, cost=9, lastNode=a
a──►d, cost=NA, lastNode=a
a──►e, cost=NA, lastNode=a
a──►f, cost=14, lastNode=a
The lowest cost is a──►b so a──►b is added to the output.
There is a connection from b──►d so the input is updated to:
a──►c, cost=9, lastNode=a
a──►d, cost=22, lastNode=b
a──►e, cost=NA, lastNode=a
a──►f, cost=14, lastNode=a
The lowest cost is a──►c so a──►c is added to the output.
Paths to d and f are cheaper via c so the input is updated to:
a──►d, cost=20, lastNode=c
a──►e, cost=NA, lastNode=a
a──►f, cost=11, lastNode=c
The lowest cost is a──►f so c──►f is added to the output.
The input is updated to:
a──►d, cost=20, lastNode=c
a──►e, cost=NA, lastNode=a
The lowest cost is a──►d so c──►d is added to the output.
There is a connection from d──►e so the input is updated to:
a──►e, cost=26, lastNode=d
Which just leaves adding d──►e to the output.
The output should now be:
[ d──►e
c──►d
c──►f
a──►c
a──►b ] </lang>
- Task
- Implement a version of Dijkstra's algorithm that outputs a set of edges depicting the shortest path to each reachable node from an origin.
- Run your program with the following directed graph starting at node a.
- Write a program which interprets the output from the above and use it to output the shortest path from node a to nodes e and f.
Vertices Number Name 1 a 2 b 3 c 4 d 5 e 6 f
| Start | End | Cost |
|---|---|---|
| a | b | 7 |
| a | c | 9 |
| a | f | 14 |
| b | c | 10 |
| b | d | 15 |
| c | d | 11 |
| c | f | 2 |
| d | e | 6 |
| e | f | 9 |
You can use numbers or names to identify vertices in your program.
- See also
11l
<lang 11l>T Edge
String start String end Int cost
F (start, end, cost)
.start = start
.end = end
.cost = cost
T Graph
[Edge] edges Set[String] vertices
F (edges)
.edges = edges.map((s, e, c) -> Edge(s, e, c))
.vertices = Set(.edges.map(e -> e.start)).union(Set(.edges.map(e -> e.end)))
F dijkstra(source, dest)
assert(source C .vertices)
V dist = Dict(.vertices, vertex -> (vertex, Float.infinity))
V previous = Dict(.vertices, vertex -> (vertex, ‘’))
dist[source] = 0
V q = copy(.vertices)
V neighbours = Dict(.vertices, vertex -> (vertex, [(String, Int)]()))
L(start, end, cost) .edges
neighbours[start].append((end, cost))
L !q.empty
V u = min(q, key' vertex -> @dist[vertex])
q.remove(u)
I dist[u] == Float.infinity | u == dest
L.break
L(v, cost) neighbours[u]
V alt = dist[u] + cost
I alt < dist[v]
dist[v] = alt
previous[v] = u
Deque[String] s
V u = dest
L previous[u] != ‘’
s.append_left(u)
u = previous[u]
s.append_left(u)
R s
V graph = Graph([(‘a’, ‘b’, 7), (‘a’, ‘c’, 9), (‘a’, ‘f’, 14), (‘b’, ‘c’, 10),
(‘b’, ‘d’, 15), (‘c’, ‘d’, 11), (‘c’, ‘f’, 2), (‘d’, ‘e’, 6),
(‘e’, ‘f’, 9)])
print(graph.dijkstra(‘a’, ‘e’))</lang>
- Output:
Deque([a, c, d, e])
Ada
This solution uses a generic package and Ada 2012 (containers, extended return statements, expression functions). The very convenient 'Img attribute is a GNAT feature. <lang Ada>private with Ada.Containers.Ordered_Maps; generic
type t_Vertex is (<>);
package Dijkstra is
type t_Graph is limited private;
-- Defining a graph (since limited private, only way to do this is to use the Build function)
type t_Edge is record
From, To : t_Vertex;
Weight : Positive;
end record;
type t_Edges is array (Integer range <>) of t_Edge;
function Build (Edges : in t_Edges; Oriented : in Boolean := True) return t_Graph;
-- Computing path and distance
type t_Path is array (Integer range <>) of t_Vertex;
function Shortest_Path (Graph : in out t_Graph;
From, To : in t_Vertex) return t_Path;
function Distance (Graph : in out t_Graph;
From, To : in t_Vertex) return Natural;
private
package Neighbor_Lists is new Ada.Containers.Ordered_Maps (Key_Type => t_Vertex, Element_Type => Positive);
type t_Vertex_Data is record
Neighbors : Neighbor_Lists.Map; -- won't be affected after build
-- Updated each time a function is called with a new source
Previous : t_Vertex;
Distance : Natural;
end record;
type t_Graph is array (t_Vertex) of t_Vertex_Data;
end Dijkstra;</lang>
<lang Ada>with Ada.Containers.Ordered_Sets; package body Dijkstra is
Infinite : constant Natural := Natural'Last;
-- ----- Graph constructor
function Build (Edges : in t_Edges; Oriented : in Boolean := True) return t_Graph is
begin
return Answer : t_Graph := (others => (Neighbors => Neighbor_Lists.Empty_Map,
Previous => t_Vertex'First,
Distance => Natural'Last)) do
for Edge of Edges loop
Answer(Edge.From).Neighbors.Insert (Key => Edge.To, New_Item => Edge.Weight);
if not Oriented then
Answer(Edge.To).Neighbors.Insert (Key => Edge.From, New_Item => Edge.Weight);
end if;
end loop;
end return;
end Build;
-- ----- Paths / distances data updating in case of computation request for a new source
procedure Update_For_Source (Graph : in out t_Graph;
From : in t_Vertex) is
function Nearer (Left, Right : in t_Vertex) return Boolean is
(Graph(Left).Distance < Graph(Right).Distance or else
(Graph(Left).Distance = Graph(Right).Distance and then Left < Right));
package Ordered is new Ada.Containers.Ordered_Sets (Element_Type => t_Vertex, "<" => Nearer);
use Ordered;
Remaining : Set := Empty_Set;
begin
-- First, let's check if vertices data are already computed for this source
if Graph(From).Distance /= 0 then
-- Reset distances and remaining vertices for a new source
for Vertex in Graph'range loop
Graph(Vertex).Distance := (if Vertex = From then 0 else Infinite);
Remaining.Insert (Vertex);
end loop;
-- ----- The Dijkstra algorithm itself
while not Remaining.Is_Empty
-- If some targets are not connected to source, at one point, the remaining
-- distances will all be infinite, hence the folllowing stop condition
and then Graph(Remaining.First_Element).Distance /= Infinite loop
declare
Nearest : constant t_Vertex := Remaining.First_Element;
procedure Update_Neighbor (Position : in Neighbor_Lists.Cursor) is
use Neighbor_Lists;
Neighbor : constant t_Vertex := Key (Position);
In_Remaining : Ordered.Cursor := Remaining.Find (Neighbor);
Try_Distance : constant Natural :=
(if In_Remaining = Ordered.No_Element
then Infinite -- vertex already reached, this distance will fail the update test below
else Graph(Nearest).Distance + Element (Position));
begin
if Try_Distance < Graph(Neighbor).Distance then
-- Update distance/path data and reorder the remaining set
Remaining.Delete (In_Remaining);
Graph(Neighbor).Distance := Try_Distance;
Graph(Neighbor).Previous := Nearest;
Remaining.Insert (Neighbor);
end if;
end Update_Neighbor;
begin
Remaining.Delete_First;
Graph(Nearest).Neighbors.Iterate (Update_Neighbor'Access);
end;
end loop;
end if;
end Update_For_Source;
-- ----- Bodies for the interfaced functions
function Shortest_Path (Graph : in out t_Graph;
From, To : in t_Vertex) return t_Path is
function Recursive_Build (From, To : in t_Vertex) return t_Path is
(if From = To then (1 => From)
else Recursive_Build(From, Graph(To).Previous) & (1 => To));
begin
Update_For_Source (Graph, From);
if Graph(To).Distance = Infinite then
raise Constraint_Error with "No path from " & From'Img & " to " & To'Img;
end if;
return Recursive_Build (From, To);
end Shortest_Path;
function Distance (Graph : in out t_Graph;
From, To : in t_Vertex) return Natural is
begin
Update_For_Source (Graph, From);
return Graph(To).Distance;
end Distance;
end Dijkstra;</lang> The testing main procedure : <lang Ada>with Ada.Text_IO; use Ada.Text_IO; with Dijkstra; procedure Test_Dijkstra is
subtype t_Tested_Vertices is Character range 'a'..'f';
package Tested is new Dijkstra (t_Vertex => t_Tested_Vertices);
use Tested;
Graph : t_Graph := Build (Edges => (('a', 'b', 7),
('a', 'c', 9),
('a', 'f', 14),
('b', 'c', 10),
('b', 'd', 15),
('c', 'd', 11),
('c', 'f', 2),
('d', 'e', 6),
('e', 'f', 9)));
procedure Display_Path (From, To : in t_Tested_Vertices) is
function Path_Image (Path : in t_Path; Start : Boolean := True) return String is
((if Start then "["
elsif Path'Length /= 0 then ","
else "") &
(if Path'Length = 0 then "]"
else Path(Path'First) & Path_Image(Path(Path'First+1..Path'Last), Start => False)));
begin
Put ("Path from '" & From & "' to '" & To & "' = ");
Put_Line (Path_Image (Shortest_Path (Graph, From, To))
& " distance =" & Distance (Graph, From, To)'Img);
exception
when others => Put_Line("no path");
end Display_Path;
begin
Display_Path ('a', 'e');
Display_Path ('a', 'f');
New_Line;
for From in t_Tested_Vertices loop
for To in t_Tested_Vertices loop
Display_Path (From, To);
end loop;
end loop;
end Test_Dijkstra;</lang>
- Output:
Path from 'a' to 'e' = [a,c,d,e] distance = 26 Path from 'a' to 'f' = [a,c,f] distance = 11 Path from 'a' to 'a' = [a] distance = 0 Path from 'a' to 'b' = [a,b] distance = 7 Path from 'a' to 'c' = [a,c] distance = 9 Path from 'a' to 'd' = [a,c,d] distance = 20 Path from 'a' to 'e' = [a,c,d,e] distance = 26 Path from 'a' to 'f' = [a,c,f] distance = 11 Path from 'b' to 'a' = no path Path from 'b' to 'b' = [b] distance = 0 Path from 'b' to 'c' = [b,c] distance = 10 Path from 'b' to 'd' = [b,d] distance = 15 Path from 'b' to 'e' = [b,d,e] distance = 21 Path from 'b' to 'f' = [b,c,f] distance = 12 Path from 'c' to 'a' = no path Path from 'c' to 'b' = no path Path from 'c' to 'c' = [c] distance = 0 Path from 'c' to 'd' = [c,d] distance = 11 Path from 'c' to 'e' = [c,d,e] distance = 17 Path from 'c' to 'f' = [c,f] distance = 2 Path from 'd' to 'a' = no path Path from 'd' to 'b' = no path Path from 'd' to 'c' = no path Path from 'd' to 'd' = [d] distance = 0 Path from 'd' to 'e' = [d,e] distance = 6 Path from 'd' to 'f' = [d,e,f] distance = 15 Path from 'e' to 'a' = no path Path from 'e' to 'b' = no path Path from 'e' to 'c' = no path Path from 'e' to 'd' = no path Path from 'e' to 'e' = [e] distance = 0 Path from 'e' to 'f' = [e,f] distance = 9 Path from 'f' to 'a' = no path Path from 'f' to 'b' = no path Path from 'f' to 'c' = no path Path from 'f' to 'd' = no path Path from 'f' to 'e' = no path Path from 'f' to 'f' = [f] distance = 0
ALGOL 68
File: prelude_dijkstras_algorithm.a68<lang algol68># -*- coding: utf-8 -*- #
COMMENT REQUIRED BY "prelude_dijkstras_algorithm.a68" CO
MODE ROUTELEN = ~; ROUTELEN route len infinity = max ~; PROC route len add = (VERTEX v, ROUTE r)ROUTELEN: route len OF v + route len OF r; # or MAX(v,r) # MODE VERTEXPAYLOAD = ~; PROC dijkstra fix value error = (STRING msg)BOOL: (put(stand error, (msg, new line)); FALSE);
- PROVIDES:#
- VERTEX*=~* #
- ROUTE*=~* #
- vertex route*=~* #
END COMMENT
MODE VALVERTEX = STRUCT(
ROUTELEN route len, FLEX[0]ROUTE route, ROUTE shortest route, VERTEXPAYLOAD vertex data
);
MODE VERTEX = REF VALVERTEX; MODE VERTEXYIELD = PROC(VERTEX)VOID; # used to "generate" VERTEX path # PRIO INIT = 1; # The same PRIOrity as +:= etc # OP INIT = (VERTEX self, VERTEXPAYLOAD vertex data)VERTEX:
self := (route len infinity, (), NIL, vertex data);
- It may be faster to preallocate "queue", rather then grow a FLEX #
OP +:= = (REF FLEX[]VERTEX in list, VERTEX rhs)REF FLEX[]VERTEX: (
[UPB in list+1]VERTEX out list; out list[:UPB in list] := in list; out list[UPB out list] := rhs; in list := out list # EXIT #
);
MODE VALROUTE = STRUCT(VERTEX from, to, ROUTELEN route len#, ROUTEPAYLOAD#); MODE ROUTE = REF VALROUTE;
OP +:= = (REF FLEX[]ROUTE in list, ROUTE rhs)REF FLEX[]ROUTE: (
[UPB in list+1]ROUTE out list; out list[:UPB in list] := in list; out list[UPB out list] := rhs; in list := out list # EXIT #
);
MODE VERTEXROUTE = UNION(VERTEX, ROUTE); MODE VERTEXROUTEYIELD = PROC(VERTEXROUTE)VOID;
- Finally: now the strong typing is in place, the task code... #
PROC vertex route gen dijkstra = (
VERTEX source, target, REF[]VALROUTE route list, VERTEXROUTEYIELD yield )VOID:(
- initialise the route len for BOTH directions on each route #
FOR this TO UPB route list DO ROUTE route = route list[this]; route OF from OF route +:= route;
- assume route lens is the same in both directions, this i.e. NO A-B gradient NOR 1-way streets #
route OF to OF route +:= (HEAP VALROUTE := (to OF route, from OF route, route len OF route)) OD;
COMMENT
Algorithium Performance "about" O(n**2)...
Optimisations:
a) bound index in [lwb queue:UPB queue] for search
b) delay adding vertices until they are actually encountered
It may be faster to preallocate "queue" vertex list, rather then grow a FLEX
END COMMENT
PROC vertex gen nearest = (REF FLEX[]VERTEX queue, VERTEXYIELD yield)VOID: (
INT vertices done := 0, lwb queue := 1;
ROUTELEN shortest route len done := -route len infinity;
WHILE vertices done <= UPB queue ANDF shortest route len done NE route len infinity DO
ROUTELEN shortest route len := route len infinity;
- skip done elements: #
FOR this FROM lwb queue TO UPB queue DO
VERTEX this vertex := queue[this];
IF NOT(shortest route len done < route len OF this vertex) THEN
lwb queue := this; # remember for next time #
break
FI
OD;
break:
- find vertex with shortest path attached #
FOR this FROM lwb queue TO UPB queue DO VERTEX this vertex := queue[this];
IF shortest route len done < route len OF this vertex ANDF
route len OF this vertex < shortest route len THEN
shortest route len := route len OF this vertex FI
OD;
- update the other vertices with shortest path found #
FOR this FROM lwb queue TO UPB queue DO VERTEX this vertex := queue[this];
IF route len OF this vertex = shortest route len THEN
vertices done +:= 1; yield(this vertex) FI
OD;
shortest route len done := shortest route len
OD
);
route len OF target := 0; FLEX[0]VERTEX queue := target;
- FOR VERTEX this vertex IN # vertex gen nearest(queue#) DO (#,
- (VERTEX this vertex)VOID: (
FOR this TO UPB route OF this vertex DO ROUTE this route = (route OF this vertex)[this];
# If this vertex has not been encountered before, then add to queue #
IF route len OF to OF this route = route len infinity THEN queue +:= to OF this route FI;
ROUTELEN route len = route len add(this vertex, this route);
IF route len < route len OF to OF this route THEN
route len OF to OF this route := route len;
shortest route OF to OF this route := this route
FI
OD;
IF this vertex IS source THEN done FI
- OD#));
IF NOT dijkstra fix value error("no path found") THEN stop FI;
- Now: generate the result #
done: (
VERTEX this vertex := source;
WHILE
yield(this vertex);
ROUTE this route = shortest route OF this vertex;
# WHILE # this route ISNT ROUTE(NIL) DO
yield(this route);
this vertex := from OF this route
OD
)
);
SKIP</lang>File: test_dijkstras_algorithm.a68<lang algol68>#!/usr/bin/a68g --script #
- -*- coding: utf-8 -*- #
CO REQUIRED BY "prelude_dijkstras_algorithm.a68" CO
MODE ROUTELEN = INT, ROUTELEN route len infinity = max int, PROC route len add = (VERTEX v, ROUTE r)ROUTELEN: route len OF v + route len OF r; # or MAX(v,r) # MODE VERTEXPAYLOAD = STRING, PROC dijkstra fix value error = (STRING msg)BOOL: (put(stand error, (msg, new line)); FALSE);
- PROVIDES:#
- VERTEX*=~* #
- ROUTE*=~* #
- vertex route*=~* #
PR READ "prelude_dijkstras_algorithm.a68" PR;
FORMAT vertex data fmt = $g$;
main:(
INT upb graph = 6, upb route list = 9;
HEAP[upb graph]VALVERTEX graph;
- name the key vertices #
FOR this TO UPB graph DO graph[this] INIT STRING("abcdef"[this]) OD;
- declare some variables of the same name #
VERTEX a := graph[1], b := graph[2], c := graph[3],
d := graph[4], e := graph[5], f := graph[6];
- define the graph #
HEAP FLEX[upb route list]VALROUTE route list := (
(a, b, 7), (a, c, 9), (a, f, 14),
(b, c, 10), (b, d, 15),
(c, d, 11), (c, f, 2),
(d, e, 6),
(e, f, 9)
);
- FOR VERTEXROUTE vertex route IN # vertex route gen dijkstra(a, e, route list#) DO #,
- (VERTEXROUTE vertex route)VOID: (
CASE vertex route IN
(VERTEX vertex): printf((vertex data fmt, vertex data OF vertex)),
(ROUTE route): printf(($" --"g(0)"-> "$, route len OF route))
ESAC
- OD #));
print(new line)
- TODO: generate random 100000 VERTEX graph test case and test performance - important #
)</lang>Output:
a --9-> c --2-> f --9-> e
Applesoft BASIC
<lang basic>100 O$ = "A" : T$ = "EF" 110 DEF FN N(P) = ASC(MID$(N$,P+(P=0),1))-64 120 DIM D(26),UNVISITED(26) 130 DIM PREVIOUS(26) : TRUE = 1 140 LET M = -1 : INFINITY = M 150 FOR I = 0 TO 26 160 LET D(I) = INFINITY : NEXT 170 FOR NE = M TO 1E38 STEP .5 180 READ C$ 190 IF LEN(C$) THEN NEXT 200 DIM C(NE),FROM(NE),T(NE) 210 DIM PC(NE) : RESTORE 220 FOR I = 0 TO NE 230 READ C(I), N$ 240 LET FROM(I) = FN N(1) 250 LET UNVISITED(FR(I)) = TRUE 260 LET T(I) = FN N(3) 270 LET UNVISITED(T(I)) = TRUE 290 NEXT 300 N$ = O$ : O = FN N(0) 310 D(O) = 0 320 FOR CV = O TO EMPTY STEP 0 330 FOR I = 0 TO NE 340 IF FROM(I) = CV THEN N = T(I) : D = D(CV) + C(I) : IF (D(N) = INFINITY) OR (D < D(N)) THEN D(N) = D : PREVIOUS(N) = CV : PC(N) = C(I) 350 NEXT I 360 LET UNVISITED(CV) = FALSE 370 LET MV = EMPTY 380 FOR I = 1 TO 26 390 IF UNVISITED(I) THEN MD = D(MV) * (MV <> INFINITY) + INFINITY * (MV = INFINITY) : IF (D(I) <> INFINITY) AND ((MD = INFINITY) OR (D(I) < MD)) THEN MV = I 400 NEXT I 410 LET CV = MV * (MD <> INF) 420 NEXT CV : M$ = CHR$(13) 430 PRINT "SHORTEST PATH"; 440 FOR I = 1 TO LEN(T$) 450 LET N$ = MID$(T$, I, 1) 460 PRINT M$ " FROM " O$; 470 PRINT " TO "; : N = FN N(0) 480 IF D(N) = INFINITY THEN PRINT N$" DOES NOT EXIST."; 490 IF D(N) <> INFINITY THEN FOR N = N TO FALSE STEP 0 : PRINT CHR$(N + 64); : IF N < > O THEN PRINT " <- "; : N = PREVIOUS(N): NEXT N 500 IF D(N) <> INFINITY THEN PRINT : PRINT " IS "; : N = FN N(0) : PRINT D(N); : H = 15 : FOR N = N TO O STEP 0: IF N < > O THEN P = PREVIOUS(N): PRINT TAB(H)CHR$(43+18*(h=15));TAB(H+2)PC(N); :N = P: H=H+5: NEXT N 510 NEXT I 600 DATA 7,A-B 610 DATA 9,A-C 620 DATA 14,A-F 630 DATA 10,B-C 640 DATA 15,B-D 650 DATA 11,C-D 660 DATA 2,C-F 670 DATA 6,D-E 680 DATA 9,E-F 690 DATA</lang> Output:
SHORTEST PATH
FROM A TO E <- D <- C <- A
IS 26 = 6 + 11 + 9
FROM A TO F <- C <- A
IS 11 = 2 + 9
Arturo
<lang rebol>define :graph [vertices, neighbours][]
initGraph: function [edges][
vs: []
ns: #[]
loop edges 'e [
[src, dst, cost]: e
vs: sort unique append vs src
vs: sort unique append vs dst
if not? key? ns src -> ns\[src]: []
ns\[src]: ns\[src] ++ @[@[dst, cost]] ] to :graph @[vs ns]
]
Inf: 1234567890
dijkstraPath: function [gr, fst, lst][
dist: #[]
prev: #[]
result: new []
notSeen: new gr\vertices
loop gr\vertices 'vertex ->
dist\[vertex]: Inf
dist\[fst]: 0
while [0 < size notSeen][
vertex1: ""
mindist: Inf
loop notSeen 'vertex [
if dist\[vertex] < mindist [
vertex1: vertex
mindist: dist\[vertex]
]
]
if vertex1 = lst -> break
'notSeen -- vertex1
if key? gr\neighbours vertex1 [
loop gr\neighbours\[vertex1] 'v [
[vertex2, cost]: v
if contains? notSeen vertex2 [
altdist: dist\[vertex1] + cost
if altdist < dist\[vertex2][
dist\[vertex2]: altdist
prev\[vertex2]: vertex1
]
]
]
]
]
vertex: lst
while [not? empty? vertex][
'result ++ vertex
vertex: (key? prev vertex)? -> prev\[vertex] -> null
]
reverse 'result
return result
]
graph: initGraph [
["a" "b" 7] ["a" "c" 9] ["a" "f" 14] ["b" "c" 10] ["b" "d" 15] ["c" "d" 11] ["c" "f" 2] ["d" "e" 6] ["e" "f" 9]
]
print ["Shortest path from 'a' to 'e': " join.with:" -> " dijkstraPath graph "a" "e"] print ["Shortest path from 'a' to 'f': " join.with:" -> " dijkstraPath graph "a" "f"]</lang>
- Output:
Shortest path from 'a' to 'e': a -> c -> d -> e Shortest path from 'a' to 'f': a -> c -> f
AutoHotkey
<lang AutoHotkey>Dijkstra(data, start){ nodes := [], dist := [], Distance := [], dist := [], prev := [], Q := [], min := "x" for each, line in StrSplit(data, "`n" , "`r") field := StrSplit(line,"`t"), nodes[field.1] := 1, nodes[field.2] := 1 , Distance[field.1,field.2] := field.3, Distance[field.2,field.1] := field.3 dist[start] := 0, prev[start] := ""
for node in nodes { if (node <> start) dist[node] := "x" , prev[node] := "" Q[node] := 1 }
while % ObjCount(Q) { u := MinDist(Q, dist).2 for node, val in Q if (node = u) { q.Remove(node) break }
for v, length in Distance[u] { alt := dist[u] + length if (alt < dist[v]) dist[v] := alt , prev[v] := u } } return [dist, prev] }
- -----------------------------------------------
MinDist(Q, dist){ for node , val in Q if A_Index=1 min := dist[node], minNode := node else min := min < dist[node] ? min : dist[node] , minNode := min < dist[node] ? minNode : node return [min,minNode] } ObjCount(Obj){ for key, val in Obj count := A_Index return count }</lang> Examples:<lang AutoHotkey>data = ( A B 7 A C 9 A F 14 B C 10 B D 15 C D 11 C F 2 D E 6 E F 9 )
nodes:=[], Distance := [] for each, line in StrSplit(data, "`n" , "`r")
field := StrSplit(line,"`t"), nodes[field.1] := 1, nodes[field.2] := 1 , Distance[field.1,field.2] := field.3 , Distance[field.2,field.1] := field.3
for node, v in nodes
nodeList .= (nodeList?"|":"") node (A_Index=1?"|":"")
Gui, add, Text,, From: Gui, add, Text, x200 yp, To: Gui, add, DDL, xs vFrom gSubmit, % nodeList Gui, add, DDL, x200 yp vTo gSubmit, % nodeList Gui, add, ListView, xs w340 r6, From|>|To|Distance Gui, add, Text, vT1 xs w340 r1 Gui, +AlwaysOnTop Gui, show Loop 4 LV_ModifyCol(A_Index, "80 Center")
Submit: Gui, Submit, NoHide GuiControl, , T1, % "" LV_Delete() if !(From && To) || (From = To)
return
res := Dijkstra(data, From) , xTo := xFrom := DirectFlight := "" , origin := to GuiControl, , T1, no routing found if !res[1, To] ; no possible route
return
Routing: Loop % objCount(nodes)
for xTo , xFrom in res.2
if (xTo = To)
{
LV_Insert(1,"", xFrom, ">" , xTo, Distance[xFrom , xTo]), To := xFrom
if (xFrom = From)
break, Routing
}
GuiControl, , T1, % "Total distance = " res.1[origin] . DirectFlight return
esc:: GuiClose: ExitApp return</lang>
Outputs:
A > C 9 C > F 2 F > E 9 Total distance = 20
C
The priority queue is implemented as a binary heap. The heap stores an index into its data array, so it can quickly update the weight of an item already in it. <lang Cafe>
- include <stdio.h>
- include <stdlib.h>
- include <limits.h>
typedef struct {
int vertex; int weight;
} edge_t;
typedef struct {
edge_t **edges; int edges_len; int edges_size; int dist; int prev; int visited;
} vertex_t;
typedef struct {
vertex_t **vertices; int vertices_len; int vertices_size;
} graph_t;
typedef struct {
int *data; int *prio; int *index; int len; int size;
} heap_t;
void add_vertex (graph_t *g, int i) {
if (g->vertices_size < i + 1) {
int size = g->vertices_size * 2 > i ? g->vertices_size * 2 : i + 4;
g->vertices = realloc(g->vertices, size * sizeof (vertex_t *));
for (int j = g->vertices_size; j < size; j++)
g->vertices[j] = NULL;
g->vertices_size = size;
}
if (!g->vertices[i]) {
g->vertices[i] = calloc(1, sizeof (vertex_t));
g->vertices_len++;
}
}
void add_edge (graph_t *g, int a, int b, int w) {
a = a - 'a';
b = b - 'a';
add_vertex(g, a);
add_vertex(g, b);
vertex_t *v = g->vertices[a];
if (v->edges_len >= v->edges_size) {
v->edges_size = v->edges_size ? v->edges_size * 2 : 4;
v->edges = realloc(v->edges, v->edges_size * sizeof (edge_t *));
}
edge_t *e = calloc(1, sizeof (edge_t));
e->vertex = b;
e->weight = w;
v->edges[v->edges_len++] = e;
}
heap_t *create_heap (int n) {
heap_t *h = calloc(1, sizeof (heap_t)); h->data = calloc(n + 1, sizeof (int)); h->prio = calloc(n + 1, sizeof (int)); h->index = calloc(n, sizeof (int)); return h;
}
void push_heap (heap_t *h, int v, int p) {
int i = h->index[v] == 0 ? ++h->len : h->index[v];
int j = i / 2;
while (i > 1) {
if (h->prio[j] < p)
break;
h->data[i] = h->data[j];
h->prio[i] = h->prio[j];
h->index[h->data[i]] = i;
i = j;
j = j / 2;
}
h->data[i] = v;
h->prio[i] = p;
h->index[v] = i;
}
int min (heap_t *h, int i, int j, int k) {
int m = i;
if (j <= h->len && h->prio[j] < h->prio[m])
m = j;
if (k <= h->len && h->prio[k] < h->prio[m])
m = k;
return m;
}
int pop_heap (heap_t *h) {
int v = h->data[1];
int i = 1;
while (1) {
int j = min(h, h->len, 2 * i, 2 * i + 1);
if (j == h->len)
break;
h->data[i] = h->data[j];
h->prio[i] = h->prio[j];
h->index[h->data[i]] = i;
i = j;
}
h->data[i] = h->data[h->len];
h->prio[i] = h->prio[h->len];
h->index[h->data[i]] = i;
h->len--;
return v;
}
void dijkstra (graph_t *g, int a, int b) {
int i, j;
a = a - 'a';
b = b - 'a';
for (i = 0; i < g->vertices_len; i++) {
vertex_t *v = g->vertices[i];
v->dist = INT_MAX;
v->prev = 0;
v->visited = 0;
}
vertex_t *v = g->vertices[a];
v->dist = 0;
heap_t *h = create_heap(g->vertices_len);
push_heap(h, a, v->dist);
while (h->len) {
i = pop_heap(h);
if (i == b)
break;
v = g->vertices[i];
v->visited = 1;
for (j = 0; j < v->edges_len; j++) {
edge_t *e = v->edges[j];
vertex_t *u = g->vertices[e->vertex];
if (!u->visited && v->dist + e->weight <= u->dist) {
u->prev = i;
u->dist = v->dist + e->weight;
push_heap(h, e->vertex, u->dist);
}
}
}
}
void print_path (graph_t *g, int i) {
int n, j;
vertex_t *v, *u;
i = i - 'a';
v = g->vertices[i];
if (v->dist == INT_MAX) {
printf("no path\n");
return;
}
for (n = 1, u = v; u->dist; u = g->vertices[u->prev], n++)
;
char *path = malloc(n);
path[n - 1] = 'a' + i;
for (j = 0, u = v; u->dist; u = g->vertices[u->prev], j++)
path[n - j - 2] = 'a' + u->prev;
printf("%d %.*s\n", v->dist, n, path);
}
int main () {
graph_t *g = calloc(1, sizeof (graph_t)); add_edge(g, 'a', 'b', 7); add_edge(g, 'a', 'c', 9); add_edge(g, 'a', 'f', 14); add_edge(g, 'b', 'c', 10); add_edge(g, 'b', 'd', 15); add_edge(g, 'c', 'd', 11); add_edge(g, 'c', 'f', 2); add_edge(g, 'd', 'e', 6); add_edge(g, 'e', 'f', 9); dijkstra(g, 'a', 'e'); print_path(g, 'e'); return 0;
} </lang> output 26 acde
C#
<lang csharp>using static System.Linq.Enumerable; using static System.String; using static System.Console; using System.Collections.Generic; using System; using EdgeList = System.Collections.Generic.List<(int node, double weight)>;
public static class Dijkstra {
public static void Main() {
Graph graph = new Graph(6);
Func<char, int> id = c => c - 'a';
Func<int , char> name = i => (char)(i + 'a');
foreach (var (start, end, cost) in new [] {
('a', 'b', 7),
('a', 'c', 9),
('a', 'f', 14),
('b', 'c', 10),
('b', 'd', 15),
('c', 'd', 11),
('c', 'f', 2),
('d', 'e', 6),
('e', 'f', 9),
}) {
graph.AddEdge(id(start), id(end), cost);
}
var path = graph.FindPath(id('a'));
for (int d = id('b'); d <= id('f'); d++) {
WriteLine(Join(" -> ", Path(id('a'), d).Select(p => $"{name(p.node)}({p.distance})").Reverse()));
}
IEnumerable<(double distance, int node)> Path(int start, int destination) {
yield return (path[destination].distance, destination);
for (int i = destination; i != start; i = path[i].prev) {
yield return (path[path[i].prev].distance, path[i].prev);
}
}
}
}
sealed class Graph {
private readonly List<EdgeList> adjacency;
public Graph(int vertexCount) => adjacency = Range(0, vertexCount).Select(v => new EdgeList()).ToList();
public int Count => adjacency.Count; public bool HasEdge(int s, int e) => adjacency[s].Any(p => p.node == e); public bool RemoveEdge(int s, int e) => adjacency[s].RemoveAll(p => p.node == e) > 0;
public bool AddEdge(int s, int e, double weight) {
if (HasEdge(s, e)) return false;
adjacency[s].Add((e, weight));
return true;
}
public (double distance, int prev)[] FindPath(int start) {
var info = Range(0, adjacency.Count).Select(i => (distance: double.PositiveInfinity, prev: i)).ToArray();
info[start].distance = 0;
var visited = new System.Collections.BitArray(adjacency.Count);
var heap = new Heap<(int node, double distance)>((a, b) => a.distance.CompareTo(b.distance));
heap.Push((start, 0));
while (heap.Count > 0) {
var current = heap.Pop();
if (visited[current.node]) continue;
var edges = adjacency[current.node];
for (int n = 0; n < edges.Count; n++) {
int v = edges[n].node;
if (visited[v]) continue;
double alt = info[current.node].distance + edges[n].weight;
if (alt < info[v].distance) {
info[v] = (alt, current.node);
heap.Push((v, alt));
}
}
visited[current.node] = true;
}
return info;
}
}
sealed class Heap<T> {
private readonly IComparer<T> comparer;
private readonly List<T> list = new List<T> { default };
public Heap() : this(default(IComparer<T>)) { }
public Heap(IComparer<T> comparer) {
this.comparer = comparer ?? Comparer<T>.Default;
}
public Heap(Comparison<T> comparison) : this(Comparer<T>.Create(comparison)) { }
public int Count => list.Count - 1;
public void Push(T element) {
list.Add(element);
SiftUp(list.Count - 1);
}
public T Pop() {
T result = list[1];
list[1] = list[list.Count - 1];
list.RemoveAt(list.Count - 1);
SiftDown(1);
return result;
}
private static int Parent(int i) => i / 2; private static int Left(int i) => i * 2; private static int Right(int i) => i * 2 + 1;
private void SiftUp(int i) {
while (i > 1) {
int parent = Parent(i);
if (comparer.Compare(list[i], list[parent]) > 0) return;
(list[parent], list[i]) = (list[i], list[parent]);
i = parent;
}
}
private void SiftDown(int i) {
for (int left = Left(i); left < list.Count; left = Left(i)) {
int smallest = comparer.Compare(list[left], list[i]) <= 0 ? left : i;
int right = Right(i);
if (right < list.Count && comparer.Compare(list[right], list[smallest]) <= 0) smallest = right;
if (smallest == i) return;
(list[i], list[smallest]) = (list[smallest], list[i]);
i = smallest;
}
}
}</lang>
- Output:
a(0) -> b(7) a(0) -> c(9) a(0) -> c(9) -> d(20) a(0) -> c(9) -> d(20) -> e(26) a(0) -> c(9) -> f(11)
C++
(Modified from LiteratePrograms, which is MIT/X11 licensed.)
Solution follows Dijkstra's algorithm as described elsewhere. Data like min-distance, previous node, neighbors, are kept in separate data structures instead of part of the vertex. We number the vertexes starting from 0, and represent the graph using an adjacency list (vector whose i'th element is the vector of neighbors that vertex i has edges to) for simplicity.
For the priority queue of vertexes, we use a self-balancing binary search tree (std::set), which should bound time complexity by O(E log V). Although C++ has heaps, without knowing the index of an element it would take linear time to find it to re-order it for a changed weight. It is not easy to keep the index of vertexes in the heap because the heap operations are opaque without callbacks. On the other hand, using a self-balancing binary search tree is efficient because it has the same log(n) complexity for insertion and removal of the head element as a binary heap. In addition, a self-balancing binary search tree also allows us to find and remove any other element in log(n) time, allowing us to perform the decrease-key step in logarithmic time by removing and re-inserting.
We do not need to keep track of whether a vertex is "done" ("visited") as in the Wikipedia description, since re-reaching such a vertex will always fail the relaxation condition (when re-reaching a "done" vertex, the new distance will never be less than it was originally), so it will be skipped anyway.
The time complexity of this algorithm is O(E log V), as described on Wikipedia. Each vertex is added to the priority queue at most once (re-ordering doesn't count as adding), because once it's in the priority queue, we only re-order it, never add it again. And when it's popped from the priority queue, that means we already have the real minimum distance to this vertex, so the relaxation condition will always fail in the future for this vertex, and it will never be added to the priority queue again. Therefore, we will only pop each vertex at most once from the priority queue, and the size of the priority queue is bounded by V (the number of vertexes).
The outer loop executes once for each element popped from the priority queue, so it will execute at most once for each vertex, so at most V times. Each iteration of the outer loop executes one pop from the priority queue, which has time complexity O(log V). The inner loop executes at most once for each directed edge, since each directed edge has one originating vertex, and there is only at most one iteration of the outer loop for each vertex. Each iteration of the inner loop potentially performs one push or re-order on the priority queue (where re-order is a pop and a push), which has complexity O(log V). There is also the O(V) complexity for initializing the data structures. Combining these, we have a complexity of O(V log V + E log V), and assuming this is a connected graph, V <= E+1 = O(E), so we can write it as O(E log V).
<lang cpp>#include <iostream>
- include <vector>
- include <string>
- include <list>
- include <limits> // for numeric_limits
- include <set>
- include <utility> // for pair
- include <algorithm>
- include <iterator>
typedef int vertex_t;
typedef double weight_t;
const weight_t max_weight = std::numeric_limits<double>::infinity();
struct neighbor {
vertex_t target;
weight_t weight;
neighbor(vertex_t arg_target, weight_t arg_weight)
: target(arg_target), weight(arg_weight) { }
};
typedef std::vector<std::vector<neighbor> > adjacency_list_t;
void DijkstraComputePaths(vertex_t source,
const adjacency_list_t &adjacency_list,
std::vector<weight_t> &min_distance,
std::vector<vertex_t> &previous)
{
int n = adjacency_list.size(); min_distance.clear(); min_distance.resize(n, max_weight); min_distance[source] = 0; previous.clear(); previous.resize(n, -1); std::set<std::pair<weight_t, vertex_t> > vertex_queue; vertex_queue.insert(std::make_pair(min_distance[source], source));
while (!vertex_queue.empty())
{
weight_t dist = vertex_queue.begin()->first;
vertex_t u = vertex_queue.begin()->second;
vertex_queue.erase(vertex_queue.begin());
// Visit each edge exiting u
const std::vector<neighbor> &neighbors = adjacency_list[u];
for (std::vector<neighbor>::const_iterator neighbor_iter = neighbors.begin();
neighbor_iter != neighbors.end();
neighbor_iter++)
{
vertex_t v = neighbor_iter->target;
weight_t weight = neighbor_iter->weight;
weight_t distance_through_u = dist + weight;
if (distance_through_u < min_distance[v]) { vertex_queue.erase(std::make_pair(min_distance[v], v));
min_distance[v] = distance_through_u; previous[v] = u; vertex_queue.insert(std::make_pair(min_distance[v], v));
}
} }
}
std::list<vertex_t> DijkstraGetShortestPathTo(
vertex_t vertex, const std::vector<vertex_t> &previous)
{
std::list<vertex_t> path;
for ( ; vertex != -1; vertex = previous[vertex])
path.push_front(vertex);
return path;
}
int main()
{
// remember to insert edges both ways for an undirected graph adjacency_list_t adjacency_list(6); // 0 = a adjacency_list[0].push_back(neighbor(1, 7)); adjacency_list[0].push_back(neighbor(2, 9)); adjacency_list[0].push_back(neighbor(5, 14)); // 1 = b adjacency_list[1].push_back(neighbor(0, 7)); adjacency_list[1].push_back(neighbor(2, 10)); adjacency_list[1].push_back(neighbor(3, 15)); // 2 = c adjacency_list[2].push_back(neighbor(0, 9)); adjacency_list[2].push_back(neighbor(1, 10)); adjacency_list[2].push_back(neighbor(3, 11)); adjacency_list[2].push_back(neighbor(5, 2)); // 3 = d adjacency_list[3].push_back(neighbor(1, 15)); adjacency_list[3].push_back(neighbor(2, 11)); adjacency_list[3].push_back(neighbor(4, 6)); // 4 = e adjacency_list[4].push_back(neighbor(3, 6)); adjacency_list[4].push_back(neighbor(5, 9)); // 5 = f adjacency_list[5].push_back(neighbor(0, 14)); adjacency_list[5].push_back(neighbor(2, 2)); adjacency_list[5].push_back(neighbor(4, 9));
std::vector<weight_t> min_distance; std::vector<vertex_t> previous; DijkstraComputePaths(0, adjacency_list, min_distance, previous); std::cout << "Distance from 0 to 4: " << min_distance[4] << std::endl; std::list<vertex_t> path = DijkstraGetShortestPathTo(4, previous); std::cout << "Path : "; std::copy(path.begin(), path.end(), std::ostream_iterator<vertex_t>(std::cout, " ")); std::cout << std::endl;
return 0;
}</lang>
Note that it is possible to use C++ built-in heaps (or the abstract std::priority_queue datatype) to implement this without changing the time complexity. Although the previous section noted that, without knowing the position of the element in the heap, it would take linear time to search for it in order to re-order it, the trick here is that we can insert the new updated element (with the vertex and updated lower distance), and simply leave the old element (with the vertex and old higher distance) in the priority queue without removing it, thereby eliminating the need to find it.
Since we now leave multiple elements with the same vertex in the priority queue, in order to ensure we still only process a vertex's edges only once, we add a check when we retrieve an element from the priority queue, to check whether its distance is greater than the known minimum distance to that vertex. If this element is the most updated version for this vertex (i.e. the vertex's minimum distance has not been decreased since this element was added to the priority queue), then its distance must be equal to the current known minimum distance, since we only update the minimum distance in the decrease-key step. So if the element's distance is greater, we know that this is not the most updated version for this vertex -- i.e. we have already processed the edges for this vertex -- and we should ignore it.
The only downside to this strategy is that many old "garbage" elements will be left in the priority queue, increasing its size, and thus also increasing the time it takes to push and pop, as well as increasing the number of times we have to pop. However, we argue that the time complexity remains the same.
The main difference with the time complexity analysis for the previous algorithm is that here, we may add a vertex to the priority queue more than once. However, it is still true that the inner loop executes at most once for each directed edge. This is because in the outer loop, we added a check to ignore vertexes that we've already processed, so we will still only proceed down to the processing the edges at most once for each vertex. Therefore, the number of times that push is done on the priority queue (which happens at most once per iteration of the inner loop) is bounded by E, and the size of the priority queue is also bounded by E.
The number of times the outer loop executes (the number of times an element is popped from the priority queue) is bounded by E, and in each iteration, the popping operation takes time complexity O(log E). The number of times the inner loop executes is also bounded by E, and the pushing operation inside it also takes time complexity O(log E). So in total, the time complexity is O(E log E). But not that, for a simple graph, E < V^2, so log E < 2 log V = O(log V). So O(E log E) can also be written as O(E log V), which is the same as for the preceding algorithm.
<lang cpp>#include <iostream>
- include <vector>
- include <string>
- include <list>
- include <limits> // for numeric_limits
- include <queue>
- include <utility> // for pair
- include <algorithm>
- include <iterator>
typedef int vertex_t;
typedef double weight_t;
const weight_t max_weight = std::numeric_limits<double>::infinity();
struct neighbor {
vertex_t target;
weight_t weight;
neighbor(vertex_t arg_target, weight_t arg_weight)
: target(arg_target), weight(arg_weight) { }
};
typedef std::vector<std::vector<neighbor> > adjacency_list_t; typedef std::pair<weight_t, vertex_t> weight_vertex_pair_t;
void DijkstraComputePaths(vertex_t source,
const adjacency_list_t &adjacency_list,
std::vector<weight_t> &min_distance,
std::vector<vertex_t> &previous)
{
int n = adjacency_list.size(); min_distance.clear(); min_distance.resize(n, max_weight); min_distance[source] = 0; previous.clear(); previous.resize(n, -1); // we use greater instead of less to turn max-heap into min-heap std::priority_queue<weight_vertex_pair_t,
std::vector<weight_vertex_pair_t>, std::greater<weight_vertex_pair_t> > vertex_queue;
vertex_queue.push(std::make_pair(min_distance[source], source));
while (!vertex_queue.empty())
{
weight_t dist = vertex_queue.top().first;
vertex_t u = vertex_queue.top().second;
vertex_queue.pop();
// Because we leave old copies of the vertex in the priority queue // (with outdated higher distances), we need to ignore it when we come // across it again, by checking its distance against the minimum distance if (dist > min_distance[u]) continue;
// Visit each edge exiting u
const std::vector<neighbor> &neighbors = adjacency_list[u];
for (std::vector<neighbor>::const_iterator neighbor_iter = neighbors.begin();
neighbor_iter != neighbors.end();
neighbor_iter++)
{
vertex_t v = neighbor_iter->target;
weight_t weight = neighbor_iter->weight;
weight_t distance_through_u = dist + weight;
if (distance_through_u < min_distance[v]) { min_distance[v] = distance_through_u; previous[v] = u; vertex_queue.push(std::make_pair(min_distance[v], v));
}
} }
}
std::list<vertex_t> DijkstraGetShortestPathTo(
vertex_t vertex, const std::vector<vertex_t> &previous)
{
std::list<vertex_t> path;
for ( ; vertex != -1; vertex = previous[vertex])
path.push_front(vertex);
return path;
}
int main()
{
// remember to insert edges both ways for an undirected graph adjacency_list_t adjacency_list(6); // 0 = a adjacency_list[0].push_back(neighbor(1, 7)); adjacency_list[0].push_back(neighbor(2, 9)); adjacency_list[0].push_back(neighbor(5, 14)); // 1 = b adjacency_list[1].push_back(neighbor(0, 7)); adjacency_list[1].push_back(neighbor(2, 10)); adjacency_list[1].push_back(neighbor(3, 15)); // 2 = c adjacency_list[2].push_back(neighbor(0, 9)); adjacency_list[2].push_back(neighbor(1, 10)); adjacency_list[2].push_back(neighbor(3, 11)); adjacency_list[2].push_back(neighbor(5, 2)); // 3 = d adjacency_list[3].push_back(neighbor(1, 15)); adjacency_list[3].push_back(neighbor(2, 11)); adjacency_list[3].push_back(neighbor(4, 6)); // 4 = e adjacency_list[4].push_back(neighbor(3, 6)); adjacency_list[4].push_back(neighbor(5, 9)); // 5 = f adjacency_list[5].push_back(neighbor(0, 14)); adjacency_list[5].push_back(neighbor(2, 2)); adjacency_list[5].push_back(neighbor(4, 9));
std::vector<weight_t> min_distance; std::vector<vertex_t> previous; DijkstraComputePaths(0, adjacency_list, min_distance, previous); std::cout << "Distance from 0 to 4: " << min_distance[4] << std::endl; std::list<vertex_t> path = DijkstraGetShortestPathTo(4, previous); std::cout << "Path : "; std::copy(path.begin(), path.end(), std::ostream_iterator<vertex_t>(std::cout, " ")); std::cout << std::endl;
return 0;
}</lang>
CafeOBJ
<lang CafeOBJ> " Save this file as DijkstraRosetta.cafe. To run the file type CafeOBJ> in DijkstraRosetta.cafe at the CafeOBJ command prompt.
CafeOBJ is primarily a first order specification language which can also be used as a functional programming language. Being first order, we make no use higher order functions such as map. There is a minimal library of basic types such as natural numbers, integers, floating point number, and character string. There are no libraries for arrays, lists, trees, graphs. Hence the user written list module.
Dijkstra's algorithm repeatedly chooses the nearest vertex and relaxes the edges leaving it, terminating when no more vertices are accessible from the origin.
Input
A directed positively weighted graph. The graph is represented as a list of 4tuples containing directed edges of the form (beginning, end, edgeDist,pathDist)
The tuple (start, start,0,0) means there is zero distance from start to start.
Ouput 1) a list of 4-tuples with each tuple represents a node N, its source, length of the conecting edge to N, and the shortest distance from the some starting node to N . 2) a list of nodes on the shortest path from a chosen start to some cosen end node.
Note needs a bit more work to exacly match the specified Rosetta Dijkstra task. "
-- some system settings -- Most important is memoization (memo) which stores the value of a function instead of recomputing it each time the function is called. full reset set step off set print fancy set stats on set verbose off set quiet on set memo on
-- A module defining a simple parameterized list. mod! LIST(T :: TRIV) principal-sort List { [Elt < List ] op nil : -> List op (_:_) : List List -> List {memo assoc id: nil} op reverse_ : List -> List op head_ : List -> Elt var e : Elt var l : List eq reverse nil = nil . eq reverse (e : l) = (reverse l) : e . eq head e : l = e . }
-- Main module
mod! DIJKSTRA {
-- A four tuple used to store graph and shortest distance
-- start, end, edgeDist,pathDist
pr(LIST(4TUPLE(INT,INT,INT,INT)))
-- A list of integers used to store final shortest path.
pr(LIST(INT) *{sort List -> PathList})
[List < PathList]
op dijkstra___ : Int List List -> List
op exploreNeighbours___ : Int List List -> 4Tuple {memo}
ops inf finished : -> Int
op currDist__ : Int List -> Int
op relax__ : List List -> List
op connectedTo__ : Int List -> Bool
op nextNode2Explore_ : List -> 4Tuple
op connectedList___ : List Int List -> List
op unvisitedList__ : List List -> List
op shortestPath__ : Int List -> PathList
op SP___ : 4Tuple 4Tuple List -> PathList
op shortestDist__ : Int List -> Int
op SD____ : Int Int List List -> Int
vars x y z n eD pD eD1 pD1 eD2 pD2 source currentVertex startVertex : Int vars graph permanent xs : List vars t t1 t2 : 4Tuple
eq inf = 500 . eq finished = -1 .
-- Main dijkstra function eq dijkstra startVertex graph permanent =
if (exploreNeighbours startVertex permanent graph) == << finished ; finished ; finished ; finished >> then permanent
else (dijkstra startVertex graph ( ((exploreNeighbours startVertex permanent graph) : permanent))) fi .
eq exploreNeighbours startVertex permanent graph = (nextNode2Explore (relax (unvisitedList (connectedList graph startVertex permanent) permanent) permanent )) .
-- nextNode2Explore takes a list of records and returns a record with the minimum 4th element else finished eq nextNode2Explore nil = << finished ; finished ; finished ; finished >> . eq nextNode2Explore (t1 : nil) = t1 . eq nextNode2Explore (t2 : (t1 : xs)) = if (4* t1) < (4* t2) then t1 else nextNode2Explore (t2 : xs) fi .
-- relaxes all edges leaving y eq relax nil permanent = nil . eq relax (<< x ; y ; eD ; pD >> : xs) permanent = if (currDist x permanent) < pD then << y ; x ; eD ; ((currDist x permanent) + eD) >> : (relax xs permanent) else << y ; x ; eD ; pD >> : (relax xs permanent) fi .
-- Get the current best distance for a particulare vertex n.
eq currDist n nil = inf .
eq currDist n (t : permanent) = if ((1* t) == n) then (4* t ) else (currDist n permanent) fi .
eq connectedTo z nil = false .
eq connectedTo z ((<< x ; y ; eD ; pD >>) : xs) = if (x == z) then true else (connectedTo z xs) fi .
eq connectedList nil source permanent = nil .
eq connectedList (t : graph) source permanent = if (connectedTo source permanent) then
(t : (connectedList graph source permanent))
else (connectedList graph source permanent) fi .
eq unvisitedList nil permanent = nil .
eq unvisitedList (t : graph) permanent = if not(connectedTo (2* t) permanent) then (t : (unvisitedList graph permanent)) else (unvisitedList graph permanent) fi .
-- To get the shortest path we used the above dijkstra function.
-- From a given start to a given end we need to trace the path from the finish to the start and then reverse the output.
var pList : PathList
var currentTuple : 4Tuple
vars start end : Int
eq SP start end nil = nil .
eq SP start end (currentTuple : pList) = if (end == (1* currentTuple)) then ( end : (SP start (2* currentTuple) pList)) else (SP start end pList) fi .
-- The graph to be traversed
op DirectedRosetta : -> List
eq DirectedRosetta = ( << 1 ; 2 ; 7 ; inf >> :
<< 1 ; 3 ; 9 ; inf >> :
<< 1 ; 6 ; 14 ; inf >> :
<< 2 ; 3 ; 10 ; inf >> :
<< 2 ; 4 ; 15 ; inf >> :
<< 3 ; 4 ; 11 ; inf >> :
<< 3 ; 6 ; 2 ; inf >> :
<< 4 ; 5 ; 6 ; inf >> :
<< 5 ; 6 ; 9 ; inf >>) .
-- A set of possible strating points ops oneStart twoStart threeStart fourStart fiveStart sixStart : -> List eq oneStart = << 1 ; 1 ; 0 ; 0 >> . eq twoStart = << 2 ; 2 ; 0 ; 0 >> . eq threeStart = << 3 ; 3 ; 0 ; 0 >> . eq fourStart = << 4 ; 4 ; 0 ; 0 >> . eq fiveStart = << 5 ; 5 ; 0 ; 0 >> . eq sixStart = << 6 ; 6 ; 0 ; 0 >> .
} -- End module
-- We must open the module in the CafeOBJ interpreter open DIJKSTRA . --> All shortest distances starting from a(1) red dijkstra 1 DirectedRosetta oneStart . -- Gives -- ((((<< 5 ; 4 ; 6 ; 26 >>) : (<< 4 ; 3 ; 11 ; 20 >>)) : ((<< 6 ; 3 ; 2 ; 11 >>) : (<< 3 ; 1 ; 9 ; 9 >>))) : ((<< 2 ; 1 ; 7 ; 7 >>) : (<< 1 ; 1 ; 0 ; 0 >>))):List
--> Shortest path from a(1) to e(5) red reverse (SP 1 5 (dijkstra 1 DirectedRosetta oneStart)) . -- Gives -- ((1 : 3) : (4 : 5)):PathList
--> Shortest path from a(1) to f(6) red reverse (SP 1 6 (dijkstra 1 DirectedRosetta oneStart)) . -- Gives -- ((1 : 3) : 6):PathList eof </lang>
Clojure
<lang clojure> (declare neighbours
process-neighbour
prepare-costs
get-next-node
unwind-path
all-shortest-paths)
- Main algorithm
(defn dijkstra
"Given two nodes A and B, and graph, finds shortest path from point A to point B. Given one node and graph, finds all shortest paths to all other nodes.
Graph example: {1 {2 7 3 9 6 14}
2 {1 7 3 10 4 15}
3 {1 9 2 10 4 11 6 2}
4 {2 15 3 11 5 6}
5 {6 9 4 6}
6 {1 14 3 2 5 9}}
^ ^ ^
| | |
node label | |
neighbour label--- |
edge cost------
From example in Wikipedia: https://en.wikipedia.org/wiki/Dijkstra's_algorithm
Output example: [20 [1 3 6 5]]
^ ^
| |
shortest path cost |
shortest path---"
([a b graph]
(loop [costs (prepare-costs a graph)
unvisited (set (keys graph))]
(let [current-node (get-next-node costs unvisited)
current-cost (first (costs current-node))]
(cond (nil? current-node)
(all-shortest-paths a costs)
(= current-node b)
[current-cost (unwind-path a b costs)]
:else
(recur (reduce (partial process-neighbour
current-node
current-cost)
costs
(filter (comp unvisited first)
(neighbours current-node graph costs)))
(disj unvisited current-node))))))
([a graph] (dijkstra a nil graph)))
- Implementation details
(defn prepare-costs
"For given start node A ang graph prepare map of costs to start with
(assign maximum value for all nodes and zero for starting one).
Also save info about most advantageous parent.
Example output: {2 [2147483647 7], 6 [2147483647 14]}
^ ^ ^
| | |
node | |
cost----- |
parent---------------"
[start graph]
(assoc (zipmap (keys graph)
(repeat [Integer/MAX_VALUE nil]))
start [0 start]))
(defn neighbours
"Get given node's neighbours along with their own costs and costs of corresponding edges.
Example output is: {1 [7 10] 2 [4 15]}
^ ^ ^
| | |
neighbour node label | |
neighbour cost --- |
edge cost ------"
[node graph costs]
(->> (graph node)
(map (fn neighbour edge-cost
[neighbour [(first (costs neighbour)) edge-cost]]))
(into {})))
(defn process-neighbour
[parent
parent-cost
costs
[neighbour [old-cost edge-cost]]]
(let [new-cost (+ parent-cost edge-cost)]
(if (< new-cost old-cost)
(assoc costs
neighbour
[new-cost parent])
costs)))
(defn get-next-node [costs unvisited]
(->> costs
(filter (comp unvisited first))
(sort-by (comp first second))
ffirst))
(defn unwind-path
"Restore path from A to B based on costs data"
[a b costs]
(letfn [(f [a b costs]
(when-not (= a b)
(cons b (f a (second (costs b)) costs))))]
(cons a (reverse (f a b costs)))))
(defn all-shortest-paths
"Get shortest paths for all nodes, along with their costs"
[start costs]
(let [paths (->> (keys costs)
(remove #{start})
(map (fn [n] [n (unwind-path start n costs)])))]
(into (hash-map)
(map (fn n p
[n [(first (costs n)) p]])
paths))))
- Utils
(require '[clojure.pprint :refer [print-table]])
(defn print-solution [solution]
(print-table
(map (fn [[node [cost path]]]
{'node node 'cost cost 'path path})
solution)))
- Solutions
- Task 1. Implement a version of Dijkstra's algorithm that outputs a set of edges depicting the shortest path to each reachable node from an origin.
- see above
- Task 2. Run your program with the following directed graph starting at node a.
- Edges
- Start End Cost
- a b 7
- a c 9
- a f 14
- b c 10
- b d 15
- c d 11
- c f 2
- d e 6
- e f 9
(def rosetta-graph
'{a {b 7 c 9 f 14}
b {c 10 d 15}
c {d 11 f 2}
d {e 6}
e {f 9}
f {}})
(def task-2-solution
(dijkstra 'a rosetta-graph))
(print-solution task-2-solution)
- Output
- | node | cost | path |
- |------+------+-----------|
- | b | 7 | (a b) |
- | c | 9 | (a c) |
- | d | 20 | (a c d) |
- | e | 26 | (a c d e) |
- | f | 11 | (a c f) |
- Task 3. Write a program which interprets the output from the above and use it to output the shortest path from node a to nodes e and f
(print-solution (select-keys task-2-solution '[e f]))
- Output
- | node | cost | path |
- |------+------+-----------|
- | e | 26 | (a c d e) |
- | f | 11 | (a c f) |
</lang>
Commodore BASIC
This should work on any Commodore 8-bit BASIC from V2 on; with the given sample data, it even runs on an unexpanded VIC-20.
(The program outputs the shortest path to each node in the graph, including E and F, so I assume that meets the requirements of Task item 3.)
<lang basic>100 NV=0: REM NUMBER OF VERTICES 110 READ N$:IF N$<>"" THEN NV=NV+1:GOTO 110 120 NE=0: REM NUMBER OF EDGES 130 READ N1:IF N1 >= 0 THEN READ N2,W:NE=NE+1:GOTO 130 140 DIM VN$(NV-1),VD(NV-1,2): REM VERTEX NAMES AND DATA 150 DIM ED(NE-1,2): REM EDGE DATA 160 RESTORE 170 FOR I=0 TO NV-1 180 : READ VN$(I): REM VERTEX NAME 190 : VD(I,0) = -1: REM DISTANCE = INFINITY 200 : VD(I,1) = 0: REM NOT YET VISITED 210 : VD(I,2) = -1: REM NO PREV VERTEX YET 220 NEXT I 230 READ N$: REM SKIP SENTINEL 240 FOR I=0 TO NE-1 250 : READ ED(I,0),ED(I,1),ED(I,2): REM EDGE FROM, TO, WEIGHT 260 NEXT I 270 READ N1: REM SKIP SENTINEL 280 READ O: REM ORIGIN VERTEX 290 : 300 REM BEGIN DIJKSTRA'S 310 VD(O,0) = 0: REM DISTANCE TO ORIGIN IS 0 320 CV = 0: REM CURRENT VERTEX IS ORIGIN 330 FOR I=0 TO NE-1 340 : IF ED(I,0)<>CV THEN 390: REM SKIP EDGES NOT FROM CURRENT 350 : N=ED(I,1): REM NEIGHBOR VERTEX 360 : D=VD(CV,0) + ED(I,2): REM TOTAL DISTANCE TO NEIGHBOR THROUGH THIS PATH 370 : REM IF PATH THRU CV < DISTANCE, UPDATE DISTANCE AND PREV VERTEX 380 : IF (VD(N,0)=-1) OR (D<VD(N,0)) THEN VD(N,0) = D:VD(N,2)=CV 390 NEXT I 400 VD(CV,1)=1: REM CURRENT VERTEX HAS BEEN VISITED 410 MV=-1: REM VERTEX WITH MINIMUM DISTANCE SEEN 420 FOR I=0 TO NV-1 430 : IF VD(I,1) THEN 470: REM SKIP VISITED VERTICES 440 : REM IF THIS IS THE SMALLEST DISTANCE SEEN, REMEMBER IT 450 : MD=-1:IF MV > -1 THEN MD=VD(MV,0) 460 : IF ( VD(I,0)<>-1 ) AND ( ( MD=-1 ) OR ( VD(I,0)<MD ) ) THEN MV=I 470 NEXT I 480 IF MD=-1 THEN 510: REM END IF ALL VERTICES VISITED OR AT INFINITY 490 CV=MV 500 GOTO 330 510 PRINT "SHORTEST PATH TO EACH VERTEX FROM "VN$(O)":";CHR$(13) 520 FOR I=0 TO NV-1 530 : IF I=0 THEN 600 540 : PRINT VN$(I)":"VD(I,0)"("; 550 : IF VD(I,0)=-1 THEN 600 560 : N=I 570 : PRINT VN$(N); 580 : IF N<>O THEN PRINT "←";:N=VD(N,2):GOTO 570 590 : PRINT ")" 600 NEXT I 610 DATA A,B,C,D,E,F,"" 620 DATA 0,1,7 630 DATA 0,2,9 640 DATA 0,5,14 650 DATA 1,2,10 660 DATA 1,3,15 670 DATA 2,3,11 680 DATA 2,5,2 690 DATA 3,4,6 700 DATA 4,5,9 710 DATA -1 720 DATA 0</lang>
- Output:
Paths are printed right-to-left mainly because PETSCII includes a left-facing arrow and not a right-facing one:
SHORTEST PATH TO EACH VERTEX FROM A: B: 7 (B←A) C: 9 (C←A) D: 20 (D←C←A) E: 26 (E←D←C←A) F: 11 (F←C←A)
Common Lisp
<lang lisp> (defparameter *w* '((a (a b . 7) (a c . 9) (a f . 14))
(b (b c . 10) (b d . 15))
(c (c d . 11) (c f . 2))
(d (d e . 6))
(e (e f . 9))))
(defvar *r* nil)
(defun dijkstra-short-path (i g)
(setf *r* nil) (paths i g 0 `(,i)) (car (sort *r* #'< :key #'cadr)))
(defun paths (c g z v)
(if (eql c g) (push `(,(reverse v) ,z) *r*)
(loop for a in (nodes c) for b = (cadr a) do
(unless (member b v)
(paths b g (+ (cddr a) z) (cons b v))))))
(defun nodes (c)
(sort (cdr (assoc c *w*)) #'< :key #'cddr))
</lang>
- Output:
> (dijkstra-short-path 'a 'e) ((A C D E) 26)
<lang lisp> (defvar *r* nil)
(defun dijkstra-short-paths (z w)
(loop for (a b) in (loop for v on z nconc
(loop for e in (cdr v)
collect `(,(car v) ,e)))
do (setf *r* nil) (paths w a b 0 `(,a))
(format t "~{Path: ~A Distance: ~A~}~%"
(car (sort *r* #'< :key #'cadr)))))
(defun paths (w c g z v)
(if (eql c g) (push `(,(reverse v) ,z) *r*)
(loop for a in (sort (cdr (assoc c w)) #'< :key #'cddr)
for b = (cadr a) do (unless (member b v)
(paths w b g (+ (cddr a) z)
(cons b v))))))
</lang>
- Output:
> (dijkstra-short-paths
'(a b c d e f)
'((a (a b . 7) (a c . 9) (a f . 14))
(b (b c . 10) (b d . 15))
(c (c d . 11) (c f . 2))
(d (d e . 6))
(e (e f . 9))))
Path: (A B) Distance: 7
Path: (A C) Distance: 9
Path: (A C D) Distance: 20
Path: (A C D E) Distance: 26
Path: (A C F) Distance: 11
Path: (B C) Distance: 10
Path: (B D) Distance: 15
Path: (B D E) Distance: 21
Path: (B C F) Distance: 12
Path: (C D) Distance: 11
Path: (C D E) Distance: 17
Path: (C F) Distance: 2
Path: (D E) Distance: 6
Path: (D E F) Distance: 15
Path: (E F) Distance: 9
NIL
D
The algorithm and the important data structures are essentially the same as in the C++ version, so the same comments apply (built-in D associative arrays are unsorted). <lang d>import std.stdio, std.typecons, std.algorithm, std.container;
alias Vertex = string; alias Weight = int;
struct Neighbor {
Vertex target; Weight weight;
}
alias AdjacencyMap = Neighbor[][Vertex];
pure dijkstraComputePaths(Vertex source, Vertex target, AdjacencyMap adjacencyMap){
Weight[Vertex] minDistance; Vertex[Vertex] previous;
foreach(v, neighs; adjacencyMap){
minDistance[v] = Weight.max;
foreach(n; neighs) minDistance[n.target] = Weight.max;
}
minDistance[source] = 0; auto vertexQueue = redBlackTree(tuple(minDistance[source], source));
foreach(_, u; vertexQueue){
if (u == target)
break;
// Visit each edge exiting u.
foreach(n; adjacencyMap.get(u, null)){
const v = n.target;
const distanceThroughU = minDistance[u] + n.weight;
if(distanceThroughU < minDistance[v]){
vertexQueue.removeKey(tuple(minDistance[v], v));
minDistance[v] = distanceThroughU;
previous[v] = u;
vertexQueue.insert(tuple(minDistance[v], v));
}
}
}
return tuple(minDistance, previous);
}
pure dijkstraGetShortestPathTo(Vertex v, Vertex[Vertex] previous){
Vertex[] path = [v];
while (v in previous) {
v = previous[v];
if (v == path[$ - 1])
break;
path ~= v;
}
path.reverse(); return path;
}
void main() {
immutable arcs = [tuple("a", "b", 7),
tuple("a", "c", 9),
tuple("a", "f", 14),
tuple("b", "c", 10),
tuple("b", "d", 15),
tuple("c", "d", 11),
tuple("c", "f", 2),
tuple("d", "e", 6),
tuple("e", "f", 9)];
AdjacencyMap adj;
foreach (immutable arc; arcs) {
adj[arc[0]] ~= Neighbor(arc[1], arc[2]);
// Add this if you want an undirected graph:
//adj[arc[1]] ~= Neighbor(arc[0], arc[2]);
}
const minDist_prev = dijkstraComputePaths("a", "e", adj);
const minDistance = minDist_prev[0];
const previous = minDist_prev[1];
writeln(`Distance from "a" to "e": `, minDistance["e"]);
writeln("Path: ", dijkstraGetShortestPathTo("e", previous));
}</lang>
- Output:
Distance from "a" to "e": 26 Path: ["a", "c", "d", "e"]
Delphi
A console program written in Delphi 7. It runs from the Windows command prompt.
An infinite distance is here represented by -1, which complicates the code when comparing distances, but ensures that infinity can't be equalled or exceeded by any finite distance. <lang delphi>program Rosetta_Dijkstra_Console;
{$APPTYPE CONSOLE}
uses SysUtils; // for printing the result
// Conventional values (any negative values would do) const
INFINITY = -1; NO_VERTEX = -2;
const
NR_VERTICES = 6;
// DISTANCE_MATRIX[u, v] = length of directed edge from u to v, or -1 if no such edge exists. // A simple way to represent a directed graph with not many vertices. const DISTANCE_MATRIX : array [0..(NR_VERTICES - 1), 0..(NR_VERTICES - 1)] of integer = ((-1, 7, 9, -1, -1, -1),
(-1, -1, 10, 15, -1, -1), (-1, -1, -1, 11, -1, 2), (-1, -1, -1, -1, 6, -1), (-1, -1, -1, -1, -1, 9), (-1, -1, -1, -1, -1, -1));
type TVertex = record
Distance : integer; // distance from vertex 0; infinity if a path has not yet been found Previous : integer; // previous vertex in the path from vertex 0 Visited : boolean; // as defined in the algorithm
end;
// For distances x and y, test whether x < y, using the convention that -1 means infinity. function IsLess( x, y : integer) : boolean; begin
result := (x <> INFINITY)
and ( (y = INFINITY) or (x < y) );
end;
// Main routine var
v : array [0..NR_VERTICES - 1] of TVertex; // array of vertices c : integer; // index of current vertex j : integer; // loop counter trialDistance : integer; minDistance : integer; // Variables for printing the result p : integer; lineOut : string;
begin
// Initialize the vertices for j := 0 to NR_VERTICES - 1 do begin v[j].Distance := INFINITY; v[j].Previous := NO_VERTEX; v[j].Visited := false; end;
// Start with vertex 0 as the current vertex c := 0; v[c].Distance := 0;
// Main loop of Dijkstra's algorithm repeat
// Work through unvisited neighbours of the current vertex, updating them where possible.
// "Neighbour" means the end of a directed edge from the current vertex.
// Note that v[c].Distance is always finite.
for j := 0 to NR_VERTICES - 1 do begin
if (not v[j].Visited) and (DISTANCE_MATRIX[c, j] >= 0) then begin
trialDistance := v[c].Distance + DISTANCE_MATRIX[c, j];
if IsLess( trialDistance, v[j].Distance) then begin
v[j].Distance := trialDistance;
v[j].Previous := c;
end;
end;
end;
// When all neighbours have been tested, mark the current vertex as visited. v[c].Visited := true;
// The new current vertex is the unvisited vertex with the smallest finite distance.
// If there is no such vertex, the algorithm is finished.
c := NO_VERTEX;
minDistance := INFINITY;
for j := 0 to NR_VERTICES - 1 do begin
if (not v[j].Visited) and IsLess( v[j].Distance, minDistance) then begin
minDistance := v[j].Distance;
c := j;
end;
end;
until (c = NO_VERTEX);
// Print the result
for j := 0 to NR_VERTICES - 1 do begin
if (v[j].Distance = INFINITY) then begin
// The algorithm never found a path to v[j]
lineOut := SysUtils.Format( '%2d: inaccessible', [j]);
end
else begin
// Build up the path of vertices, working backwards from v[j]
lineOut := SysUtils.Format( '%2d', [j]);
p := v[j].Previous;
while (p <> NO_VERTEX) do begin
lineOut := SysUtils.Format( '%2d --> ', [p]) + lineOut;
p := v[p].Previous;
end;
// Print the path of vertices, preceded by distance from vertex 0
lineOut := SysUtils.Format( '%2d: distance = %3d, ', [j, v[j].Distance]) + lineOut;
end;
WriteLn( lineOut);
end;
end.</lang>
- Output:
0: distance = 0, 0 1: distance = 7, 0 --> 1 2: distance = 9, 0 --> 2 3: distance = 20, 0 --> 2 --> 3 4: distance = 26, 0 --> 2 --> 3 --> 4 5: distance = 11, 0 --> 2 --> 5
Emacs Lisp
<lang lisp>
- Path format
- (start-point end-point previous-point distance)
(setq path-list `( (a b ,nil 7) (a c ,nil 9) (a f ,nil 14) (b c ,nil 10) (b d ,nil 15) (c d ,nil 11) (c f ,nil 2) (d e ,nil 6) (e f ,nil 9) ))
(defun calculate-shortest-path ()
(let ((shortest-path '())
(head-point (nth 0 (nth 0 path-list))))
(defun combine-new-path (path1 path2)
(list (nth 0 path1) (nth 1 path2) (nth 0 path2)
(+ (nth 3 path1) (nth 3 path2))) )
(defun find-shortest-path (start end)
(seq-find (lambda (item)
(and (eq (nth 0 item) start) (eq (nth 1 item) end))) shortest-path)
)
(defun add-shortest-path (item)
(add-to-list 'shortest-path item) )
(defun process-path (path)
(if (eq head-point (nth 0 path))
(add-to-list 'shortest-path path) (progn (dolist (spath shortest-path) (when (eq (nth 1 spath) (nth 0 path)) (let* ((new-path (combine-new-path spath path)) (spath-found (find-shortest-path (nth 0 new-path) (nth 1 new-path)))) (if spath-found (when (< (nth 3 new-path) (nth 3 spath-found)) (setcdr (nthcdr 1 spath-found) (list (nth 2 new-path) (nth 3 new-path))) )
(add-shortest-path new-path)) ) ) ) ) ) )
(defun find-shortest-route (start end)
(let ((point-list '())
(end-point end) path-found) (add-to-list 'point-list end) (catch 'no-more-path (while 't (setq path-found (find-shortest-path start end-point)) (if (or (not path-found) (not (nth 2 path-found))) (throw 'no-more-path nil) (progn (add-to-list 'point-list (nth 2 path-found)) (setq end-point (nth 2 path-found)) ) ) ) ) (add-to-list 'point-list start) )
)
(defun show-shortest-path (start end)
(let ((path-found (find-shortest-path start end))
(route-found (find-shortest-route start end))) (if path-found (progn (message "shortest distance: %s" (nth 3 path-found)) (message "shortest route: %s" route-found) ) (message "shortest path not found") ) )
(message "--") )
;; Process each path
(dolist (path path-list)
(process-path path) )
(message "from %s to %s:" 'a 'e)
(show-shortest-path 'a 'e)
(message "from %s to %s:" 'a 'f)
(show-shortest-path 'a 'f)
)
)
(calculate-shortest-path) </lang>
outputs:
from a to e: shortest distance: 26 shortest route: (a c d e) -- from a to f: shortest distance: 11 shortest route: (a c f) --
Erlang
<lang erlang> -module(dijkstra). -include_lib("eunit/include/eunit.hrl"). -export([dijkstrafy/3]).
% just hide away recursion so we have a nice interface dijkstrafy(Graph, Start, End) when is_map(Graph) -> shortest_path(Graph, [{0, [Start]}], End, #{}).
shortest_path(_Graph, [], _End, _Visited) -> % if we're not going anywhere, it's time to start going back {0, []}; shortest_path(_Graph, [{Cost, [End | _] = Path} | _ ], End, _Visited) -> % this is the base case, and finally returns the distance and the path {Cost, lists:reverse(Path)}; shortest_path(Graph, [{Cost, [Node | _ ] = Path} | Routes], End, Visited) -> % this is the recursive case. % here we build a list of new "unvisited" routes, where the stucture is % a tuple of cost, then a list of paths taken to get to that cost from the "Start" NewRoutes = [{Cost + NewCost, [NewNode | Path]} || {NewCost, NewNode} <- maps:get(Node, Graph), not maps:get(NewNode, Visited, false)], shortest_path( Graph, % add the routes we ripped off earlier onto the new routes % that we want to visit. sort the list of routes to get the % shortest routes (lowest cost) at the beginning. % Erlangs sort is already good enough, and it will sort the % tuples by the number at the beginning of each (the cost). lists:sort(NewRoutes ++ Routes), End, Visited#{Node => true} ).
basic_test() -> Graph = #{ a => [{7,b},{9,c},{14,f}], b => [{7,a},{10,c},{15,d}], c => [{10,b},{9,c},{11,d},{2,f}], d => [{15,b},{6,e},{11,c}], e => [{9,f},{6,d}], f => [{14,f},{2,c},{9,e}] }, {Cost, Path} = dijkstrafy(Graph, a, e), {20,[a,c,f,e]} = {Cost, Path}, io:format(user, "The total cost was ~p and the path was: ", [Cost]), io:format(user, "~w~n", [Path]). </lang>
- Output:
$ ./rebar3 eunit ===> Verifying dependencies... ===> Compiling dijkstra ===> Performing EUnit tests... The total cost was 20 and the path was: [a,c,f,e] Test passed.
F#
Dijkstra's algorithm
<lang fsharp> //Dijkstra's algorithm: Nigel Galloway, August 5th., 2018 [<CustomEquality;CustomComparison>] type Dijkstra<'N,'G when 'G:comparison>={toN:'N;cost:Option<'G>;fromN:'N}
override g.Equals n =match n with| :? Dijkstra<'N,'G> as n->n.cost=g.cost|_->false
override g.GetHashCode() = hash g.cost
interface System.IComparable with
member n.CompareTo g =
match g with
| :? Dijkstra<'N,'G> as n when n.cost=None -> (-1)
| :? Dijkstra<'N,'G> when n.cost=None -> 1
| :? Dijkstra<'N,'G> as g -> compare n.cost g.cost
| _-> invalidArg "n" "expecting type Dijkstra<'N,'G>"
let inline Dijkstra N G y =
let rec fN l f=
if List.isEmpty l then f
else let n=List.min l
if n.cost=None then f else
fN(l|>List.choose(fun n'->if n'.toN=n.toN then None else match n.cost,n'.cost,Map.tryFind (n.toN,n'.toN) G with
|Some g,None,Some wg ->Some {toN=n'.toN;cost=Some(g+wg);fromN=n.toN}
|Some g,Some g',Some wg when g+wg<g'->Some {toN=n'.toN;cost=Some(g+wg);fromN=n.toN}
|_ ->Some n'))((n.fromN,n.toN)::f)
let r = fN (N|>List.map(fun n->{toN=n;cost=(Map.tryFind(y,n)G);fromN=y})) []
(fun n->let rec fN z l=match List.tryFind(fun (_,g)->g=z) r with
|Some(n',g') when y=n'->Some(n'::g'::l)
|Some(n',g') ->fN n' (g'::l)
|_ ->None
fN n [])
</lang>
The Task
<lang fsharp> type Node= |A|B|C|D|E|F let G=Map[((A,B),7);((A,C),9);((A,F),14);((B,C),10);((B,D),15);((C,D),11);((C,F),2);((D,E),6);((E,F),9)] let paths=Dijkstra [B;C;D;E;F] G A printfn "%A" (paths E) printfn "%A" (paths F) </lang>
- Output:
Some [A; C; D; E] Some [A; C; F]
Go
<lang go>package main
import ( "container/heap" "fmt" )
// A PriorityQueue implements heap.Interface and holds Items. type PriorityQueue struct { items []Vertex m map[Vertex]int // value to index pr map[Vertex]int // value to priority }
func (pq *PriorityQueue) Len() int { return len(pq.items) } func (pq *PriorityQueue) Less(i, j int) bool { return pq.pr[pq.items[i]] < pq.pr[pq.items[j]] } func (pq *PriorityQueue) Swap(i, j int) { pq.items[i], pq.items[j] = pq.items[j], pq.items[i] pq.m[pq.items[i]] = i pq.m[pq.items[j]] = j } func (pq *PriorityQueue) Push(x interface{}) { n := len(pq.items) item := x.(Vertex) pq.m[item] = n pq.items = append(pq.items, item) } func (pq *PriorityQueue) Pop() interface{} { old := pq.items n := len(old) item := old[n-1] pq.m[item] = -1 pq.items = old[0 : n-1] return item }
// update modifies the priority of an item in the queue. func (pq *PriorityQueue) update(item Vertex, priority int) { pq.pr[item] = priority heap.Fix(pq, pq.m[item]) } func (pq *PriorityQueue) addWithPriority(item Vertex, priority int) { heap.Push(pq, item) pq.update(item, priority) }
const ( Infinity = int(^uint(0) >> 1) Uninitialized = -1 )
func Dijkstra(g Graph, source Vertex) (dist map[Vertex]int, prev map[Vertex]Vertex) { vs := g.Vertices() dist = make(map[Vertex]int, len(vs)) prev = make(map[Vertex]Vertex, len(vs)) sid := source dist[sid] = 0 q := &PriorityQueue{ items: make([]Vertex, 0, len(vs)), m: make(map[Vertex]int, len(vs)), pr: make(map[Vertex]int, len(vs)), } for _, v := range vs { if v != sid { dist[v] = Infinity } prev[v] = Uninitialized q.addWithPriority(v, dist[v]) } for len(q.items) != 0 { u := heap.Pop(q).(Vertex) for _, v := range g.Neighbors(u) { alt := dist[u] + g.Weight(u, v) if alt < dist[v] { dist[v] = alt prev[v] = u q.update(v, alt) } } } return dist, prev }
// A Graph is the interface implemented by graphs that // this algorithm can run on. type Graph interface { Vertices() []Vertex Neighbors(v Vertex) []Vertex Weight(u, v Vertex) int }
// Nonnegative integer ID of vertex type Vertex int
// sg is a graph of strings that satisfies the Graph interface. type sg struct { ids map[string]Vertex names map[Vertex]string edges map[Vertex]map[Vertex]int }
func newsg(ids map[string]Vertex) sg { g := sg{ids: ids} g.names = make(map[Vertex]string, len(ids)) for k, v := range ids { g.names[v] = k } g.edges = make(map[Vertex]map[Vertex]int) return g } func (g sg) edge(u, v string, w int) { if _, ok := g.edges[g.ids[u]]; !ok { g.edges[g.ids[u]] = make(map[Vertex]int) } g.edges[g.ids[u]][g.ids[v]] = w } func (g sg) path(v Vertex, prev map[Vertex]Vertex) (s string) { s = g.names[v] for prev[v] >= 0 { v = prev[v] s = g.names[v] + s } return s } func (g sg) Vertices() []Vertex { vs := make([]Vertex, 0, len(g.ids)) for _, v := range g.ids { vs = append(vs, v) } return vs } func (g sg) Neighbors(u Vertex) []Vertex { vs := make([]Vertex, 0, len(g.edges[u])) for v := range g.edges[u] { vs = append(vs, v) } return vs } func (g sg) Weight(u, v Vertex) int { return g.edges[u][v] }
func main() { g := newsg(map[string]Vertex{ "a": 1, "b": 2, "c": 3, "d": 4, "e": 5, "f": 6, }) g.edge("a", "b", 7) g.edge("a", "c", 9) g.edge("a", "f", 14) g.edge("b", "c", 10) g.edge("b", "d", 15) g.edge("c", "d", 11) g.edge("c", "f", 2) g.edge("d", "e", 6) g.edge("e", "f", 9)
dist, prev := Dijkstra(g, g.ids["a"]) fmt.Printf("Distance to %s: %d, Path: %s\n", "e", dist[g.ids["e"]], g.path(g.ids["e"], prev)) fmt.Printf("Distance to %s: %d, Path: %s\n", "f", dist[g.ids["f"]], g.path(g.ids["f"], prev)) }</lang> Runable on the Go playground.
- Output:
Distance to e: 26, Path: acde Distance to f: 11, Path: acf
Haskell
Translation of the C++ solution, and all the complexities are the same as in the C++ solution. In particular, we again use a self-balancing binary search tree (Data.Set) to implement the priority queue, which results in an optimal complexity.
<lang haskell>
{-# LANGUAGE FlexibleContexts #-}
import Data.Array
import Data.Array.MArray
import Data.Array.ST
import Control.Monad.ST
import Control.Monad (foldM)
import Data.Set as S
dijkstra :: (Ix v, Num w, Ord w, Bounded w) => v -> v -> Array v [(v,w)] -> (Array v w, Array v v) dijkstra src invalid_index adj_list = runST $ do
min_distance <- newSTArray b maxBound
writeArray min_distance src 0
previous <- newSTArray b invalid_index
let aux vertex_queue =
case S.minView vertex_queue of
Nothing -> return ()
Just ((dist, u), vertex_queue') ->
let edges = adj_list ! u
f vertex_queue (v, weight) = do
let dist_thru_u = dist + weight
old_dist <- readArray min_distance v
if dist_thru_u >= old_dist then
return vertex_queue
else do
let vertex_queue' = S.delete (old_dist, v) vertex_queue
writeArray min_distance v dist_thru_u
writeArray previous v u
return $ S.insert (dist_thru_u, v) vertex_queue'
in
foldM f vertex_queue' edges >>= aux -- note that aux is being called within its own definition (i.e. aux is recursive). The foldM only iterates on the neighbours of v, it does not execute the while loop itself in Dijkstra's
aux (S.singleton (0, src))
m <- freeze min_distance
p <- freeze previous
return (m, p)
where b = bounds adj_list
newSTArray :: Ix i => (i,i) -> e -> ST s (STArray s i e)
newSTArray = newArray
shortest_path_to :: (Ix v) => v -> v -> Array v v -> [v] shortest_path_to target invalid_index previous =
aux target [] where
aux vertex acc | vertex == invalid_index = acc
| otherwise = aux (previous ! vertex) (vertex : acc)
adj_list :: Array Char [(Char, Int)] adj_list = listArray ('a', 'f') [ [('b',7), ('c',9), ('f',14)],
[('a',7), ('c',10), ('d',15)],
[('a',9), ('b',10), ('d',11), ('f',2)],
[('b',15), ('c',11), ('e',6)],
[('d',6), ('f',9)],
[('a',14), ('c',2), ('e',9)] ]
main :: IO () main = do
let (min_distance, previous) = dijkstra 'a' ' ' adj_list putStrLn $ "Distance from a to e: " ++ show (min_distance ! 'e') let path = shortest_path_to 'e' ' ' previous putStrLn $ "Path: " ++ show path</lang>
Huginn
<lang huginn>import Algorithms as algo; import Mathematics as math; import Text as text;
class Edge { _to = none; _name = none; _cost = none; constructor( to_, name_, cost_ ) { _to = to_; _name = name_; _cost = real( cost_ ); } to_string() { return ( "{}<{}>".format( _name, _cost ) ); } }
class Path { _id = none; _from = none; _cost = none; _names = none; constructor( toName_, ids_, names_ ) { _id = ids_[toName_]; _names = names_; _cost = math.INFINITY; } less( other_ ) { return ( _cost < other_._cost ); } update( from_, cost_ ) { _from = from_; _cost = cost_; } to_string() { return ( "{} via {} at cost {}".format( _names[_id], _from != none ? _names[_from] : none, _cost ) ); } }
class Graph { _neighbours = []; _ids = {}; _names = []; add_node( name_ ) { if ( name_ ∉ _ids ) { _ids[name_] = size( _names ); _names.push( name_ ); } } add_edge( from_, to_, cost_ ) { assert( ( from_ ∈ _ids ) && ( to_ ∈ _ids ) ); from = _ids[from_]; to = _ids[to_]; if ( from >= size( _neighbours ) ) { _neighbours.resize( from + 1, [] ); } _neighbours[from].push( Edge( to, to_, cost_ ) ); } shortest_paths( from_ ) { assert( from_ ∈ _ids ); from = _ids[from_]; paths = algo.materialize( algo.map( _names, @[_ids, _names]( name ) { Path( name, _ids, _names ); } ), list ); paths[from].update( none, 0.0 ); todo = algo.sorted( paths, @(x){-x._cost;} ); while ( size( todo ) > 0 ) { node = todo[-1]._id; todo.resize( size( todo ) - 1, none ); if ( node >= size( _neighbours ) ) { continue; } neighbours = _neighbours[node]; for ( n : neighbours ) { newCost = n._cost + paths[node]._cost; if ( newCost < paths[n._to]._cost ) { paths[n._to].update( node, newCost ); } } todo = algo.sorted( todo, @(x){-x._cost;} ); } return ( paths ); } path( paths_, to_ ) { assert( to_ ∈ _ids ); to = _ids[to_]; p = [to_]; while ( paths_[to]._from != none ) { to = paths_[to]._from; p.push( _names[to] ); } return ( algo.materialize( algo.reversed( p ), list ) ); } to_string() { s = ""; for ( i, n : algo.enumerate( _neighbours ) ) { s += "{} -> {}\n".format( _names[i], n ); } } }
main() { g = Graph(); confStr = input(); if ( confStr == none ) { return ( 1 ); } conf = algo.materialize( algo.map( text.split( confStr ), integer ), tuple ); assert( size( conf ) == 2 ); for ( _ : algo.range( conf[0] ) ) { line = input(); if ( line == none ) { return ( 1 ); } g.add_node( line.strip() ); } for ( _ : algo.range( conf[1] ) ) { line = input(); if ( line == none ) { return ( 1 ); } g.add_edge( algo.materialize( text.split( line.strip() ), tuple )... ); } print( string( g ) ); paths = g.shortest_paths( "a" ); for ( p : paths ) { print( "{}\n".format( p ) ); } print( "{}\n".format( g.path( paths, "e" ) ) ); print( "{}\n".format( g.path( paths, "f" ) ) ); }</lang>
Sample run via:
cat ~/graph.g | ./dijkstra.hgn
, output:
a -> [b<7.0>, c<9.0>, f<14.0>] b -> [c<10.0>, d<15.0>] c -> [d<11.0>, f<2.0>] d -> [e<6.0>] e -> [f<9.0>] a via none at cost 0.0 b via a at cost 7.0 c via a at cost 9.0 d via c at cost 20.0 e via d at cost 26.0 f via c at cost 11.0 [a, c, d, e] [a, c, f]
Icon and Unicon
This Unicon-only solution is an adaptation of the Unicon parallel maze solver found in Maze solving. It searches paths in the graph in parallel until all possible shortest paths from the start node to the finish node have been discovered and then outputs the shortest path.
<lang unicon>procedure main(A)
graph := getGraph()
repeat {
writes("What is the start node? ")
start := \graph.nodes[read()] | stop()
writes("What is the finish node? ")
finish := read() | stop()
QMouse(graph,start,finish)
waitForCompletion() # block until all quantum mice have finished
showPath(getBestMouse(),start.name,finish)
cleanGraph(graph)
}
end
procedure getGraph()
graph := Graph(table(),table())
write("Enter edges as 'n1,n2,weight' (blank line terminates)")
repeat {
if *(line := trim(read())) = 0 then break
line ? {
n1 := 1(tab(upto(',')),move(1))
n2 := 1(tab(upto(',')),move(1))
w := tab(0)
/graph.nodes[n1] := Node(n1,set())
/graph.nodes[n2] := Node(n2,set())
insert(graph.nodes[n1].targets,graph.nodes[n2])
graph.weights[n1||":"||n2] := w
}
}
return graph
end
procedure showPath(mouse,start,finish)
if \mouse then {
path := mouse.getPath()
writes("Weight: ",path.weight," -> ")
every writes(" ",!path.nodes)
write("\n")
}
else write("No path from ",start," to ",finish,"\n")
end
- A "Quantum-mouse" for traversing graphs. Each mouse lives for just
- one node but can spawn additional mice to search adjoining nodes.
global qMice, goodMice, region, qMiceEmpty
record Graph(nodes,weights) record Node(name,targets,weight) record Path(weight, nodes)
class QMouse(graph, loc, finish, path)
method getPath(); return path; end
method atEnd(); return (finish == loc.name); end
method visit(n) # Visit if we don't already have a cheaper route to n
newWeight := path.weight + graph.weights[loc.name||":"||n.name]
critical region[n]: if /n.weight | (newWeight < n.weight) then {
n.weight := newWeight
unlock(region[n])
return n
}
end
initially (g, l, f, p)
initial { # Construct critical region mutexes and completion condvar
qMiceEmpty := condvar()
region := table()
every region[n := !g.nodes] := mutex()
qMice := mutex(set())
cleanGraph(g)
}
graph := g
loc := l
finish := f
/p := Path(0,[])
path := Path(p.weight,copy(p.nodes))
if *path.nodes > 0 then
path.weight +:= g.weights[path.nodes[-1]||":"||loc.name]
put(path.nodes, loc.name)
insert(qMice,self)
thread {
if atEnd() then insert(goodMice, self) # This mouse found a finish
every QMouse(g,visit(!loc.targets),f,path)
delete(qMice, self) # Kill this mouse
if *qMice=0 then signal(qMiceEmpty) # All mice are dead
}
end
procedure cleanGraph(graph)
every (!graph.nodes).weight := &null goodMice := mutex(set())
end
procedure getBestMouse()
every mouse := !goodMice do { # Locate shortest path
weight := mouse.getPath().weight
/minPathWeight := weight
if minPathWeight >=:= weight then bestMouse := mouse
}
return bestMouse
end
procedure waitForCompletion()
critical qMiceEmpty: while *qMice > 0 do wait(qMiceEmpty)
end</lang>
Sample run:
-> dSolve Enter edges as 'n1,n2,weight' (blank line terminates) a,b,7 a,c,9 a,f,14 b,c,10 b,d,15 c,d,11 c,f,2 d,e,6 e,f,9 What is the start node? a What is the finish node? f Weight: 11 -> a c f What is the start node? a What is the finish node? e Weight: 26 -> a c d e What is the start node? f What is the finish node? a No path from f to a What is the start node? ->
J
<lang J> NB. verbs and adverb parse_table=: ;:@:(LF&= [;._2 -.&CR) mp=: $:~ :(+/ .*) NB. matrix product min=: <./ NB. minimum Index=: (i.`)(`:6) NB. Index adverb
dijkstra=: dyad define
'LINK WEIGHT'=. , (0 _ ,. 2) <;.3 y
'SOURCE SINK'=. |: LINK
FRONTIER=. , < {. x
GOAL=. {: x
enumerate=. 2&([\)&.>
while. FRONTIER do.
PATH_MASK=. FRONTIER (+./@:(-:"1/)&:>"0 _~ enumerate)~ LINK
I=. PATH_MASK min Index@:mp WEIGHTS
PATH=. I >@{ FRONTIER
STATE=. {: PATH
if. STATE -: GOAL do. PATH return. end.
FRONTIER=. (<<< I) { FRONTIER NB. elision
ADJACENCIES=. (STATE = SOURCE) # SINK
FRONTIER=. FRONTIER , PATH <@,"1 0 ADJACENCIES
end.
EMPTY
)
NB. The specific problem
INPUT=: noun define a b 7 a c 9 a f 14 b c 10 b d 15 c d 11 c f 2 d e 6 e f 9 )
T=: parse_table INPUT NAMED_LINKS=: _ 2 {. T NODES=: ~. , NAMED_LINKS NB. vector of boxed names NUMBERED_LINKS=: NODES i. NAMED_LINKS WEIGHTS=: _ ".&> _ _1 {. T GRAPH=: NUMBERED_LINKS ,. WEIGHTS NB. GRAPH is the numerical representation
TERMINALS=: NODES (i. ;:) 'a e'
NODES {~ TERMINALS dijkstra GRAPH
Note 'Output' ┌─┬─┬─┬─┐ │a│c│d│e│ └─┴─┴─┴─┘
TERMINALS and GRAPH are integer arrays:
TERMINALS
0 5
GRAPH
0 1 7 0 2 9 0 3 14 1 2 10 1 4 15 2 4 11 2 3 2 4 5 6 5 3 9 ) </lang>
J: Alternative Implementation
<lang j>vertices=: ;:'a b c d e f' edges=:|: ;:;._2]0 :0
a b 7 a c 9 a f 14 b c 10 b d 15 c d 11 c f 2 d e 6 e f 9
)
shortest_path=:1 :0
NB. x: path endpoints, m: vertex labels, y: edges (starts,ends,:costs)
terminals=. m i. x
starts=. m i. 0{y
ends=. m i. 1{y
tolls=. _&".@> 2{y
C=. tolls (starts,&.>ends)}_$~2##m
bestprice=. (<terminals){ (<. <./ .+/~)^:_ C
best=. i.0
if. _>bestprice do.
paths=. ,.{.terminals
goal=. {:terminals
costs=. ,0
while. #costs do.
next=. ({:paths){C
keep=. (_>next)*bestprice>:next+costs
rep=. +/"1 keep
paths=. (rep#"1 paths),(#m)|I.,keep
costs=. (rep#"1 costs)+keep #&, next
if. #j=. I. goal = {:paths do.
best=. best, (bestprice=j{costs)# <"1 j{|:paths
end.
toss=. <<<j,I.bestprice<:costs
paths=. toss {"1 paths
costs=. toss { costs
end.
end.
best {L:0 _ m
)</lang>
Example use:
<lang J> (;:'a e') vertices shortest_path edges ┌─────────┐ │┌─┬─┬─┬─┐│ ││a│c│d│e││ │└─┴─┴─┴─┘│ └─────────┘</lang>
This version finds all shortest paths, and for this example completes in two thirds the time of the other J implementation.
This algorithm first translates the graph representation to a cost connection matrix, with infinite cost for unconnected nodes. Then we use a summing variation on transitive closure to find minimal connection costs for all nodes, and extract our best price from that. If our desired nodes are connected, we then search for paths which satisfy this best (minimal) price constraint: We repeatedly find all connections from our frontier, tracking path cost and discarding paths which have a cost which exceeds our best price. When a path reaches the end node, it is removed and remembered.
Java
Algorithm is derived from Wikipedia section 'Using a priority queue'. This implementation finds the single path from a source to all reachable vertices. Building the graph from a set of edges takes O(E log V) for each pass. Vertices are stored in a TreeSet (self-balancing binary search tree) instead of a PriorityQueue (a binary heap) in order to get O(log n) performance for removal of any element, not just the head. Decreasing the distance of a vertex is accomplished by removing it from the tree and later re-inserting it. <lang java> import java.io.*; import java.util.*;
public class Dijkstra {
private static final Graph.Edge[] GRAPH = {
new Graph.Edge("a", "b", 7),
new Graph.Edge("a", "c", 9),
new Graph.Edge("a", "f", 14),
new Graph.Edge("b", "c", 10),
new Graph.Edge("b", "d", 15),
new Graph.Edge("c", "d", 11),
new Graph.Edge("c", "f", 2),
new Graph.Edge("d", "e", 6),
new Graph.Edge("e", "f", 9),
};
private static final String START = "a";
private static final String END = "e";
public static void main(String[] args) {
Graph g = new Graph(GRAPH);
g.dijkstra(START);
g.printPath(END);
//g.printAllPaths();
}
}
class Graph {
private final Map<String, Vertex> graph; // mapping of vertex names to Vertex objects, built from a set of Edges
/** One edge of the graph (only used by Graph constructor) */
public static class Edge {
public final String v1, v2;
public final int dist;
public Edge(String v1, String v2, int dist) {
this.v1 = v1;
this.v2 = v2;
this.dist = dist;
}
}
/** One vertex of the graph, complete with mappings to neighbouring vertices */
public static class Vertex implements Comparable<Vertex>{
public final String name; public int dist = Integer.MAX_VALUE; // MAX_VALUE assumed to be infinity public Vertex previous = null; public final Map<Vertex, Integer> neighbours = new HashMap<>();
public Vertex(String name) { this.name = name; }
private void printPath() { if (this == this.previous) { System.out.printf("%s", this.name); } else if (this.previous == null) { System.out.printf("%s(unreached)", this.name); } else { this.previous.printPath(); System.out.printf(" -> %s(%d)", this.name, this.dist); } }
public int compareTo(Vertex other) { if (dist == other.dist) return name.compareTo(other.name);
return Integer.compare(dist, other.dist); }
@Override public String toString() { return "(" + name + ", " + dist + ")"; } }
/** Builds a graph from a set of edges */
public Graph(Edge[] edges) {
graph = new HashMap<>(edges.length);
//one pass to find all vertices
for (Edge e : edges) {
if (!graph.containsKey(e.v1)) graph.put(e.v1, new Vertex(e.v1));
if (!graph.containsKey(e.v2)) graph.put(e.v2, new Vertex(e.v2));
}
//another pass to set neighbouring vertices
for (Edge e : edges) {
graph.get(e.v1).neighbours.put(graph.get(e.v2), e.dist);
//graph.get(e.v2).neighbours.put(graph.get(e.v1), e.dist); // also do this for an undirected graph
}
}
/** Runs dijkstra using a specified source vertex */
public void dijkstra(String startName) {
if (!graph.containsKey(startName)) {
System.err.printf("Graph doesn't contain start vertex \"%s\"\n", startName);
return;
}
final Vertex source = graph.get(startName);
NavigableSet<Vertex> q = new TreeSet<>();
// set-up vertices
for (Vertex v : graph.values()) {
v.previous = v == source ? source : null;
v.dist = v == source ? 0 : Integer.MAX_VALUE;
q.add(v);
}
dijkstra(q);
}
/** Implementation of dijkstra's algorithm using a binary heap. */
private void dijkstra(final NavigableSet<Vertex> q) {
Vertex u, v;
while (!q.isEmpty()) {
u = q.pollFirst(); // vertex with shortest distance (first iteration will return source)
if (u.dist == Integer.MAX_VALUE) break; // we can ignore u (and any other remaining vertices) since they are unreachable
//look at distances to each neighbour
for (Map.Entry<Vertex, Integer> a : u.neighbours.entrySet()) {
v = a.getKey(); //the neighbour in this iteration
final int alternateDist = u.dist + a.getValue();
if (alternateDist < v.dist) { // shorter path to neighbour found
q.remove(v);
v.dist = alternateDist;
v.previous = u;
q.add(v);
}
}
}
}
/** Prints a path from the source to the specified vertex */
public void printPath(String endName) {
if (!graph.containsKey(endName)) {
System.err.printf("Graph doesn't contain end vertex \"%s\"\n", endName);
return;
}
graph.get(endName).printPath();
System.out.println();
}
/** Prints the path from the source to every vertex (output order is not guaranteed) */
public void printAllPaths() {
for (Vertex v : graph.values()) {
v.printPath();
System.out.println();
}
}
}</lang>
- Output:
a -> c(9) -> d(20) -> e(26)
JavaScript
Using the wp:Dijkstra's_algorithm#Pseudocode <lang javascript> const dijkstra = (edges,source,target) => {
const Q = new Set(),
prev = {},
dist = {},
adj = {}
const vertex_with_min_dist = (Q,dist) => {
let min_distance = Infinity,
u = null
for (let v of Q) {
if (dist[v] < min_distance) {
min_distance = dist[v]
u = v
}
}
return u
}
for (let i=0;i<edges.length;i++) {
let v1 = edges[i][0],
v2 = edges[i][1],
len = edges[i][2]
Q.add(v1)
Q.add(v2)
dist[v1] = Infinity
dist[v2] = Infinity
if (adj[v1] === undefined) adj[v1] = {}
if (adj[v2] === undefined) adj[v2] = {}
adj[v1][v2] = len
adj[v2][v1] = len
}
dist[source] = 0
while (Q.size) {
let u = vertex_with_min_dist(Q,dist),
neighbors = Object.keys(adj[u]).filter(v=>Q.has(v)) //Neighbor still in Q
Q.delete(u)
if (u===target) break //Break when the target has been found
for (let v of neighbors) {
let alt = dist[u] + adj[u][v]
if (alt < dist[v]) {
dist[v] = alt
prev[v] = u
}
}
}
{
let u = target,
S = [u],
len = 0
while (prev[u] !== undefined) {
S.unshift(prev[u])
len += adj[u][prev[u]]
u = prev[u]
}
return [S,len]
}
}
//Testing algorithm let graph = [] graph.push(["a", "b", 7]) graph.push(["a", "c", 9]) graph.push(["a", "f", 14]) graph.push(["b", "c", 10]) graph.push(["b", "d", 15]) graph.push(["c", "d", 11]) graph.push(["c", "f", 2]) graph.push(["d", "e", 6]) graph.push(["e", "f", 9])
let [path,length] = dijkstra(graph, "a", "e"); console.log(path) //[ 'a', 'c', 'f', 'e' ] console.log(length) //20 </lang>
jq
Works with gojq, the Go implementation of jq (*)
For this entry, the graph will be represented by a JSON object with keys, $k, such that `.[$k]` describes the immediate neighbors of node $k, with `.[$k][$n]` being the distance from node $k to node $n. A JSON object of this form will be referred to as a Graph.
Not all nodes of the graph need be top-level keys of the graph object.
The function `dijkstra($startname)` fulfills the first part of the task. It produces the final state of a "scratchpad" in the form of a JSON object with key:value pairs of the form `node: {prev, dist}`. In its final state, `dist` is the shortest distance from the node identified by the key to the node identified by `node`. The `readout` function is the function envisioned by the third part of the task.
Preliminaries <lang jq># (*) If using gojq, uncomment the following line:
- def keys_unsorted: keys;
- remove the first occurrence of $x from the input array
def rm($x):
index($x) as $ix | if $ix then .[:$ix] + .[$ix+1:] else . end;
- Input: a Graph
- Output: a (possibly empty) stream of the neighbors of $node
- that are also in the array $ary
def neighbors($node; $ary:
.[$node] | select(.) | keys_unsorted[] | . as $n | select($ary | index($n));
- Input: a Graph
def vertices:
[keys_unsorted[], (.[] | keys_unsorted[])] | unique;
- Input: a Graph
- Output: the final version of the scratchpad
def dijkstra($startname):
. as $graph
| vertices as $Q
# scratchpad: { node: { prev, dist} }
| reduce $Q[] as $v ({};
. + { ($v): {prev: null, dist: infinite}} )
| .[$startname].dist = 0
| { scratchpad: ., $Q }
| until( .Q|length == 0;
.scratchpad as $scratchpad
| ( .Q | min_by($scratchpad[.].dist)) as $u
| .Q |= rm($u)
| .Q as $Q
# for each neighbor v of u still in Q:
| reduce ($graph|neighbors($u; $Q)) as $v (.;
(.scratchpad[$u].dist + $graph[$u][$v]) as $alt
| if $alt < .scratchpad[$v].dist
then .scratchpad[$v].dist = $alt
| .scratchpad[$v].prev = $u
else . end ) )
| .scratchpad ;
- Input: a Graph
- Output: the scratchpad
def Dijkstra($startname):
if .[$startname] == null then "The graph does not contain start vertex \(startname)" else dijkstra($startname) end;
- Input: scratchpad, i.e. a dictionary with key:value pairs of the form:
- node: {prev, dist}
- Output: an array, being
- [optimal path from $node to $n, optimal distance from $node to $n]
def readout($node):
. as $in | $node | [recurse($in[.].prev; .)] | [reverse, $in[$node].dist] ;
- Input: a graph
- Output: [path, value]
def Dijkstra($startname; $endname):
Dijkstra($startname) | readout($endname) ;
</lang> The Task <lang jq>def GRAPH: {
"a": {"b": 7, "c": 9, "f": 14},
"b": {"c": 10, "d": 15},
"c": {"d": 11, "f": 2},
"d": {"e": 6},
"e": {"f": 9}
};
- To produce the final version of the scratchpad:
- GRAPH | Dijkstra("a")
"\nThe shortest paths from a to e and to f:", (GRAPH | Dijkstra("a"; "e", "f") | .[0])</lang>
- Output:
The shortest paths from a to e and to f: ["a","c","d","e"] ["a","c","f"]
Julia
<lang julia>struct Digraph{T <: Real,U}
edges::Dict{Tuple{U,U},T}
verts::Set{U}
end
function Digraph(edges::Vector{Tuple{U,U,T}}) where {T <: Real,U}
vnames = Set{U}(v for edge in edges for v in edge[1:2])
adjmat = Dict((edge[1], edge[2]) => edge[3] for edge in edges)
return Digraph(adjmat, vnames)
end
vertices(g::Digraph) = g.verts edges(g::Digraph) = g.edges
neighbours(g::Digraph, v) = Set((b, c) for ((a, b), c) in edges(g) if a == v)
function dijkstrapath(g::Digraph{T,U}, source::U, dest::U) where {T, U}
@assert source ∈ vertices(g) "$source is not a vertex in the graph"
# Easy case if source == dest return [source], 0 end # Initialize variables inf = typemax(T) dist = Dict(v => inf for v in vertices(g)) prev = Dict(v => v for v in vertices(g)) dist[source] = 0 Q = copy(vertices(g)) neigh = Dict(v => neighbours(g, v) for v in vertices(g))
# Main loop
while !isempty(Q)
u = reduce((x, y) -> dist[x] < dist[y] ? x : y, Q)
pop!(Q, u)
if dist[u] == inf || u == dest break end
for (v, cost) in neigh[u]
alt = dist[u] + cost
if alt < dist[v]
dist[v] = alt
prev[v] = u
end
end
end
# Return path
rst, cost = U[], dist[dest]
if prev[dest] == dest
return rst, cost
else
while dest != source
unshift!(rst, dest)
dest = prev[dest]
end
unshift!(rst, dest)
return rst, cost
end
end
- testgraph = [("a", "b", 1), ("b", "e", 2), ("a", "e", 4)]
testgraph = [("a", "b", 7), ("a", "c", 9), ("a", "f", 14), ("b", "c", 10),
("b", "d", 15), ("c", "d", 11), ("c", "f", 2), ("d", "e", 6),
("e", "f", 9)]
g = Digraph(testgraph) src, dst = "a", "e" path, cost = dijkstrapath(g, src, dst) println("Shortest path from $src to $dst: ", isempty(path) ? "no possible path" : join(path, " → "), " (cost $cost)")
- Print all possible paths
@printf("\n%4s | %3s | %s\n", "src", "dst", "path") @printf("----------------\n") for src in vertices(g), dst in vertices(g)
path, cost = dijkstrapath(g, src, dst)
@printf("%4s | %3s | %s\n", src, dst, isempty(path) ? "no possible path" : join(path, " → ") * " ($cost)")
end</lang>
- Output:
Shortest path from a to e: a → c → d → e (cost 26) src | dst | path ---------------- f | f | f (0) f | c | no possible path f | e | no possible path f | b | no possible path f | a | no possible path f | d | no possible path c | f | c → f (2) c | c | c (0) c | e | c → d → e (17) c | b | no possible path c | a | no possible path c | d | c → d (11) e | f | e → f (9) e | c | no possible path e | e | e (0) e | b | no possible path e | a | no possible path e | d | no possible path b | f | b → c → f (12) b | c | b → c (10) b | e | b → d → e (21) b | b | b (0) b | a | no possible path b | d | b → d (15) a | f | a → c → f (11) a | c | a → c (9) a | e | a → c → d → e (26) a | b | a → b (7) a | a | a (0) a | d | a → c → d (20) d | f | d → e → f (15) d | c | no possible path d | e | d → e (6) d | b | no possible path d | a | no possible path d | d | d (0)
Kotlin
<lang scala>// version 1.1.51
import java.util.TreeSet
class Edge(val v1: String, val v2: String, val dist: Int)
/** One vertex of the graph, complete with mappings to neighbouring vertices */
class Vertex(val name: String) : Comparable<Vertex> {
var dist = Int.MAX_VALUE // MAX_VALUE assumed to be infinity var previous: Vertex? = null val neighbours = HashMap<Vertex, Int>()
fun printPath() {
if (this == previous) {
print(name)
}
else if (previous == null) {
print("$name(unreached)")
}
else {
previous!!.printPath()
print(" -> $name($dist)")
}
}
override fun compareTo(other: Vertex): Int {
if (dist == other.dist) return name.compareTo(other.name)
return dist.compareTo(other.dist)
}
override fun toString() = "($name, $dist)"
}
class Graph(
val edges: List<Edge>, val directed: Boolean, val showAllPaths: Boolean = false
) {
// mapping of vertex names to Vertex objects, built from a set of Edges private val graph = HashMap<String, Vertex>(edges.size)
init {
// one pass to find all vertices
for (e in edges) {
if (!graph.containsKey(e.v1)) graph.put(e.v1, Vertex(e.v1))
if (!graph.containsKey(e.v2)) graph.put(e.v2, Vertex(e.v2))
}
// another pass to set neighbouring vertices
for (e in edges) {
graph[e.v1]!!.neighbours.put(graph[e.v2]!!, e.dist)
// also do this for an undirected graph if applicable
if (!directed) graph[e.v2]!!.neighbours.put(graph[e.v1]!!, e.dist)
}
}
/** Runs dijkstra using a specified source vertex */
fun dijkstra(startName: String) {
if (!graph.containsKey(startName)) {
println("Graph doesn't contain start vertex '$startName'")
return
}
val source = graph[startName]
val q = TreeSet<Vertex>()
// set-up vertices
for (v in graph.values) {
v.previous = if (v == source) source else null
v.dist = if (v == source) 0 else Int.MAX_VALUE
q.add(v)
}
dijkstra(q) }
/** Implementation of dijkstra's algorithm using a binary heap */
private fun dijkstra(q: TreeSet<Vertex>) {
while (!q.isEmpty()) {
// vertex with shortest distance (first iteration will return source)
val u = q.pollFirst()
// if distance is infinite we can ignore 'u' (and any other remaining vertices)
// since they are unreachable
if (u.dist == Int.MAX_VALUE) break
//look at distances to each neighbour
for (a in u.neighbours) {
val v = a.key // the neighbour in this iteration
val alternateDist = u.dist + a.value
if (alternateDist < v.dist) { // shorter path to neighbour found
q.remove(v)
v.dist = alternateDist
v.previous = u
q.add(v)
}
}
}
}
/** Prints a path from the source to the specified vertex */
fun printPath(endName: String) {
if (!graph.containsKey(endName)) {
println("Graph doesn't contain end vertex '$endName'")
return
}
print(if (directed) "Directed : " else "Undirected : ")
graph[endName]!!.printPath()
println()
if (showAllPaths) printAllPaths() else println()
}
/** Prints the path from the source to every vertex (output order is not guaranteed) */
private fun printAllPaths() {
for (v in graph.values) {
v.printPath()
println()
}
println()
}
}
val GRAPH = listOf(
Edge("a", "b", 7),
Edge("a", "c", 9),
Edge("a", "f", 14),
Edge("b", "c", 10),
Edge("b", "d", 15),
Edge("c", "d", 11),
Edge("c", "f", 2),
Edge("d", "e", 6),
Edge("e", "f", 9)
)
const val START = "a" const val END = "e"
fun main(args: Array<String>) {
with (Graph(GRAPH, true)) { // directed
dijkstra(START)
printPath(END)
}
with (Graph(GRAPH, false)) { // undirected
dijkstra(START)
printPath(END)
}
}</lang>
- Output:
Directed : a -> c(9) -> d(20) -> e(26) Undirected : a -> c(9) -> f(11) -> e(20)
Lua
Hopefully the variable names here make the process as clear as possible... <lang Lua>-- Graph definition local edges = {
a = {b = 7, c = 9, f = 14},
b = {c = 10, d = 15},
c = {d = 11, f = 2},
d = {e = 6},
e = {f = 9}
}
-- Fill in paths in the opposite direction to the stated edges function complete (graph)
for node, edges in pairs(graph) do
for edge, distance in pairs(edges) do
if not graph[edge] then graph[edge] = {} end
graph[edge][node] = distance
end
end
end
-- Create path string from table of previous nodes function follow (trail, destination)
local path, nextStep = destination, trail[destination]
while nextStep do
path = nextStep .. " " .. path
nextStep = trail[nextStep]
end
return path
end
-- Find the shortest path between the current and destination nodes function dijkstra (graph, current, destination, directed)
if not directed then complete(graph) end
local unvisited, distanceTo, trail = {}, {}, {}
local nearest, nextNode, tentative
for node, edgeDists in pairs(graph) do
if node == current then
distanceTo[node] = 0
trail[current] = false
else
distanceTo[node] = math.huge
unvisited[node] = true
end
end
repeat
nearest = math.huge
for neighbour, pathDist in pairs(graph[current]) do
if unvisited[neighbour] then
tentative = distanceTo[current] + pathDist
if tentative < distanceTo[neighbour] then
distanceTo[neighbour] = tentative
trail[neighbour] = current
end
if tentative < nearest then
nearest = tentative
nextNode = neighbour
end
end
end
unvisited[current] = false
current = nextNode
until unvisited[destination] == false or nearest == math.huge
return distanceTo[destination], follow(trail, destination)
end
-- Main procedure print("Directed:", dijkstra(edges, "a", "e", true)) print("Undirected:", dijkstra(edges, "a", "e", false))</lang>
- Output:
Directed: 26 a c d e Undirected: 20 a c f e
M2000 Interpreter
<lang M2000 Interpreter> Module Dijkstra`s_algorithm { const max_number=1.E+306 GetArr=lambda (n, val)->{ dim d(n)=val =d() } term=("",0) Edges=(("a", ("b",7),("c",9),("f",14)),("b",("c",10),("d",15)),("c",("d",11),("f",2)),("d",("e",6)),("e",("f", 9)),("f",term)) Document Doc$="Graph:"+{ } ShowGraph() Doc$="Paths"+{ } Print "Paths" For from_here=0 to 5 pa=GetArr(len(Edges), -1) d=GetArr(len(Edges), max_number) Inventory S=1,2,3,4,5,6 return d, from_here:=0 RemoveMin=Lambda S, d, max_number-> { ss=each(S) min=max_number p=0 while ss val=d#val(eval(S,ss^)-1) if min>val then let min=val : p=ss^ end while =s(p!) ' use p as index not key Delete S, eval(s,p) } Show_Distance_and_Path$=lambda$ d, pa, from_here, max_number (n) -> { ret1$=chr$(from_here+asc("a"))+" to "+chr$(n+asc("a")) if d#val(n) =max_number then =ret1$+ " No Path" :exit let ret$="", mm=n, m=n repeat n=m ret$+=chr$(asc("a")+n) m=pa#val(n) until from_here=n =ret1$+format$("{0::-4} {1}",d#val(mm),strrev$(ret$)) } while len(s)>0 u=RemoveMin() rem Print u, chr$(u-1+asc("a")) Relaxed() end while For i=0 to len(d)-1 line$=Show_Distance_and_Path$(i) Print line$ doc$=line$+{ } next next Clipboard Doc$ End Sub Relaxed() local vertex=Edges#val(u-1), i local e=Len(vertex)-1, edge=(,), val for i=1 to e edge=vertex#val(i) if edge#val$(0)<>"" then val=Asc(edge#val$(0))-Asc("a") if d#val(val)>edge#val(1)+d#val(u-1) then return d, val:=edge#val(1)+d#val(u-1) : Return Pa, val:=u-1 end if next end sub Sub ShowGraph() Print "Graph" local i for i=1 to len(Edges) show_edges(i) next end sub Sub show_edges(n) n-- local vertex=Edges#val(n), line$ local e=each(vertex 2 to end), v2=(,) While e v2=array(e) line$=vertex#val$(0)+if$(v2#val$(0)<>""->"->"+v2#val$(0)+format$(" {0::-2}",v2#val(1)),"") Print line$ Doc$=line$+{ } end while end sub } Dijkstra`s_algorithm </lang>
- Output:
Graph: a->b 7 a->c 9 a->f 14 b->c 10 b->d 15 c->d 11 c->f 2 d->e 6 e->f 9 f Paths a to a 0 a a to b 7 ab a to c 9 ac a to d 20 acd a to e 26 acde a to f 11 acf b to a No Path b to b 0 b b to c 10 bc b to d 15 bd b to e 21 bde b to f 12 bcf c to a No Path c to b No Path c to c 0 c c to d 11 cd c to e 17 cde c to f 2 cf d to a No Path d to b No Path d to c No Path d to d 0 d d to e 6 de d to f 15 def e to a No Path e to b No Path e to c No Path e to d No Path e to e 0 e e to f 9 ef f to a No Path f to b No Path f to c No Path f to d No Path f to e No Path f to f 0 f
Maple
<lang maple>restart: with(GraphTheory): G:=Digraph([a,b,c,d,e,f],{[[a,b],7],[[a,c],9],[[a,f],14],[[b,c],10],[[b,d],15],[[c,d],11],[[c,f],2],[[d,e],6],[[e,f],9]}): DijkstrasAlgorithm(G,a);
- [[[a], 0], [[a, b], 7], [[a, c], 9], [[a, c, d], 20], [[a, c, d, e], 26], [[a, c, f], 11]]</lang>
Mathematica/Wolfram Language
<lang Mathematica>bd = Graph[{"a" \[DirectedEdge] "b", "a" \[DirectedEdge] "c",
"b" \[DirectedEdge] "c", "b" \[DirectedEdge] "d",
"c" \[DirectedEdge] "d", "d" \[DirectedEdge] "e",
"a" \[DirectedEdge] "f", "c" \[DirectedEdge] "f",
"e" \[DirectedEdge] "f"},
EdgeWeight -> {7, 9, 10, 15, 11, 6, 14, 2, 9},
VertexLabels -> "Name", VertexLabelStyle -> Directive[Black, 20],
ImagePadding -> 20]
FindShortestPath[bd, "a", "e", Method -> "Dijkstra"] -> {"a", "c", "d", "e"}</lang> File:Mma dijkstra2.PNG
Maxima
<lang maxima>load(graphs)$ g: create_graph([[1, "a"], [2, "b"], [3, "c"], [4, "d"], [5, "e"], [6, "f"]],
[[[1, 2], 7], [[1, 3], 9], [[1, 6], 14], [[2, 3], 10], [[2, 4], 15], [[3, 4], 11], [[3, 6], 2], [[4, 5], 6], [[5, 6], 9]], directed)$
shortest_weighted_path(1, 5, g); /* [26, [1, 3, 4, 5]] */</lang>
Nim
Translation of Wikipedia pseudo-code. <lang Nim>
- Dijkstra algorithm.
from algorithm import reverse import sets import strformat import tables
type
Edge = tuple[src, dst: string; cost: int] Graph = object vertices: HashSet[string] neighbours: Table[string, seq[tuple[dst: string, cost: float]]]
- ---------------------------------------------------------------------------------------------------
proc initGraph(edges: openArray[Edge]): Graph =
## Initialize a graph from an edge list. ## Use floats for costs in order to compare to Inf value.
for (src, dst, cost) in edges: result.vertices.incl(src) result.vertices.incl(dst) result.neighbours.mgetOrPut(src, @[]).add((dst, cost.toFloat))
- ---------------------------------------------------------------------------------------------------
proc dijkstraPath(graph: Graph; first, last: string): seq[string] =
## Find the path from "first" to "last" which minimizes the cost.
var dist = initTable[string, float]() var previous = initTable[string, string]() var notSeen = graph.vertices for vertex in graph.vertices: dist[vertex] = Inf dist[first] = 0
while notSeen.card > 0:
# Search vertex with minimal distance.
var vertex1: string
var mindist = Inf
for vertex in notSeen:
if dist[vertex] < mindist:
vertex1 = vertex
mindist = dist[vertex]
if vertex1 == last:
break
notSeen.excl(vertex1)
# Find shortest paths to neighbours.
for (vertex2, cost) in graph.neighbours.getOrDefault(vertex1):
if vertex2 in notSeen:
let altdist = dist[vertex1] + cost
if altdist < dist[vertex2]:
# Found a shorter path to go to vertex2.
dist[vertex2] = altdist
previous[vertex2] = vertex1 # To go to vertex2, go through vertex1.
# Build the path. var vertex = last while vertex.len > 0: result.add(vertex) vertex = previous.getOrDefault(vertex) result.reverse()
- ---------------------------------------------------------------------------------------------------
proc printPath(path: seq[string]) =
## Print a path.
stdout.write(fmt"Shortest path from '{path[0]}' to '{path[^1]}': {path[0]}")
for i in 1..path.high:
stdout.write(fmt" → {path[i]}")
stdout.write('\n')
- ---------------------------------------------------------------------------------------------------
let graph = initGraph([("a", "b", 7), ("a", "c", 9), ("a", "f", 14),
("b", "c", 10), ("b", "d", 15), ("c", "d", 11),
("c", "f", 2), ("d", "e", 6), ("e", "f", 9)])
printPath(graph.dijkstraPath("a", "e")) printPath(graph.dijkstraPath("a", "f")) </lang>
- Output:
Shortest path from 'a' to 'e': a → c → d → e Shortest path from 'a' to 'f': a → c → f
OCaml
Just a straightforward implementation of the pseudo-code from the Wikipedia article:
<lang ocaml>let list_vertices graph =
List.fold_left (fun acc ((a, b), _) -> let acc = if List.mem b acc then acc else b::acc in let acc = if List.mem a acc then acc else a::acc in acc ) [] graph
let neighbors v =
List.fold_left (fun acc ((a, b), d) -> if a = v then (b, d)::acc else acc ) []
let remove_from v lst =
let rec aux acc = function [] -> failwith "remove_from" | x::xs -> if x = v then List.rev_append acc xs else aux (x::acc) xs in aux [] lst
let with_smallest_distance q dist =
match q with
| [] -> assert false
| x::xs ->
let rec aux distance v = function
| x::xs ->
let d = Hashtbl.find dist x in
if d < distance
then aux d x xs
else aux distance v xs
| [] -> (v, distance)
in
aux (Hashtbl.find dist x) x xs
let dijkstra max_val zero add graph source target =
let vertices = list_vertices graph in
let dist_between u v =
try List.assoc (u, v) graph
with _ -> zero
in
let dist = Hashtbl.create 1 in
let previous = Hashtbl.create 1 in
List.iter (fun v -> (* initializations *)
Hashtbl.add dist v max_val (* unknown distance function from source to v *)
) vertices;
Hashtbl.replace dist source zero; (* distance from source to source *)
let rec loop = function [] -> ()
| q ->
let u, dist_u =
with_smallest_distance q dist in (* vertex in q with smallest distance in dist *)
if dist_u = max_val then
failwith "vertices inaccessible"; (* all remaining vertices are inaccessible from source *)
if u = target then () else begin
let q = remove_from u q in
List.iter (fun (v, d) ->
if List.mem v q then begin
let alt = add dist_u (dist_between u v) in
let dist_v = Hashtbl.find dist v in
if alt < dist_v then begin (* relax (u,v,a) *)
Hashtbl.replace dist v alt;
Hashtbl.replace previous v u; (* previous node in optimal path from source *)
end
end
) (neighbors u graph);
loop q
end
in
loop vertices;
let s = ref [] in
let u = ref target in
while Hashtbl.mem previous !u do
s := !u :: !s;
u := Hashtbl.find previous !u
done;
(source :: !s)
let () =
let graph =
[ ("a", "b"), 7;
("a", "c"), 9;
("a", "f"), 14;
("b", "c"), 10;
("b", "d"), 15;
("c", "d"), 11;
("c", "f"), 2;
("d", "e"), 6;
("e", "f"), 9; ]
in
let p = dijkstra max_int 0 (+) graph "a" "e" in
print_endline (String.concat " -> " p)</lang>
Output:
a -> c -> d -> e
Translation of the C++ solution, and all the complexities are the same as in the C++ solution. In particular, we again use a self-balancing binary search tree (Set) to implement the priority queue, which results in an optimal complexity.
<lang ocaml>type vertex = int
type weight = float
type neighbor = vertex * weight
module VertexSet = Set.Make(struct type t = weight * vertex let compare = compare end)
let dijkstra (src:vertex) (adj_list:neighbor list array) : weight array * vertex array =
let n = Array.length adj_list in
let min_distance = Array.make n infinity in
min_distance.(src) <- 0.;
let previous = Array.make n (-1) in
let rec aux vertex_queue =
if not (VertexSet.is_empty vertex_queue) then
let dist, u = VertexSet.min_elt vertex_queue in
let vertex_queue' = VertexSet.remove (dist, u) vertex_queue in
let edges = adj_list.(u) in
let f vertex_queue (v, weight) =
let dist_thru_u = dist +. weight in
if dist_thru_u >= min_distance.(v) then
vertex_queue
else begin
let vertex_queue' = VertexSet.remove (min_distance.(v), v) vertex_queue in
min_distance.(v) <- dist_thru_u;
previous.(v) <- u;
VertexSet.add (min_distance.(v), v) vertex_queue'
end
in
aux (List.fold_left f vertex_queue' edges)
in
aux (VertexSet.singleton (min_distance.(src), src));
min_distance, previous
let shortest_path_to (target : vertex) (previous : vertex array) : vertex list =
let rec aux target acc =
if target = -1 then
acc
else
aux previous.(target) (target :: acc)
in
aux target []
let adj_list =
[| [(1, 7.); (2, 9.); (5, 14.)]; (* 0 = a *)
[(0, 7.); (2, 10.); (3, 15.)]; (* 1 = b *)
[(0, 9.); (1, 10.); (3, 11.); (5, 2.)]; (* 2 = c *)
[(1, 15.); (2, 11.); (4, 6.)]; (* 3 = d *)
[(3, 6.); (5, 9.)]; (* 4 = e *)
[(0, 14.); (2, 2.); (4, 9.)] (* 5 = f *)
|]
let () =
let min_distance, previous = dijkstra 0 adj_list in Printf.printf "Distance from 0 to 4: %f\n" min_distance.(4); let path = shortest_path_to 4 previous in print_string "Path: "; List.iter (Printf.printf "%d, ") path; print_newline ()</lang>
PARI/GP
Basic, inefficient implementation. Takes an n×n matrix representing distance between nodes (a 0-1 matrix if you just want to count number of steps) and a number in 1..n representing the starting node, which defaults to 1 if not given. <lang parigp>shortestPath(G, startAt=1)={ my(n=#G[,1],dist=vector(n,i,9e99),prev=dist,Q=2^n-1); dist[startAt]=0; while(Q, my(t=vecmin(vecextract(dist,Q)),u); if(t==9e99, break); for(i=1,#v,if(dist[i]==t && bittest(Q,i-1), u=i; break)); Q-=1<<(u-1); for(i=1,n, if(!G[u,i],next); my(alt=dist[u]+G[u,i]); if (alt < dist[i], dist[i]=alt; prev[i]=u; ) ) ); dist };</lang>
Pascal
Classic algorithm like this has to have a Pascal implementation...
<lang pascal>program dijkstra(output);
type
{ We dynamically build the list of vertices from the edge list,
just to avoid repeating ourselves in the graph input. Vertices are linked
together via their `next` pointers to form a list of all vertices (sorted by
name), while the `previous` pointer indicates the previous vertex along the
shortest path to this one. }
vertex = record
name: char;
visited: boolean;
distance: integer;
previous: ^vertex;
next: ^vertex;
end;
vptr = ^vertex;
{ The graph is specified as an array of these }
edge_desc = record
source: char;
dest: char;
weight: integer;
end;
const
{ the input graph }
edges: array of edge_desc = (
(source:'a'; dest:'b'; weight:7),
(source:'a'; dest:'c'; weight:9),
(source:'a'; dest:'f'; weight:14),
(source:'b'; dest:'c'; weight:10),
(source:'b'; dest:'d'; weight:15),
(source:'c'; dest:'d'; weight:11),
(source:'c'; dest:'f'; weight:2),
(source:'d'; dest:'e'; weight:6),
(source:'e'; dest:'f'; weight:9)
);
{ find the shortest path to all nodes starting from this one }
origin: char = 'a';
var
head_vertex: vptr = nil; curr, next, closest: vptr; vtx: vptr; dist: integer; edge: edge_desc; done: boolean = false;
{ allocate a new vertex node with the given name and `next` pointer } function new_vertex(key: char; next: vptr): vptr;
var vtx: vptr; begin new(vtx); vtx^.name := key; vtx^.visited := false; vtx^.distance := maxint; vtx^.previous := nil; vtx^.next := next; new_vertex := vtx; end;
{ look up a vertex by name; create it if needed }
function find_or_make_vertex(key: char): vptr; var
vtx, prev, found: vptr; done: boolean;
begin
found := nil;
if head_vertex = nil then
head_vertex := new_vertex(key, nil)
else if head_vertex^.name > key then
head_vertex := new_vertex(key, head_vertex);
if head_vertex^.name = key then
found := head_vertex
else begin
prev := head_vertex;
vtx := head_vertex^.next;
done := false;
while not done do
if vtx = nil then
done := true
else if vtx^.name >= key then
done := true
else begin
prev := vtx;
vtx := vtx^.next
end;
if vtx <> nil then
if vtx^.name = key then
found := vtx;
if found = nil then begin
prev^.next := new_vertex(key, vtx);
found := prev^.next;
end
end;
find_or_make_vertex := found
end;
{ display the path to a vertex indicated by its `previous` pointer chain } procedure write_path(vtx: vptr);
begin
if vtx <> nil then begin
if vtx^.previous <> nil then begin
write_path(vtx^.previous);
write('→');
end;
write(vtx^.name);
end;
end;
begin
curr := find_or_make_vertex(origin);
curr^.distance := 0;
curr^.previous := nil;
while not done do begin
for edge in edges do begin
if edge.source = curr^.name then begin
next := find_or_make_vertex(edge.dest);
dist := curr^.distance + edge.weight;
if dist < next^.distance then begin
next^.distance := dist;
next^.previous := curr;
end
end
end;
curr^.visited := true;
closest := nil;
vtx := head_vertex;
while vtx <> nil do begin
if not vtx^.visited then
if closest = nil then
closest := vtx
else if vtx^.distance < closest^.distance then
closest := vtx;
vtx := vtx^.next;
end;
if closest = nil then
done := true
else if closest^.distance = maxint then
done := true;
curr := closest;
end;
writeln('Shortest path to each vertex from ', origin, ':');
vtx := head_vertex;
while vtx <> nil do begin
write(vtx^.name, ':', vtx^.distance);
if vtx^.distance > 0 then begin
write(' (');
write_path(vtx);
write(')');
end;
writeln();
vtx := vtx^.next;
end
end.</lang>
- Output:
Shortest path to each vertex from a: a:0 b:7 (a→b) c:9 (a→c) d:20 (a→c→d) e:26 (a→c→d→e) f:11 (a→c→f)
Pascal (alternative)
This alternative Pascal version is based directly on Dijkstra's 1959 paper. It was written independently of the VBA version, q.v. for references and for the algorithm in Dijkstra's own words.
The problem as stated by Dijkstra is to find the shortest path from a given node P to a given node Q. A common variant, which is implemented here and requires only a small change in the algorithm, is to find the shortest path from P to every other node.
Dijkstra divides the nodes into three sets A, B, and C. Nodes in set C have no path length assigned yet; nodes in set B have a provisional path length; nodes in set A have a final path length. As the algorithm proceeds, nodes are transferred from C to B to A. The algorithm stops when Q has been transferred to A (in the original problem) or when all nodes have been transferred to A (in the variant). Nodes in set A are called "visited" in many descriptions of the algorithm.
The original paper also has a division of the branches (edges) into three sets I, II, and III, but the program posted here does not make use of these.
Almost every online description of the algorithm introduces the concept of infinite path length. There is no mention of this in Dijkstra's paper, and it doesn't seem to be necessary. <lang pascal> program Dijkstra_console; // Demo of Dijkstra's algorithm. // Free Pascal (Lazarus), console application.
uses SysUtils; type
TNodeSet = (setA, setB, setC); TNode = record NodeSet : TNodeSet; PrevIndex : integer; // previous node in path leading to this node PathLength : integer; // total length of path to this node end;
const // Rosetta code task
NR_NODES = 6;
START_INDEX = 0;
NODE_NAMES: array [0..NR_NODES - 1] of string = ('a','b','c','d','e','f');
// LENGTHS[j,k] = length of branch j -> k, or -1 if no such branch exists.
LENGTHS : array [0..NR_NODES - 1] of array [0..NR_NODES - 1] of integer
= ((-1, 7, 9,-1,-1,14),
(-1,-1,10,15,-1,-1),
(-1,-1,-1,11,-1, 2),
(-1,-1,-1,-1, 6,-1),
(-1,-1,-1,-1,-1, 9),
(-1,-1,-1,-1,-1,-1));
var
nodes : array [0..NR_NODES - 1] of TNode; j, j_min, k : integer; lastToSetA, nrInSetA: integer; branchLength, trialLength, minLength : integer; lineOut : string;
begin
// Initialize nodes: all in set C for j := 0 to NR_NODES - 1 do begin nodes[j].NodeSet := setC; // No need to initialize PrevIndex and PathLength, as they are // not used until a value has been assigned by the algorithm. end;
// Begin by transferring the start node to set A nodes[START_INDEX].NodeSet := setA; nodes[START_INDEX].PathLength := 0; nrInSetA := 1; lastToSetA := START_INDEX;
// Transfer nodes to set A one at a time, until all have been transferred while (nrInSetA < NR_NODES) do begin
// Step 1: Work through branches leading from the node that was most recently
// transferred to set A, and deal with end nodes in set B or set C.
for j := 0 to NR_NODES - 1 do begin
branchLength := LENGTHS[ lastToSetA, j];
if (branchLength >= 0) then begin
// If the end node is in set B, and the path to the end node via lastToSetA
// is shorter than the existing path, then update the path.
if (nodes[j].NodeSet = setB) then begin
trialLength := nodes[lastToSetA].PathLength + branchLength;
if (trialLength < nodes[j].PathLength) then begin
nodes[j].PrevIndex := lastToSetA;
nodes[j].PathLength := trialLength;
end;
end
// If the end node is in set C, transfer it to set B.
else if (nodes[j].NodeSet = setC) then begin
nodes[j].NodeSet := setB;
nodes[j].PrevIndex := lastToSetA;
nodes[j].PathLength := nodes[lastToSetA].PathLength + branchLength;
end;
end;
end;
// Step 2: Find the node in set B with the smallest path length,
// and transfer that node to set A.
// (Note that set B cannot be empty at this point.)
minLength := -1; // just to stop compiler warning "might not have been initialized"
j_min := -1; // index of node with smallest path length; will become >= 0
for j := 0 to NR_NODES - 1 do begin
if (nodes[j].NodeSet = setB) then begin
if (j_min < 0) or (nodes[j].PathLength < minLength) then begin
j_min := j;
minLength := nodes[j].PathLength;
end;
end;
end;
nodes[j_min].NodeSet := setA;
inc( nrInSetA);
lastToSetA := j_min;
end;
// Write result to console
WriteLn( SysUtils.Format( 'Shortest paths from node %s:', [NODE_NAMES[START_INDEX]]));
for j := 0 to NR_NODES - 1 do begin
if (j <> START_INDEX) then begin
k := j;
lineOut := NODE_NAMES[k];
repeat
k := nodes[k].PrevIndex;
lineOut := NODE_NAMES[k] + ' -> ' + lineOut;
until (k = START_INDEX);
lineOut := SysUtils.Format( '%3s: length %3d, ',
[NODE_NAMES[j], nodes[j].PathLength]) + lineOut;
WriteLn( lineOut);
end;
end;
end. </lang>
- Output:
Shortest paths from node a: b: length 7, a -> b c: length 9, a -> c d: length 20, a -> c -> d e: length 26, a -> c -> d -> e f: length 11, a -> c -> f
Perl
<lang perl>use strict; use warnings; use constant True => 1;
sub add_edge {
my ($g, $a, $b, $weight) = @_;
$g->{$a} ||= {name => $a};
$g->{$b} ||= {name => $b};
push @{$g->{$a}{edges}}, {weight => $weight, vertex => $g->{$b}};
}
sub push_priority {
my ($a, $v) = @_;
my $i = 0;
my $j = $#{$a};
while ($i <= $j) {
my $k = int(($i + $j) / 2);
if ($a->[$k]{dist} >= $v->{dist}) { $j = $k - 1 }
else { $i = $k + 1 }
}
splice @$a, $i, 0, $v;
}
sub dijkstra {
my ($g, $a, $b) = @_;
for my $v (values %$g) {
$v->{dist} = 10e7; # arbitrary large value
delete @$v{'prev', 'visited'}
}
$g->{$a}{dist} = 0;
my $h = [];
push_priority($h, $g->{$a});
while () {
my $v = shift @$h;
last if !$v or $v->{name} eq $b;
$v->{visited} = True;
for my $e (@{$v->{edges}}) {
my $u = $e->{vertex};
if (!$u->{visited} && $v->{dist} + $e->{weight} <= $u->{dist}) {
$u->{prev} = $v;
$u->{dist} = $v->{dist} + $e->{weight};
push_priority($h, $u);
}
}
}
}
my $g = {}; add_edge($g, @$_) for
(['a', 'b', 7], ['a', 'c', 9], ['a', 'f', 14], ['b', 'c', 10], ['b', 'd', 15], ['c', 'd', 11], ['c', 'f', 2], ['d', 'e', 6], ['e', 'f', 9]);
dijkstra($g, 'a', 'e');
my $v = $g->{e}; my @a; while ($v) {
push @a, $v->{name};
$v = $v->{prev};
} my $path = join , reverse @a; print "$g->{e}{dist} $path\n";</lang>
- Output:
26 acde
Phix
I didn't really copy any other code/pseudocode, just followed the basic concept of (update costs) (select lowest cost unvisited) until target reached.
Selects the shortest path from A to B only. As for time complexity, it looks plenty efficient enough to me, though it clearly is O(V^2).
Written after the task was changed to be a directed graph, and shows the correct solution for that.
with javascript_semantics --requires("1.0.2") -- (builtin E renamed as EULER) --enum A,B,C,D,E,F constant A=1, B=2, C=3, D=4, E=5, F=6 -- (or use this) constant edges = {{A,B,7}, {A,C,9}, {A,F,14}, {B,C,10}, {B,D,15}, {C,D,11}, {C,F,2}, {D,E,6}, {E,F,9}} sequence visited, cost, from procedure reset() visited = repeat(0,6) cost = repeat(0,6) from = repeat(0,6) end procedure function backtrack(integer finish,start) sequence res = {finish} while finish!=start do finish = from[finish] res = prepend(res,finish) end while return res end function function shortest_path(integer start, integer finish) integer estart, eend, ecost, ncost, mincost while 1 do visited[start] = 1 for i=1 to length(edges) do {estart,eend,ecost} = edges[i] if estart=start then ncost = cost[start]+ecost if visited[eend]=0 then if from[eend]=0 or cost[eend]>ncost then cost[eend] = ncost from[eend] = start end if elsif cost[eend]>ncost then ?9/0 -- sanity check end if end if end for mincost = 0 for i=1 to length(visited) do if visited[i]=0 and from[i]!=0 then if mincost=0 or cost[i]<mincost then start = i mincost = cost[start] end if end if end for if visited[start] then return -1 end if if start=finish then return cost[finish] end if end while end function function AFi(integer i) -- output helper return 'A'+i-1 end function procedure test(sequence testset) integer start, finish, ecost, len string epath, path for i=1 to length(testset) do {start,finish,ecost,epath} = testset[i] reset() len = shortest_path(start,finish) path = iff(len=-1?"no path found":join(apply(backtrack(finish,start),AFi),"")) printf(1,"%c->%c: length %d:%s (expected %d:%s)\n",{AFi(start),AFi(finish),len,path,ecost,epath}) end for end procedure test({{A,E,26,"ACDE"},{A,F,11,"ACF"},{F,A,-1,"none"}})
- Output:
A->E: length 26:ACDE (expected 26:ACDE) A->F: length 11:ACF (expected 11:ACF) F->A: length -1:no path found (expected -1:none)
PHP
There are parts of this algorithm that could be optimized which have been marked TODO. <lang PHP> <?php function dijkstra($graph_array, $source, $target) {
$vertices = array();
$neighbours = array();
foreach ($graph_array as $edge) {
array_push($vertices, $edge[0], $edge[1]);
$neighbours[$edge[0]][] = array("end" => $edge[1], "cost" => $edge[2]);
$neighbours[$edge[1]][] = array("end" => $edge[0], "cost" => $edge[2]);
}
$vertices = array_unique($vertices);
foreach ($vertices as $vertex) {
$dist[$vertex] = INF;
$previous[$vertex] = NULL;
}
$dist[$source] = 0;
$Q = $vertices;
while (count($Q) > 0) {
// TODO - Find faster way to get minimum
$min = INF;
foreach ($Q as $vertex){
if ($dist[$vertex] < $min) {
$min = $dist[$vertex];
$u = $vertex;
}
}
$Q = array_diff($Q, array($u));
if ($dist[$u] == INF or $u == $target) {
break;
}
if (isset($neighbours[$u])) {
foreach ($neighbours[$u] as $arr) {
$alt = $dist[$u] + $arr["cost"];
if ($alt < $dist[$arr["end"]]) {
$dist[$arr["end"]] = $alt;
$previous[$arr["end"]] = $u;
}
}
}
}
$path = array();
$u = $target;
while (isset($previous[$u])) {
array_unshift($path, $u);
$u = $previous[$u];
}
array_unshift($path, $u);
return $path;
}
$graph_array = array(
array("a", "b", 7),
array("a", "c", 9),
array("a", "f", 14),
array("b", "c", 10),
array("b", "d", 15),
array("c", "d", 11),
array("c", "f", 2),
array("d", "e", 6),
array("e", "f", 9)
);
$path = dijkstra($graph_array, "a", "e");
echo "path is: ".implode(", ", $path)."\n"; </lang> Output is:
path is: a, c, f, e
PicoLisp
Following the Wikipedia algorithm: <lang PicoLisp>(de neighbor (X Y Cost)
(push (prop X 'neighbors) (cons Y Cost)) (push (prop Y 'neighbors) (cons X Cost)) )
(de dijkstra (Curr Dest)
(let Cost 0
(until (== Curr Dest)
(let (Min T Next)
(for N (; Curr neighbors)
(with (car N)
(let D (+ Cost (cdr N))
(unless (and (: distance) (>= D @))
(=: distance D) ) )
(when (> Min (: distance))
(setq Min (: distance) Next This) )
(del (asoq Curr (: neighbors)) (:: neighbors)) ) )
(setq Curr Next Cost Min) ) )
Cost ) )</lang>
Test: <lang PicoLisp>(neighbor 'a 'b 7) (neighbor 'a 'c 9) (neighbor 'a 'f 14) (neighbor 'b 'c 10) (neighbor 'b 'd 15) (neighbor 'c 'd 11) (neighbor 'c 'f 2) (neighbor 'd 'e 6) (neighbor 'e 'f 9)
(dijkstra 'a 'e)</lang> Output:
-> 20
Prolog
An implementation of Dijkstra's algorithm in Prolog
Dijkstra's algorithm starts with a set of all unvisited nodes, assigning an initial distance value for each as infinite. It then attempts to minimise the distance for each node from the origin.
Starting at the origin (distance 0), the algorithm checks each neighbor's distance value and if larger than the current path distance, replaces the neighboring node's distance value. It then marks the current node as visited, and repeats the process for each of the neighbors. When the current node becomes the destination, the distance to the origin is known.
This implementation is a slight variation on Dijkstra, which lends itself to Prolog's strengths while retaining approximate algorithmic equivalence.
Prolog is not good at modifying memory in place, but is quite good at handling facts, pattern matching, recursion and backtracking to find all possible solutions.
A dynamic database predicate, namely:
rpath([target|reversed_path], distance)
stores the currently known shortest distance and best path to a destination from the origin. Since the path is a reversed list, the first item in the list is the destination node, and the predicate is efficiently matched.
Instead of using unvisited flags on nodes, we test whether neighbors are already in the traversed path. This achieves the same thing as 'visited' flags, but in a way that is more efficient for Prolog.
After the graph traversal is complete, we are left with a single rpath/2 predicate for each reachable node, containing the shortest path and distance from the origin.
Subtle differences
1) Dijkstra visits each node only once, starting with the origin. This algorithm:
- arbitrarily selects a node (Qi) neighboring origin (o), and for that node
- if o->Qi is the shortest known path:
- update path and distance
- traverse Qi
- if o->Qi is not the shortest, select the next node.
It is possible therefore, contrary to Dijkstra, that we may visit a node more than once whilst discovering a shorter path. It is also possible that the first path we choose is already the shortest eliminating processing.
2) As traversal spreads outwards, the path is built as a list of traversed nodes.
- We use this list to ensure that we do not loop endlessly.
- This path is recorded as the shortest if the distance is indeed shorter than a known path.
- Leaf nodes in the traversal tree are processed completely before the origin node processing
is completed.
- This implies that the first stage in our algorithm involves allocating each node
in the traversal tree a path and 'shortest known distance from origin' value.
- ...Which is arguably better than assigning an initial 'infinite distance' value.
We could possibly improve our algorithm by processing the neighbor with the shortest distance first, rather than an arbitrary selection as is currently the case. There is nothing though, to suggest that the eventual shortest path found would necessarily follow the shortest initial path, unless the target node is already the closest neighbor.
<lang prolog>%___________________________________________________________________________
- -dynamic
rpath/2. % A reversed path
edge(a,b,7). edge(a,c,9). edge(b,c,10). edge(b,d,15). edge(c,d,11). edge(d,e,6). edge(a,f,14). edge(c,f,2). edge(e,f,9).
path(From,To,Dist) :- edge(To,From,Dist). path(From,To,Dist) :- edge(From,To,Dist).
shorterPath([H|Path], Dist) :- % path < stored path? replace it rpath([H|T], D), !, Dist < D, % match target node [H|_] retract(rpath([H|_],_)), writef('%w is closer than %w\n', Path], [H|T), assert(rpath([H|Path], Dist)). shorterPath(Path, Dist) :- % Otherwise store a new path writef('New path:%w\n', [Path]), assert(rpath(Path,Dist)).
traverse(From, Path, Dist) :- % traverse all reachable nodes path(From, T, D), % For each neighbor not(memberchk(T, Path)), % which is unvisited shorterPath([T,From|Path], Dist+D), % Update shortest path and distance traverse(T,[From|Path],Dist+D). % Then traverse the neighbor
traverse(From) :- retractall(rpath(_,_)), % Remove solutions traverse(From,[],0). % Traverse from origin traverse(_).
go(From, To) :- traverse(From), % Find all distances rpath([To|RPath], Dist)-> % If the target was reached reverse([To|RPath], Path), % Report the path and distance Distance is round(Dist), writef('Shortest path is %w with distance %w = %w\n', [Path, Dist, Distance]); writef('There is no route from %w to %w\n', [From, To]). </lang> for example:
?- go(a,e). New path:[b,a] New path:[c,b,a] New path:[d,c,b,a] New path:[e,d,c,b,a] New path:[f,e,d,c,b,a] [f,c,b,a] is closer than [f,e,d,c,b,a] [e,f,c,b,a] is closer than [e,d,c,b,a] [d,b,a] is closer than [d,c,b,a] [c,a] is closer than [c,b,a] [d,c,a] is closer than [d,b,a] [e,d,c,a] is closer than [e,f,c,b,a] [f,c,a] is closer than [f,c,b,a] [e,f,c,a] is closer than [e,d,c,a] Shortest path is [a,c,f,e] with distance 0+9+2+9 = 20 true.
Python
Starts from the wp:Dijkstra's_algorithm#Pseudocode recognising that their function dist_between is what this task calls cost; and that their action decrease-key v in Q at their line 24 should be omitted if their Q is a set as stated in their line 9. The wp back-tracking pseudocode also misses a final insert of u at the beginning of S that must occur after exiting their while loop.
Note: q could be changed to be a priority queue instead of a set as mentioned here. <lang python>from collections import namedtuple, deque from pprint import pprint as pp
inf = float('inf')
Edge = namedtuple('Edge', ['start', 'end', 'cost'])
class Graph():
def __init__(self, edges):
self.edges = [Edge(*edge) for edge in edges]
# print(dir(self.edges[0]))
self.vertices = {e.start for e in self.edges} | {e.end for e in self.edges}
def dijkstra(self, source, dest):
assert source in self.vertices
dist = {vertex: inf for vertex in self.vertices}
previous = {vertex: None for vertex in self.vertices}
dist[source] = 0
q = self.vertices.copy()
neighbours = {vertex: set() for vertex in self.vertices}
for start, end, cost in self.edges:
neighbours[start].add((end, cost))
neighbours[end].add((start, cost))
#pp(neighbours)
while q:
# pp(q)
u = min(q, key=lambda vertex: dist[vertex])
q.remove(u)
if dist[u] == inf or u == dest:
break
for v, cost in neighbours[u]:
alt = dist[u] + cost
if alt < dist[v]: # Relax (u,v,a)
dist[v] = alt
previous[v] = u
#pp(previous)
s, u = deque(), dest
while previous[u]:
s.appendleft(u)
u = previous[u]
s.appendleft(u)
return s
graph = Graph([("a", "b", 7), ("a", "c", 9), ("a", "f", 14), ("b", "c", 10),
("b", "d", 15), ("c", "d", 11), ("c", "f", 2), ("d", "e", 6),
("e", "f", 9)])
pp(graph.dijkstra("a", "e"))</lang>
- Output:
deque(['a', 'c', 'f', 'e'])
Racket
<lang racket>
- lang racket
(require (planet jaymccarthy/dijkstra:1:2))
(define edges
'([a . ((b 7)(c 9)(f 14))] [b . ((c 10)(d 15))] [c . ((d 11)(f 2))] [d . ((e 6))] [e . ((f 9))]))
(define (node-edges n)
(cond [(assoc n edges) => rest] ['()]))
(define edge-weight second) (define edge-end first)
(match/values (shortest-path node-edges edge-weight edge-end 'a (λ(n) (eq? n 'e)))
[(dists prevs)
(displayln (~a "Distances from a: " (for/list ([(n d) dists]) (list n d))))
(displayln (~a "Shortest path: "
(let loop ([path '(e)])
(cond [(eq? (first path) 'a) path]
[(loop (cons (hash-ref prevs (first path)) path))]))))])
</lang> Output: <lang racket> Distances from a: ((b 7) (d 20) (a 0) (c 9) (f 11) (e 26)) Shortest path: (a c d e) </lang>
Raku
(formerly Perl 6)
<lang perl6>class Graph {
has (%.edges, %.nodes);
method new(*@args){
my (%edges, %nodes);
for @args {
%edges{.[0] ~ .[1]} = $_;
%nodes{.[0]}.push( .[0] ~ .[1] );
%nodes{.[1]}.push( .[0] ~ .[1] );
}
self.bless(edges => %edges, nodes => %nodes);
}
method neighbours ($source) {
my (%neighbours, $edges);
$edges = self.nodes{$source};
for @$edges -> $x {
for self.edges{$x}[0..1] -> $y {
if $y ne $source {
%neighbours{$y} = self.edges{$x}
}
}
}
return %neighbours
}
method dijkstra ($source, $dest) {
my (%node_data, $v, $u);
my @q = self.nodes.keys;
for self.nodes.keys {
%node_data{$_}{'dist'} = Inf;
%node_data{$_}{'prev'} = ;
}
%node_data{$source}{'dist'} = 0;
while @q {
# %node_data.perl.say;
my ($mindist, $idx) =
@((map {[%node_data{@q[$_]}{'dist'},$_]},^@q).min(*[0]));
$u = @q[$idx];
if $mindist eq Inf {
return ()
}
elsif $u eq $dest {
my @s;
while %node_data{$u}{'prev'} {
@s.unshift($u);
$u = %node_data{$u}{'prev'}
}
@s.unshift($source);
return @s;
}
else {
@q.splice($idx,1);
}
for self.neighbours($u).kv -> $v, $edge {
my $alt = %node_data{$u}{'dist'} + $edge[2];
if $alt < %node_data{$v}{'dist'} {
%node_data{$v}{'dist'} = $alt;
%node_data{$v}{'prev'} = $u
}
}
}
}
}
my $a = Graph.new([
["a", "b", 7], ["a", "c", 9], ["a", "f", 14], ["b", "c", 10], ["b", "d", 15], ["c", "d", 11], ["c", "f", 2], ["d", "e", 6], ["e", "f", 9]
]).dijkstra('a', 'e').say;</lang>
- Output:
[a c f e]
REXX
Some program features are:
- elimination of null edges
- elimination of duplicates (the cheapest path is chosen)
- a test for a no path found condition
- use of memoization
<lang rexx>/*REXX program determines the least costly path between two vertices given a list. */ $.= copies(9, digits() ) /*edge cost: indicates doesn't exist. */ xList= '!. @. $. beg fin bestP best$ xx yy' /*common EXPOSEd variables for subs. */ @abc= 'abcdefghijklmnopqrstuvwxyz' /*list of all the possible vertices. */ verts= 0; edges= 0 /*the number of vertices and also edges*/
do #=1 for length(@abc); _= substr(@abc, #, 1)
call value translate(_), #; @@.#= _
end /*#*/
call def$ a b 7 /*define an edge and its cost. */ call def$ a c 9 /* " " " " " " */ call def$ a f 14 /* " " " " " " */ call def$ b c 10 /* " " " " " " */ call def$ b d 15 /* " " " " " " */ call def$ c d 11 /* " " " " " " */ call def$ c f 2 /* " " " " " " */ call def$ d e 6 /* " " " " " " */ call def$ e f 9 /* " " " " " " */ beg= a; fin= e /*the BEGin and FINish vertexes. */ say; say 'number of edges = ' edges
say 'number of vertices = ' verts " ["left(@abc, verts)"]"
best$= $.; bestP= say; do jv=2 to verts; call paths verts, jv; end /*jv*/ @costIs= right('cost =', 16) if bestP==$. then say 'no path found.'
else say 'best path =' translate(bestP, @abc, 123456789) @costIs best$
exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ apath: parse arg pathx 1 p1 2 p2 3; Lp= length(pathx); $= $.p1.p2
if $>=best$ then return
pv= p2; do ka=3 to Lp; _= substr(pathx, ka, 1)
if $.pv._>=best$ then return
$= $ + $.pv._; if $>=best$ then return; pv= _
end /*ka*/
best$= $; bestP= pathx
return
/*──────────────────────────────────────────────────────────────────────────────────────*/ def$: parse arg xx yy $ .; if $.xx.yy<$ & $.yy.xx<$ | xx==yy then return
edges= edges + 1; verts= verts + ($.xx\==0) + ($.yy\==0)
$.xx= 0; $.yy= 0; $.xx.yy= $
say left(, 40) "cost of " @@.xx '───►' @@.yy " is " $
return
/*──────────────────────────────────────────────────────────────────────────────────────*/ paths: procedure expose (xList); parse arg xx, yy, @.
do kp=1 for xx; _= kp; !.kp= _; end /*build a path list.*/
call .path 1
return
/*──────────────────────────────────────────────────────────────────────────────────────*/ .path: procedure expose (xList); parse arg ?, _
if ?>yy then do; if @.1\==beg | @.yy\==fin then return
do #=1 for yy; _= _ || @.#; end /*#*/; call apath _
end
else do qq=1 for xx /*build vertex paths recursively*/
do kp=1 for ?-1; if @.kp==!.qq then iterate qq; end /*kp*/
@.?= !.qq; call .path ?+1 /*recursive call for next path. */
end /*qq*/
return</lang>
- output when using the internal default inputs:
cost of a ───► b is 7
cost of a ───► c is 9
cost of a ───► f is 14
cost of b ───► c is 10
cost of b ───► d is 15
cost of c ───► d is 11
cost of c ───► f is 2
cost of d ───► e is 6
cost of e ───► f is 9
number of edges = 9
number of vertices = 6 [abcdef]
best path = acde cost = 26
Ring
<lang ring>
- Project : Dijkstra's algorithm
graph = [["a", "b", 7],
["a", "c", 9],
["a", "f", 14],
["b", "c", 10],
["b", "d", 15],
["c", "d", 11],
["c", "f", 2],
["d", "e", 6],
["e", "f", 9]]
dbegin = "a" dend = "e" powlen = pow(2,len(graph)) - 1 dgraph = list(powlen) dtemp = list(powlen) lenold = 10 lennew = 0 sumold = 30 sumnew = 0
powerset(graph)
for n = 1 to len(dgraph)
dtemp[n] = str2list(substr(dgraph[n], " ", nl))
next
for n = 1 to len(dtemp)
if len(dtemp[n]) > 3 and dtemp[n][1] = dbegin and dtemp[n][len(dtemp[n])-1] = dend
flag = 1
for m = 1 to len(dtemp[n])/3-1
if dtemp[n][m*3-1] != dtemp[n][m*3+1]
flag = 0
ok
next
if flag = 1
lennew = len(dtemp[n])
if lennew <= lenold
lenold = lennew
sumnew = 0
for m = 1 to len(dtemp[n])/3
sumnew = sumnew + dtemp[n][m*3]
next
if sumnew < sumold
sumold = sumnew
gend = dtemp[n]
ok
ok
ok
ok
next str = "" see dbegin + " " + dend + " : " for m = 1 to len(gend)/3
str = str + gend[(m-1)*3 + 1] + " " + gend[(m-1)*3 + 2] + " " + gend[(m-1)*3 + 3] + " -> "
next str = left(str,len(str)-4) str = str + " cost : " + sumold + nl see str + nl
func powerset(list)
s = "{"
p = 0
for i = 2 to (2 << len(list)) - 1 step 2
s = ""
for j = 1 to len(list)
if i & (1 << j)
s = s + list[j][1] + " " + list[j][2] + " " + list[j][3] + " "
ok
next
if right(s,1) = " "
s = left(s,len(s)-1)
ok
p = p + 1
dgraph[p] = s
next
</lang> Output:
a e : a c 9 -> c d 11 -> d e 6 cost : 26
Ruby
This solution is incorrect. Since the path is directed and f is only a sink, f cannot be in the middle of a path.
(for INFINITY)
Notes for this solution:
- At every iteration, the next minimum distance node found by linear traversal of all nodes, which is inefficient.
<lang ruby>class Graph
Vertex = Struct.new(:name, :neighbours, :dist, :prev)
def initialize(graph)
@vertices = Hash.new{|h,k| h[k]=Vertex.new(k,[],Float::INFINITY)}
@edges = {}
graph.each do |(v1, v2, dist)|
@vertices[v1].neighbours << v2
@vertices[v2].neighbours << v1
@edgesv1, v2 = @edgesv2, v1 = dist
end
@dijkstra_source = nil
end
def dijkstra(source)
return if @dijkstra_source == source
q = @vertices.values
q.each do |v|
v.dist = Float::INFINITY
v.prev = nil
end
@vertices[source].dist = 0
until q.empty?
u = q.min_by {|vertex| vertex.dist}
break if u.dist == Float::INFINITY
q.delete(u)
u.neighbours.each do |v|
vv = @vertices[v]
if q.include?(vv)
alt = u.dist + @edgesu.name, v
if alt < vv.dist
vv.dist = alt
vv.prev = u.name
end
end
end
end
@dijkstra_source = source
end
def shortest_path(source, target)
dijkstra(source)
path = []
u = target
while u
path.unshift(u)
u = @vertices[u].prev
end
return path, @vertices[target].dist
end
def to_s
"#<%s vertices=%p edges=%p>" % [self.class.name, @vertices.values, @edges]
end
end
g = Graph.new([ [:a, :b, 7],
[:a, :c, 9],
[:a, :f, 14],
[:b, :c, 10],
[:b, :d, 15],
[:c, :d, 11],
[:c, :f, 2],
[:d, :e, 6],
[:e, :f, 9],
])
start, stop = :a, :e path, dist = g.shortest_path(start, stop) puts "shortest path from #{start} to #{stop} has cost #{dist}:" puts path.join(" -> ")</lang>
- Output:
shortest path from a to e has cost 20: a -> c -> f -> e
Rust
This solution uses a very bare-bones, naive implementation of an adjacency list to represent the graph.
<lang rust>use std::cmp::Ordering; use std::collections::BinaryHeap; use std::usize;
struct Grid<T> {
nodes: Vec<Node<T>>,
}
struct Node<T> {
data: T, edges: Vec<(usize,usize)>,
}
- [derive(Copy, Clone, Eq, PartialEq)]
struct State {
node: usize, cost: usize,
}
// Manually implement Ord so we get a min-heap instead of a max-heap impl Ord for State {
fn cmp(&self, other: &Self) -> Ordering {
other.cost.cmp(&self.cost)
}
}
impl PartialOrd for State {
fn partial_cmp(&self, other: &Self) -> Option<Ordering> {
Some(self.cmp(other))
}
}
type WeightedEdge = (usize, usize, usize);
impl<T> Grid<T> {
fn new() -> Self {
Grid { nodes: Vec::new() }
}
fn add_node(&mut self, data: T) -> usize {
let node = Node {
edges: Vec::new(),
data: data,
};
self.nodes.push(node);
self.nodes.len() - 1
}
fn create_edges<'a, I>(&mut self, iterator: I) where I: IntoIterator<Item=&'a WeightedEdge> {
for &(start,end,weight) in iterator.into_iter() {
self.nodes[start].edges.push((end,weight));
self.nodes[end].edges.push((start,weight));
}
}
fn find_path(&self, start: usize, end: usize) -> Option<(Vec<usize>, usize)> {
let mut dist = vec![(usize::MAX, None); self.nodes.len()];
let mut heap = BinaryHeap::new();
dist[start] = (0, None);
heap.push(State {
node: start,
cost: 0,
});
while let Some(State { node, cost }) = heap.pop() {
if node == end {
let mut path = Vec::with_capacity(dist.len() / 2);
let mut current_dist = dist[end];
path.push(end);
while let Some(prev) = current_dist.1 {
path.push(prev);
current_dist = dist[prev];
}
path.reverse();
return Some((path, cost));
}
if cost > dist[node].0 {
continue;
}
for edge in &self.nodes[node].edges {
let next = State {
node: edge.0,
cost: cost + edge.1,
};
if next.cost < dist[next.node].0 {
dist[next.node] = (next.cost, Some(node));
heap.push(next);
}
}
}
None
}
}
fn main() {
let mut grid = Grid::new();
let (a,b,c,d,e,f) = (grid.add_node("a"), grid.add_node("b"),
grid.add_node("c"), grid.add_node("d"),
grid.add_node("e"), grid.add_node("f"));
grid.create_edges(&[
(a,b,7) ,(a,c,9) ,(a,f,14),
(b,c,10),(b,d,15),(c,d,11),
(c,f,2) ,(d,e,6) ,(e,f,9) ,
]);
let (path, cost) = grid.find_path(a,e).unwrap();
print!("{}", grid.nodes[path[0]].data);
for i in path.iter().skip(1) {
print!(" -> {}", grid.nodes[*i].data);
}
println!("\nCost: {}", cost);
}</lang>
- Output:
Cost: 20 a -> c -> f -> e
SAS
Use network solver in SAS/OR: <lang sas>/* create SAS data set */ data Edges;
input Start $ End $ Cost; datalines;
a b 7 a c 9 a f 14 b c 10 b d 15 c d 11 c f 2 d e 6 e f 9
/* call OPTMODEL procedure in SAS/OR */ proc optmodel;
/* declare sets and parameters, and read input data */
set <str,str> LINKS;
num cost {LINKS};
read data Edges into LINKS=[start end] cost;
set NODES = union {<i,j> in LINKS} {i,j};
set SOURCES = {'a'};
set SINKS = {'e'};
/* <source,sink,order,from,to> */
set <str,str,num,str,str> PATHS;
/* call network solver */
solve with network /
shortpath=(source=SOURCES sink=SINKS) links=(weight=cost) out=(sppaths=PATHS);
/* write shortest path to SAS data set */ create data path from [source sink order from to]=PATHS cost[from,to];
quit;
/* print shortest path */ proc print data=path; run;</lang>
Output:
Obs source sink order from to cost 1 a e 1 a c 9 2 a e 2 c f 2 3 a e 3 e f 9
Scala
A functional implementation of Dijkstras Algorithm:
<lang scala>object Dijkstra {
type Path[Key] = (Double, List[Key])
def Dijkstra[Key](lookup: Map[Key, List[(Double, Key)]], fringe: List[Path[Key]], dest: Key, visited: Set[Key]): Path[Key] = fringe match {
case (dist, path) :: fringe_rest => path match {case key :: path_rest =>
if (key == dest) (dist, path.reverse)
else {
val paths = lookup(key).flatMap {case (d, key) => if (!visited.contains(key)) List((dist + d, key :: path)) else Nil}
val sorted_fringe = (paths ++ fringe_rest).sortWith {case ((d1, _), (d2, _)) => d1 < d2}
Dijkstra(lookup, sorted_fringe, dest, visited + key)
}
}
case Nil => (0, List())
}
def main(x: Array[String]): Unit = {
val lookup = Map(
"a" -> List((7.0, "b"), (9.0, "c"), (14.0, "f")),
"b" -> List((10.0, "c"), (15.0, "d")),
"c" -> List((11.0, "d"), (2.0, "f")),
"d" -> List((6.0, "e")),
"e" -> List((9.0, "f")),
"f" -> Nil
)
val res = Dijkstra[String](lookup, List((0, List("a"))), "e", Set())
println(res)
}
}</lang>
- Output:
(26.0,List(a, c, d, e))
An implementation based on the functional version above that uses PriorityQueue. It is made functional-look:
<lang scala>
import scala.collection.mutable
class Dijkstra[Key] {
type PathInfo = (Double, List[Key]) type Path = List[Key] type MinHeap[PathInfo] = mutable.PriorityQueue[PathInfo]
final def dijkstra(weightedGraph: Map[Key, List[(Double, Key)]],
start: Key,
dest: Key)(implicit ord: Ordering[PathInfo]): PathInfo =
dijkstraHelper(weightedGraph, mutable.PriorityQueue((0.0, List(start))), dest)
@annotation.tailrec
private final def dijkstraHelper(weightedGraph: Map[Key, List[(Double, Key)]],
fringe: MinHeap[PathInfo],
dest: Key,
visited: Set[Key] = Set.empty[Key])(implicit ord: Ordering[PathInfo]): PathInfo = {
def updateFringe(frng: MinHeap[PathInfo], currentDist: Double, currentPath: Path): MinHeap[PathInfo] =
(currentPath : @unchecked) match {
case keys @ key :: _ =>
weightedGraph(key)
.withFilter { case (_, k) => !visited.contains(k) }
.map { case (d, k) => (currentDist + d, k :: keys) } // updated PathInfo's
.foreach { p => frng.enqueue(p) }
frng
}
if (fringe.isEmpty)
(0, Nil)
else {
(fringe.dequeue() : @unchecked) match {
case (dist, path @ `dest` :: _) =>
(dist, path.reverse)
case (dist, path @ key :: _) =>
dijkstraHelper(weightedGraph, updateFringe(fringe, dist, path), dest, visited + key)
}
}
}
def main(x: Array[String]): Unit = {
val weightedGraph = Map(
"a" -> List((7.0, "b"), (9.0, "c"), (14.0, "f")),
"b" -> List((10.0, "c"), (15.0, "d")),
"c" -> List((11.0, "d"), (2.0, "f")),
"d" -> List((6.0, "e")),
"e" -> List((9.0, "f")),
"f" -> Nil
)
val res = dijkstra[String](weightedGraph, "a", "e")
println(res)
}
} </lang>
- Output:
(26.0,List(a, c, d, e))
Sidef
<lang ruby>class Graph(*args) {
struct Node {
String name,
Array edges = [],
Number dist = Inf,
prev = nil,
Bool visited = false,
}
struct Edge {
Number weight,
Node vertex,
}
has g = Hash()
method init {
args.each { |a|
self.add_edge(a...)
}
}
method get(name) {
g{name}
}
method add_edge(a, b, weight) {
g{a} ||= Node(name: a)
g{b} ||= Node(name: b)
g{a}.edges << Edge(weight, g{b})
}
method push_priority(a, v) {
var i = 0
var j = a.end
while (i <= j) {
var k = ((i + j) // 2)
if (a[k].dist >= v.dist) {
j = k-1
}
else {
i = k+1
}
}
a.insert(i, v)
}
method dijkstra(a, b) {
g{a}.dist = 0
var h = []
self.push_priority(h, g{a})
while (!h.is_empty) {
var v = h.shift
break if (v.name == b)
v.visited = true
v.edges.each { |e|
var u = e.vertex
if (!u.visited && (v.dist+e.weight <= u.dist)) {
u.prev = v
u.dist = (v.dist + e.weight)
self.push_priority(h, u)
}
}
}
}
}
var g = Graph(
["a", "b", 7], ["a", "c", 9], ["a", "f", 14], ["b", "c", 10], ["b", "d", 15], ["c", "d", 11], ["c", "f", 2], ["d", "e", 6], ["e", "f", 9],
)
g.dijkstra('a', 'e')
var v = g.get('e') var a = [] while (v != nil) {
a << v.name v = v.prev
}
var path = a.reverse.join say "#{g.get('e').dist} #{path}"</lang>
- Output:
26 acde
Swift
Uses the Heap type from the Swift Algorithm Club.
<lang swift>typealias WeightedEdge = (Int, Int, Int)
struct Grid<T> {
var nodes: [Node<T>]
mutating func addNode(data: T) -> Int {
nodes.append(Node(data: data, edges: []))
return nodes.count - 1 }
mutating func createEdges(weights: [WeightedEdge]) {
for (start, end, weight) in weights {
nodes[start].edges.append((end, weight))
nodes[end].edges.append((start, weight))
}
}
func findPath(start: Int, end: Int) -> ([Int], Int)? {
var dist = Array(repeating: (Int.max, nil as Int?), count: nodes.count)
var heap = Heap<State>(sort: { $0.cost < $1.cost })
dist[start] = (0, nil) heap.insert(State(node: start, cost: 0))
while let state = heap.remove(at: 0) {
if state.node == end {
var path = [end]
var currentDist = dist[end]
while let prev = currentDist.1 {
path.append(prev)
currentDist = dist[prev]
}
return (path.reversed(), state.cost)
}
guard state.cost <= dist[state.node].0 else {
continue
}
for edge in nodes[state.node].edges {
let next = State(node: edge.0, cost: state.cost + edge.1)
if next.cost < dist[next.node].0 {
dist[next.node] = (next.cost, state.node)
heap.insert(next)
}
}
}
return nil }
}
struct Node<T> {
var data: T var edges: [(Int, Int)]
}
struct State {
var node: Int var cost: Int
}
var grid = Grid<String>(nodes: [])
let (a, b, c, d, e, f) = (
grid.addNode(data: "a"), grid.addNode(data: "b"), grid.addNode(data: "c"), grid.addNode(data: "d"), grid.addNode(data: "e"), grid.addNode(data: "f")
)
grid.createEdges(weights: [
(a, b, 7), (a, c, 9), (a, f, 14), (b, c, 10), (b, d, 15), (c, d, 11), (c, f, 2), (d, e, 6), (e, f, 9)
])
guard let (path, cost) = grid.findPath(start: a, end: e) else {
fatalError("Could not find path")
}
print("Cost: \(cost)") print(path.map({ grid.nodes[$0].data }).joined(separator: " -> "))</lang>
- Output:
Cost: 20 a -> c -> f -> e
Tailspin
A simple algorithm that traverses through all edges at every step. <lang tailspin> data vertex <'a'..'f'>, to <vertex>
templates shortestPaths&{graph:}
@: [];
[ {to: $, distance: 0"1", path:[]} ] -> #
when <[](0)> do $@ !
otherwise
def closest: $ ... -> ..=Min&{by: :(distance:), select: :()};
$closest -> ..|@: $;
def path: [ $closest.path..., $closest.to ];
[ $... -> \(<?($.to <~=$closest.to>)> $!\),
$graph... -> \(<?($.edge(1) <=$closest.to>) ?($@shortestPaths <~[<{to: <=$.edge(2)>}>]>)> $!\)
-> { to: $.edge(2), distance: $.cost + $closest.distance, path: $path} ] -> #
end shortestPaths
def edges: [
{ edge: [vertex´'a', vertex´'b'], cost: 7"1" },
{ edge: [vertex´'a', vertex´'c'], cost: 9"1" },
{ edge: [vertex´'a', vertex´'f'], cost: 14"1" },
{ edge: [vertex´'b', vertex´'c'], cost: 10"1" },
{ edge: [vertex´'b', vertex´'d'], cost: 15"1" },
{ edge: [vertex´'c', vertex´'d'], cost: 11"1" },
{ edge: [vertex´'c', vertex´'f'], cost: 2"1" },
{ edge: [vertex´'d', vertex´'e'], cost: 6"1" },
{ edge: [vertex´'e', vertex´'f'], cost: 9"1" }];
def fromA: vertex´'a' -> shortestPaths&{graph: $edges};
$fromA... -> \(<{to:<=vertex´'e'>}> $!\) -> 'Shortest path from $.path(1); to $.to; is distance $.distance; via $.path(2..last); ' -> !OUT::write
$fromA... -> \(<{to:<=vertex´'f'>}> $!\) -> 'Shortest path from $.path(1); to $.to; is distance $.distance; via $.path(2..last); ' -> !OUT::write </lang>
Alternatively you can use relational algebra operators for the exact same algorithm <lang tailspin> data cost <"1">, distance <"1"> data vertex <'[a-f]'>, to <vertex>, from <vertex>, path <[<vertex>* VOID]>
templates shortestPaths&{graph:}
@: {||};
when <?($::count <=0>)> do $@ !
otherwise
def closest: $ ... -> ..=Min&{by: :(distance:), select: :()};
@: ($@ union {|$closest|});
def path: [ $closest.path..., $closest.to ];
-> #
end shortestPaths
def edges: {|
{ from: 'a', to: 'b', cost: 7"1" },
{ from: 'a', to: 'c', cost: 9"1" },
{ from: 'a', to: 'f', cost: 14"1" },
{ from: 'b', to: 'c', cost: 10"1" },
{ from: 'b', to: 'd', cost: 15"1" },
{ from: 'c', to: 'd', cost: 11"1" },
{ from: 'c', to: 'f', cost: 2"1" },
{ from: 'd', to: 'e', cost: 6"1" },
{ from: 'e', to: 'f', cost: 9"1" }
;
def fromA: 'a' -> shortestPaths&{graph: $edges};
($fromA matching {|{to:'e'}|})... -> 'Shortest path from $.path(1); to $.to; is distance $.distance; via $.path(2..last); ' -> !OUT::write
($fromA matching {|{to:'f'}|})... -> 'Shortest path from $.path(1); to $.to; is distance $.distance; via $.path(2..last); ' -> !OUT::write </lang>
- Output:
Shortest path from a to e is distance 26"1" via [c, d] Shortest path from a to f is distance 11"1" via [c]
Tcl
This solution is incorrect. Since the path is directed and f is only a sink, f cannot be in the middle of a path.
Note that this code traverses the entire set of unrouted nodes at each step, as this is simpler than computing the subset that are reachable at each stage.
<lang tcl>proc dijkstra {graph origin} {
# Initialize
dict for {vertex distmap} $graph {
dict set dist $vertex Inf dict set path $vertex {}
} dict set dist $origin 0 dict set path $origin [list $origin]
while {[dict size $graph]} {
# Find unhandled node with least weight set d Inf dict for {uu -} $graph { if {$d > [set dd [dict get $dist $uu]]} { set u $uu set d $dd } }
# No such node; graph must be disconnected if {$d == Inf} break
# Update the weights for nodes lead to by the node we've picked dict for {v dd} [dict get $graph $u] { if {[dict exists $graph $v]} { set alt [expr {$d + $dd}] if {$alt < [dict get $dist $v]} { dict set dist $v $alt dict set path $v [list {*}[dict get $path $u] $v] } } }
# Remove chosen node from graph still to be handled dict unset graph $u
} return [list $dist $path]
}</lang> Showing the code in use: <lang tcl>proc makeUndirectedGraph arcs {
# Assume that all nodes are connected to something
foreach arc $arcs {
lassign $arc v1 v2 cost dict set graph $v1 $v2 $cost dict set graph $v2 $v1 $cost
} return $graph
} set arcs {
{a b 7} {a c 9} {b c 10} {b d 15} {c d 11}
{d e 6} {a f 14} {c f 2} {e f 9}
} lassign [dijkstra [makeUndirectedGraph $arcs] "a"] costs path puts "path from a to e costs [dict get $costs e]" puts "route from a to e is: [join [dict get $path e] { -> }]"</lang> Output:
path from a to e costs 20 route from a to e is: a -> c -> f -> e
VBA
<lang VB>Class Branch Public from As Node '[according to Dijkstra the first Node should be closest to P] Public towards As Node Public length As Integer '[directed length!] Public distance As Integer '[from P to farthest node] Public key As String Class Node Public key As String Public correspondingBranch As Branch Const INFINITY = 32767 Private Sub Dijkstra(Nodes As Collection, Branches As Collection, P As Node, Optional Q As Node)
'Dijkstra, E. W. (1959). "A note on two problems in connexion with graphs". 'Numerische Mathematik. 1: 269–271. doi:10.1007/BF01386390. 'http://www-m3.ma.tum.de/twiki/pub/MN0506/WebHome/dijkstra.pdf 'Problem 2. Find the path of minimum total length between two given nodes 'P and Q. 'We use the fact that, if R is a node on the minimal path from P to Q, knowledge 'of the latter implies the knowledge of the minimal path from P to A. In the 'solution presented, the minimal paths from P to the other nodes are constructed 'in order of increasing length until Q is reached. 'In the course of the solution the nodes are subdivided into three sets: 'A. the nodes for which the path of minimum length from P is known; nodes 'will be added to this set in order of increasing minimum path length from node P; '[comments in square brackets are not by Dijkstra] Dim a As New Collection '[of nodes (vertices)] 'B. the nodes from which the next node to be added to set A will be selected; 'this set comprises all those nodes that are connected to at least one node of 'set A but do not yet belong to A themselves; Dim b As New Collection '[of nodes (vertices)] 'C. the remaining nodes. Dim c As New Collection '[of nodes (vertices)] 'The Branches are also subdivided into three sets: 'I the Branches occurring in the minimal paths from node P to the nodes 'in set A; Dim I As New Collection '[of Branches (edges)] 'II the Branches from which the next branch to be placed in set I will be 'selected; one and only one branch of this set will lead to each node in set B; Dim II As New Collection '[of Branches (edges)] 'III. the remaining Branches (rejected or not yet considered). Dim III As New Collection '[of Branches (edges)] Dim u As Node, R_ As Node, dist As Integer 'To start with, all nodes are in set C and all Branches are in set III. We now 'transfer node P to set A and from then onwards repeatedly perform the following 'steps. For Each n In Nodes c.Add n, n.key Next n For Each e In Branches III.Add e, e.key Next e a.Add P, P.key c.Remove P.key Set u = P Do 'Step 1. Consider all Branches r connecting the node just transferred to set A 'with nodes R in sets B or C. If node R belongs to set B, we investigate whether 'the use of branch r gives rise to a shorter path from P to R than the known 'path that uses the corresponding branch in set II. If this is not so, branch r is 'rejected; if, however, use of branch r results in a shorter connexion between P 'and R than hitherto obtained, it replaces the corresponding branch in set II 'and the latter is rejected. If the node R belongs to set C, it is added to set B and 'branch r is added to set II. For Each r In III If r.from Is u Then Set R_ = r.towards If Belongs(R_, c) Then c.Remove R_.key b.Add R_, R_.key Set R_.correspondingBranch = r If u.correspondingBranch Is Nothing Then R_.correspondingBranch.distance = r.length Else R_.correspondingBranch.distance = u.correspondingBranch.distance + r.length End If III.Remove r.key '[not mentioned by Dijkstra ...] II.Add r, r.key Else If Belongs(R_, b) Then '[initially B is empty ...] If R_.correspondingBranch.distance > u.correspondingBranch.distance + r.length Then II.Remove R_.correspondingBranch.key II.Add r, r.key Set R_.correspondingBranch = r '[needed in step 2.] R_.correspondingBranch.distance = u.correspondingBranch.distance + r.length End If End If End If End If Next r 'Step 2. Every node in set B can be connected to node P in only one way 'if we restrict ourselves to Branches from set I and one from set II. In this sense 'each node in set B has a distance from node P: the node with minimum distance 'from P is transferred from set B to set A, and the corresponding branch is transferred 'from set II to set I. We then return to step I and repeat the process 'until node Q is transferred to set A. Then the solution has been found. dist = INFINITY Set u = Nothing For Each n In b If dist > n.correspondingBranch.distance Then dist = n.correspondingBranch.distance Set u = n End If Next n b.Remove u.key a.Add u, u.key II.Remove u.correspondingBranch.key I.Add u.correspondingBranch, u.correspondingBranch.key Loop Until IIf(Q Is Nothing, a.Count = Nodes.Count, u Is Q) If Not Q Is Nothing Then GetPath Q
End Sub Private Function Belongs(n As Node, col As Collection) As Boolean
Dim obj As Node
On Error GoTo err
Belongs = True
Set obj = col(n.key)
Exit Function
err:
Belongs = False
End Function Private Sub GetPath(Target As Node)
Dim path As String
If Target.correspondingBranch Is Nothing Then
path = "no path"
Else
path = Target.key
Set u = Target
Do While Not u.correspondingBranch Is Nothing
path = u.correspondingBranch.from.key & " " & path
Set u = u.correspondingBranch.from
Loop
Debug.Print u.key, Target.key, Target.correspondingBranch.distance, path
End If
End Sub Public Sub test()
Dim a As New Node, b As New Node, c As New Node, d As New Node, e As New Node, f As New Node Dim ab As New Branch, ac As New Branch, af As New Branch, bc As New Branch, bd As New Branch Dim cd As New Branch, cf As New Branch, de As New Branch, ef As New Branch Set ab.from = a: Set ab.towards = b: ab.length = 7: ab.key = "ab": ab.distance = INFINITY Set ac.from = a: Set ac.towards = c: ac.length = 9: ac.key = "ac": ac.distance = INFINITY Set af.from = a: Set af.towards = f: af.length = 14: af.key = "af": af.distance = INFINITY Set bc.from = b: Set bc.towards = c: bc.length = 10: bc.key = "bc": bc.distance = INFINITY Set bd.from = b: Set bd.towards = d: bd.length = 15: bd.key = "bd": bd.distance = INFINITY Set cd.from = c: Set cd.towards = d: cd.length = 11: cd.key = "cd": cd.distance = INFINITY Set cf.from = c: Set cf.towards = f: cf.length = 2: cf.key = "cf": cf.distance = INFINITY Set de.from = d: Set de.towards = e: de.length = 6: de.key = "de": de.distance = INFINITY Set ef.from = e: Set ef.towards = f: ef.length = 9: ef.key = "ef": ef.distance = INFINITY a.key = "a" b.key = "b" c.key = "c" d.key = "d" e.key = "e" f.key = "f" Dim testNodes As New Collection Dim testBranches As New Collection testNodes.Add a, "a" testNodes.Add b, "b" testNodes.Add c, "c" testNodes.Add d, "d" testNodes.Add e, "e" testNodes.Add f, "f" testBranches.Add ab, "ab" testBranches.Add ac, "ac" testBranches.Add af, "af" testBranches.Add bc, "bc" testBranches.Add bd, "bd" testBranches.Add cd, "cd" testBranches.Add cf, "cf" testBranches.Add de, "de" testBranches.Add ef, "ef" Debug.Print "From", "To", "Distance", "Path" '[Call Dijkstra with target:] Dijkstra testNodes, testBranches, a, e '[Call Dijkstra without target computes paths to all reachable nodes:] Dijkstra testNodes, testBranches, a GetPath f
End Sub</lang>
- Output:
From To Distance Patha e 26 a c d e
a f 11 a c f
Wren
<lang ecmascript>import "/dynamic" for Tuple import "/trait" for Comparable import "/sort" for Cmp, Sort import "/set" for Set
var Edge = Tuple.create("Edge", ["v1", "v2", "dist"])
// One vertex of the graph, complete with mappings to neighboring vertices class Vertex is Comparable {
static map { __map } // maps a name to its corresponding Vertex
construct new(name) {
_name = name
_dist = Num.maxSafeInteger // assumed to be infinity
_prev = null
_neighbors = {}
if (!__map) __map = {}
__map[name] = this
}
name { _name }
dist { _dist }
dist=(d) { _dist = d }
prev { _prev }
prev=(v) { _prev = v }
neighbors { _neighbors }
printPath() {
if (this == _prev) {
System.write(_name)
} else if (!_prev) {
System.write("%(_name)(unreached)")
} else {
_prev.printPath()
System.write(" -> %(_name)(%(_dist))")
}
}
compare(other) {
if (_dist == other.dist) return Cmp.string.call(_name, other.name)
return Cmp.num.call(_dist, other.dist)
}
toString { "(%(_name), %(_dist))" }
}
class Graph {
construct new(edges, directed, showAllPaths) {
_graph = {}
// one pass to find all vertices
for (e in edges) {
if (!_graph.containsKey(e.v1)) _graph[e.v1] = Vertex.new(e.v1)
if (!_graph.containsKey(e.v2)) _graph[e.v2] = Vertex.new(e.v2)
}
// another pass to set neighboring vertices
for (e in edges) {
_graph[e.v1].neighbors[_graph[e.v2].name] = e.dist
// also do this for an undirected graph if applicable
if (!directed) _graph[e.v2].neighbors[_graph[e.v1].name] = e.dist
}
_showAllPaths = showAllPaths
_directed = directed
}
// Runs dijkstra using a specified source vertex
dijkstra(startName) {
if (!_graph.containsKey(startName)) {
System.print("Graph doesn't contain start vertex '%(startName)'")
return
}
var source = _graph[startName]
var q = Set.new()
// set-up vertices
for (v in _graph.values) {
v.prev = (v == source) ? source : null
v.dist = (v == source) ? 0 : Num.maxSafeInteger
q.add(v.name)
}
dijkstra_(q)
}
// Implementation of dijkstra's algorithm using a (simulated) tree set
dijkstra_(q) {
while (!q.isEmpty) {
var qq = q.toList
Sort.heap(qq)
// vertex with shortest distance (first iteration will return source)
var u = Vertex.map[qq[0]]
q.remove(qq[0])
// if distance is infinite we can ignore 'u'
// (and any other remaining vertices) since they are unreachable
if (u.dist == Num.maxSafeInteger) break
// look at distances to each neighbor
for (a in u.neighbors) {
var v = Vertex.map[a.key] // the neighbor in this iteration
var alternateDist = u.dist + a.value
if (alternateDist < v.dist) { // shorter path to neighbor found
v.dist = alternateDist
v.prev = u
}
}
}
}
// Prints the path from the source to every vertex
// (output order is not guaranteed)
printAllPaths_() {
for (v in _graph.values) {
v.printPath()
System.print()
}
System.print()
}
// Prints a path from the source to the specified vertex
printPath(endName) {
if (!_graph.containsKey(endName)) {
System.print("Graph doesn't contain end vertex '%(endName)'")
return
}
System.write(_directed ? "Directed : " : "Undirected : ")
_graph[endName].printPath()
System.print()
if (_showAllPaths) printAllPaths_() else System.print()
}
}
var GRAPH = [
Edge.new("a", "b", 7),
Edge.new("a", "c", 9),
Edge.new("a", "f", 14),
Edge.new("b", "c", 10),
Edge.new("b", "d", 15),
Edge.new("c", "d", 11),
Edge.new("c", "f", 2),
Edge.new("d", "e", 6),
Edge.new("e", "f", 9)
]
var START = "a" var END = "e"
var g = Graph.new(GRAPH, true, false) // directed g.dijkstra(START) g.printPath(END)
g = Graph.new(GRAPH, false, false) // undirected g.dijkstra(START) g.printPath(END)</lang>
- Output:
Directed : a -> c(9) -> d(20) -> e(26) Undirected : a -> c(9) -> f(11) -> e(20)
zkl
<lang zkl>const INF=(0).MAX; fcn dijkstra(graph,start,dst){
Q :=graph.copy();
prev:=graph.keys.pump(Dictionary().add.fp1(Void));
dist:=graph.keys.pump(Dictionary().add.fp1(INF));
dist[start]=0;
while(Q){
Q.reduce('wrap(min,[(v,_)],ru){
if((d:=dist[v])<min){ ru.set(v); d } else min },
INF,ru:=Ref(Void));
if(not u:=ru.value) return("Can't get there");
if(u==dst){
S:=List(); do{ S.append(u); u=prev[u]; }while(u);
return(S.reverse());
}
Q.del(u);
foreach v,len in (graph[u]){ // (neighborVertex,len to neighbor)...
alt:=dist[u] + len; if(alt<dist[v]){ dist[v]=alt; prev[v]=u; }
} }
}</lang> <lang zkl>graph:=Dictionary( // directed graph
"a", T(T("b", 7.0), T("c", 9.0), T("f",14.0)),
"b", T(T("c",10.0), T("d",15.0)),
"c", T(T("d",11.0), T("f", 2.0)),
"d", T(T("e", 6.0)),
"e", T(T("f", 9.0)),
"f", T,
); dijkstra(graph,"a","e").println(); dijkstra(graph,"e","a").println();</lang>
- Output:
L("a","c","d","e")
Can't get there
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