# Dijkstra's algorithm

*is a*

**Dijkstra's algorithm****draft**programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

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**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 nonnegative 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. It can also be used for finding costs of shortest paths from a single vertex to a single destination vertex by stopping the algorithm once the shortest path to the destination vertex has been determined.

For example, 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 IS-IS and OSPF (Open Shortest Path First).

The algorithm was extended by the A* search algorithm which uses heuristics to quickly find an approximate solution.

- Task

- Implement a version of Dijkstra's algorithm that computes a shortest path from a start vertex to an end vertex in a directed graph.
- Run your program with the following directed graph to find the shortest path from vertex
**a**to vertex**e**. - Show the output of your program.

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.

- Extra Credit

Document the specific algorithm implemented. The {{trans}} template is sufficient. Otherwise add text outside of your program or add comments within your program. This is not a requirement to explain *how* the algorithm works, but to state *which* algorithm is implemented. If your code follows an external source such as the Wikipedia pseudo-code, you can state that. You can state if it is Dijkstra's original algorithm or some more efficient variant. It is relevant to mention things like priority queues, heaps, and expected time complexity in big-O notation. If a priority queue is used, it is important to discuss how the step of decreasing the distance of a node is accomplished, and whether it is linear or logarithmic time.

- See also

## Contents

## ALGOL 68[edit]

**File: prelude_dijkstras_algorithm.a68**

# -*- 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

**File: test_dijkstras_algorithm.a68**

#!/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 #

)

**Output:**

a --9-> c --2-> f --9-> e

## AutoHotkey[edit]

Dijkstra(data, start){Examples:

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

}

data =Outputs:

(

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

A > C 9 C > F 2 F > E 9 Total distance = 20

## C[edit]

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.

#include <stdio.h>output

#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;

}

26 acde

## C++[edit]

(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).

#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;

}

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.

#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;

}

## D[edit]

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).

import std.stdio, std.typecons, std.algorithm, std.container;

alias Vertex = string;

alias Weight = int;

const struct Neighbor {

Vertex target;

Weight weight;

}

alias AdjacencyMap = Neighbor[][Vertex];

Tuple!(Weight[Vertex], Vertex[Vertex])

dijkstraComputePaths(in Vertex source,

in Vertex target,

in AdjacencyMap adjacencyMap) pure {

typeof(typeof(return).init[0]) minDistance;

foreach (immutable v, const neighs; adjacencyMap) {

minDistance[v] = Weight.max;

foreach (immutable n; neighs)

minDistance[n.target] = Weight.max;

}

minDistance[source] = 0;

alias Pair = Tuple!(Weight, Vertex);

auto vertexQueue = redBlackTree(Pair(minDistance[source], source));

typeof(typeof(return).init[1]) previous;

foreach(_, u; vertexQueue){

if (u == target)

break;

// Visit each edge exiting u.

foreach (immutable n; adjacencyMap.get(u, null)) {

const v = n.target;

const distanceThroughU = minDistance[u] + n.weight;

if (distanceThroughU < minDistance[v]) {

vertexQueue.removeKey(Pair(minDistance[v], v));

minDistance[v] = distanceThroughU;

previous[v] = u;

vertexQueue.insert(Pair(minDistance[v], v));

}

}

}

return tuple(minDistance, previous);

}

Vertex[] dijkstraGetShortestPathTo(Vertex v,

in Vertex[Vertex] previous)

pure nothrow {

auto 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));

}

- Output:

Distance from "a" to "e": 26 Path: ["a", "c", "d", "e"]

## Erlang[edit]

-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]).

- 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.

## Go[edit]

Algorithm is derived from Wikipedia section 1, titled "Algorithm," with nodes stored in a heap, which should bound time complexity by O(E log V). Decreasing the distance of a node is accomplished by removing it from the heap and then re-inserting it. Linear-time traversal to find the node in the heap is avoided by keeping the index of each node in the heap as a field of the node, and our heap operations ensure that this variable is updated appropriately whenever nodes are moved in the heap. Thus removal from the heap is accomplished in logarithmic time.

Comments in the code below refer to corresponding steps in that section of the WP page. A significant variation in step 2 is that the unvisited set is not initially populated. Instead, the unvisited set is maintained as the "tentative" set, as illustrated on the WP page in the animated image showing robot motion planning.

The heap support comes from the Go standard library. The three operations used here are each documented to have O(log n) complexity.

package main

import (

"container/heap"

"fmt"

"math"

)

// edge struct holds the bare data needed to define a graph.

type edge struct {

vert1, vert2 string

dist int

}

func main() {

// example data and parameters

graph := []edge{

{"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},

}

directed := true

start := "a"

end := "e"

findAll := false

// construct linked representation of example data

allNodes, startNode, endNode := linkGraph(graph, directed, start, end)

if directed {

fmt.Print("Directed")

} else {

fmt.Print("Undirected")

}

fmt.Printf(" graph with %d nodes, %d edges\n", len(allNodes), len(graph))

if startNode == nil {

fmt.Printf("start node %q not found in graph\n", start)

return

}

if findAll {

endNode = nil

} else if endNode == nil {

fmt.Printf("end node %q not found in graph\n", end)

return

}

// run Dijkstra's shortest path algorithm

paths := dijkstra(allNodes, startNode, endNode)

fmt.Println("Shortest path(s):")

for _, p := range paths {

fmt.Println(p.path, "length", p.length)

}

}

// node and neighbor structs hold data useful for the heap-optimized

// Dijkstra's shortest path algorithm

type node struct {

vert string // vertex name

tent int // tentative distance

prev *node // previous node in shortest path back to start

done bool // true when tent and prev represent shortest path

nbs []neighbor // edges from this vertex

rx int // heap.Remove index

}

type neighbor struct {

nd *node // node corresponding to vertex

dist int // distance to this node (from whatever node references this)

}

// linkGraph constructs a linked representation of the input graph,

// with additional fields needed by the shortest path algorithm.

//

// Return value allNodes will contain all nodes found in the input graph,

// even ones not reachable from the start node.

// Return values startNode, endNode will be nil if the specified start or

// end node names are not found in the graph.

func linkGraph(graph []edge, directed bool,

start, end string) (allNodes []*node, startNode, endNode *node) {

all := make(map[string]*node)

// one pass over graph to collect nodes and link neighbors

for _, e := range graph {

n1 := all[e.vert1]

n2 := all[e.vert2]

// add previously unseen nodes

if n1 == nil {

n1 = &node{vert: e.vert1}

all[e.vert1] = n1

}

if n2 == nil {

n2 = &node{vert: e.vert2}

all[e.vert2] = n2

}

// link neighbors

n1.nbs = append(n1.nbs, neighbor{n2, e.dist})

if !directed {

n2.nbs = append(n2.nbs, neighbor{n1, e.dist})

}

}

allNodes = make([]*node, len(all))

var n int

for _, nd := range all {

allNodes[n] = nd

n++

}

return allNodes, all[start], all[end]

}

// return type

type path struct {

path []string

length int

}

// dijkstra is a heap-enhanced version of Dijkstra's shortest path algorithm.

//

// If endNode is specified, only a single path is returned.

// If endNode is nil, paths to all nodes are returned.

//

// Note input allNodes is needed to efficiently accomplish WP steps 1 and 2.

// This initialization could be done in linkGraph, but is done here to more

// closely follow the WP algorithm.

func dijkstra(allNodes []*node, startNode, endNode *node) (pl []path) {

// WP steps 1 and 2.

for _, nd := range allNodes {

nd.tent = math.MaxInt32

nd.done = false

nd.prev = nil

nd.rx = -1

}

current := startNode

current.tent = 0

var unvis ndList

for {

// WP step 3: update tentative distances to neighbors

for _, nb := range current.nbs {

if nd := nb.nd; !nd.done {

if d := current.tent + nb.dist; d < nd.tent {

nd.tent = d

nd.prev = current

if nd.rx < 0 {

heap.Push(&unvis, nd)

} else {

heap.Fix(&unvis, nd.rx)

}

}

}

}

// WP step 4: mark current node visited, record path and distance

current.done = true

if endNode == nil || current == endNode {

// record path and distance for return value

distance := current.tent

// recover path by tracing prev links,

var p []string

for ; current != nil; current = current.prev {

p = append(p, current.vert)

}

// then reverse list

for i := (len(p) + 1) / 2; i > 0; i-- {

p[i-1], p[len(p)-i] = p[len(p)-i], p[i-1]

}

pl = append(pl, path{p, distance}) // pl is return value

// WP step 5 (case of end node reached)

if endNode != nil {

return

}

}

if len(unvis) == 0 {

break // WP step 5 (case of no more reachable nodes)

}

// WP step 6: new current is node with smallest tentative distance

current = heap.Pop(&unvis).(*node)

}

return

}

// ndList implements container/heap

type ndList []*node

func (n ndList) Len() int { return len(n) }

func (n ndList) Less(i, j int) bool { return n[i].tent < n[j].tent }

func (n ndList) Swap(i, j int) {

n[i], n[j] = n[j], n[i]

n[i].rx = i

n[j].rx = j

}

func (n *ndList) Push(x interface{}) {

nd := x.(*node)

nd.rx = len(*n)

*n = append(*n, nd)

}

func (n *ndList) Pop() interface{} {

s := *n

last := len(s) - 1

r := s[last]

*n = s[:last]

r.rx = -1

return r

}

- Output:

Directed graph with 6 nodes, 9 edges Shortest path(s): [a c d e] length 26

## Haskell[edit]

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.

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

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

## Icon and Unicon[edit]

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.

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

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[edit]

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

)

### J: Alternative Implementation[edit]

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

)

Example use:

(;:'a e') vertices shortest_path edges

┌─────────┐

│┌─┬─┬─┬─┐│

││a│c│d│e││

│└─┴─┴─┴─┘│

└─────────┘

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[edit]

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.

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();

}

}

}

- Output:

a -> c(9) -> d(20) -> e(26)

## Lua[edit]

Hopefully the variable names here make the process as clear as possible...

-- 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))

- Output:

Directed: 26 a c d e Undirected: 20 a c f e

## Mathematica[edit]

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"}

## Maxima[edit]

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]] */

## OCaml[edit]

Just a straightforward implementation of the pseudo-code from the Wikipedia article:

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)

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.

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 ()

## PARI/GP[edit]

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.

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

};

## Perl[edit]

use strict;

use warnings;

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} = 9999999;

delete $v->{prev};

delete $v->{visited};

}

$g->{$a}{dist} = 0;

my $h = [];

push_priority($h, $g->{$a});

while (1) {

my $v = shift @$h;

last if !$v || $v->{name} eq $b;

$v->{visited} = 1;

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, "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");

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";

output:

26 acde

## Perl 6[edit]

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;

- Output:

a c f e

## Phix[edit]

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.

enum A,B,C,D,E,F

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

function AFs(sequence s) -- output helper

string res = ""

for i=1 to length(s) do

res &= AFi(s[i])

end for

return res

end function

procedure test(sequence testset)

integer start,finish,ecost

integer len

string epath,path

for i=1 to length(testset) do

{start,finish,ecost,epath} = testset[i]

reset()

len = shortest_path(start,finish)

if len=-1 then

path = "no path found"

else

path = AFs(backtrack(finish,start))

end if

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[edit]

There are parts of this algorithm that could be optimized which have been marked TODO.

<?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";

Output is:

path is: a, c, f, e

## PicoLisp[edit]

Following the Wikipedia algorithm:

(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 ) )

Test:

(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)

Output:

-> 20

## Prolog[edit]

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.

%___________________________________________________________________________

:-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', [[H|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]).

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[edit]

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.

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 = edges2 = [Edge(*edge) for edge in edges]

self.vertices = set(sum(([e.start, e.end] for e in edges2), []))

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))

#pp(neighbours)

while 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"))

- Output:

['a', 'c', 'd', 'e']

## Racket[edit]

#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))]))))])

Output:

Distances from a: ((b 7) (d 20) (a 0) (c 9) (f 11) (e 26))

Shortest path: (a c d e)

## REXX[edit]

Some program features are:

- elimination of null edges
- elimination of duplications (the cheapest path is chosen)
- a test for a
*no path found*condition - use of memoization

/*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.*/

@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

**output** when using the (internal) defaults:

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

## Ruby[edit]

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.

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

@edges[[v1, v2]] = @edges[[v2, 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 + @edges[[u.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(" -> ")

- Output:

shortest path from a to e has cost 20: a -> c -> f -> e

## Rust[edit]

This solution uses a very bare-bones, naive implementation of an adjacency list to represent the graph.

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);

}

- Output:

Cost: 20 a -> c -> f -> e

## SAS[edit]

Use network solver in SAS/OR:

/* 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;

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[edit]

A functional implementation of Dijkstras Algorithm:

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)

}

}

- Output:

(26.0,List(a, c, d, e))

## Sidef[edit]

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}"

- Output:

26 acde

## Tcl[edit]

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.

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]

}

Showing the code in use:

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] { -> }]"

Output:

path from a to e costs 20 route from a to e is: a -> c -> f -> e

## zkl[edit]

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; }

}

}

}

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();

- Output:

L("a","c","d","e") Can't get there