# Dijkstra's algorithm

Dijkstra's algorithm is a 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.
 This page uses content from Wikipedia. The current wikipedia article is at Dijkstra's algorithm. The original RosettaCode article was extracted from the wikipedia article № 295012245 of 17:56, 7 June 2009‎‎ . The list of authors can be seen in the page history. As with Rosetta Code, the pre 5 June 2009 text of Wikipedia is available under the GNU FDL. (See links for details on variance)

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).

1. Implement a version of Dijkstra's algorithm that computes a shortest path from a start vertex to an end vertex in a directed graph.
2. Run your program with the following directed graph to find the shortest path from vertex "a" to vertex "e."
3. Show the output of your program.
Vertices
Number Name
1 a
2 b
3 c
4 d
5 e
6 f
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

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 pseudocode, 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.

## ALGOL 68

Works with: ALGOL 68 version Revision 1 - one minor extension to language used - PRAGMA READ, similar to C's #include directive.
Works with: ALGOL 68G version Any - tested with release algol68g-2.6.
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

`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}`
Examples:
`data =(A	B	7A	C	9A	F	14B	C	10B	D	15C	D	11C	F	2D	E	6E	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, % nodeListGui, add, DDL, x200 yp vTo gSubmit, % nodeListGui, add, ListView, xs w340 r6, From|>|To|DistanceGui, add, Text, vT1 xs w340 r1Gui, +AlwaysOnTopGui, showLoop 4	LV_ModifyCol(A_Index, "80 Center") Submit:Gui, Submit, NoHideGuiControl, , T1, % ""LV_Delete()if !(From && To) || (From = To)    returnres := Dijkstra(data, From)	, 	xTo := xFrom := DirectFlight := "" , origin := toGuiControl, , T1, no routing foundif !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] . DirectFlightreturn esc::GuiClose:ExitAppreturn`
Outputs:
```A	>	C	9
C	>	F	2
F	>	E	9
Total distance = 20```

## C

Standard binary heap-as-priority queue affair. Only that each node links back to its heap position for easier update.

There are two `main()` functions to choose from (look for `#define BIG_EXAMPLE`), one is for task example, the other is a much heavier duty test case.

`#include <stdio.h>#include <stdlib.h>#include <string.h> //#define BIG_EXAMPLE typedef struct node_t node_t, *heap_t;typedef struct edge_t edge_t;struct edge_t {	node_t *nd;	/* target of this edge */	edge_t *sibling;/* for singly linked list */	int len;	/* edge cost */};struct node_t {	edge_t *edge;	/* singly linked list of edges */	node_t *via;	/* where previous node is in shortest path */	double dist;	/* distance from origining node */	char name[8];	/* the, er, name */	int heap_idx;	/* link to heap position for updating distance */};  /* --- edge management --- */#ifdef BIG_EXAMPLE#	define BLOCK_SIZE (1024 * 32 - 1)#else#	define BLOCK_SIZE 15#endifedge_t *edge_root = 0, *e_next = 0; /* Don't mind the memory management stuff, they are besides the point.   Pretend e_next = malloc(sizeof(edge_t)) */void add_edge(node_t *a, node_t *b, double d){	if (e_next == edge_root) {		edge_root = malloc(sizeof(edge_t) * (BLOCK_SIZE + 1));		edge_root[BLOCK_SIZE].sibling = e_next;		e_next = edge_root + BLOCK_SIZE;	}	--e_next; 	e_next->nd = b;	e_next->len = d;	e_next->sibling = a->edge;	a->edge = e_next;} void free_edges(){	for (; edge_root; edge_root = e_next) {		e_next = edge_root[BLOCK_SIZE].sibling;		free(edge_root);	}} /* --- priority queue stuff --- */heap_t *heap;int heap_len; void set_dist(node_t *nd, node_t *via, double d){	int i, j; 	/* already knew better path */	if (nd->via && d >= nd->dist) return; 	/* find existing heap entry, or create a new one */	nd->dist = d;	nd->via = via; 	i = nd->heap_idx;	if (!i) i = ++heap_len; 	/* upheap */	for (; i > 1 && nd->dist < heap[j = i/2]->dist; i = j)		(heap[i] = heap[j])->heap_idx = i; 	heap[i] = nd;	nd->heap_idx = i;} node_t * pop_queue(){	node_t *nd, *tmp;	int i, j; 	if (!heap_len) return 0; 	/* remove leading element, pull tail element there and downheap */	nd = heap[1];	tmp = heap[heap_len--]; 	for (i = 1; i < heap_len && (j = i * 2) <= heap_len; i = j) {		if (j < heap_len && heap[j]->dist > heap[j+1]->dist) j++; 		if (heap[j]->dist >= tmp->dist) break;		(heap[i] = heap[j])->heap_idx = i;	} 	heap[i] = tmp;	tmp->heap_idx = i; 	return nd;} /* --- Dijkstra stuff; unreachable nodes will never make into the queue --- */void calc_all(node_t *start){	node_t *lead;	edge_t *e; 	set_dist(start, start, 0);	while ((lead = pop_queue()))		for (e = lead->edge; e; e = e->sibling)			set_dist(e->nd, lead, lead->dist + e->len);} void show_path(node_t *nd){	if (nd->via == nd)		printf("%s", nd->name);	else if (!nd->via)		printf("%s(unreached)", nd->name);	else {		show_path(nd->via);		printf("-> %s(%g) ", nd->name, nd->dist);	}} int main(void){#ifndef BIG_EXAMPLE	int i; #	define N_NODES ('f' - 'a' + 1)	node_t *nodes = calloc(sizeof(node_t), N_NODES); 	for (i = 0; i < N_NODES; i++)		sprintf(nodes[i].name, "%c", 'a' + i); #	define E(a, b, c) add_edge(nodes + (a - 'a'), nodes + (b - 'a'), c)	E('a', 'b', 7);	E('a', 'c', 9); E('a', 'f', 14);	E('b', 'c', 10);E('b', 'd', 15);E('c', 'd', 11);	E('c', 'f', 2); E('d', 'e', 6);	E('e', 'f', 9);#	undef E #else /* BIG_EXAMPLE */	int i, j, c; #	define N_NODES 4000	node_t *nodes = calloc(sizeof(node_t), N_NODES); 	for (i = 0; i < N_NODES; i++)		sprintf(nodes[i].name, "%d", i + 1); 	/* given any pair of nodes, there's about 50% chance they are not	   connected; if connected, the cost is randomly chosen between 0	   and 49 (inclusive! see output for consequences) */	for (i = 0; i < N_NODES; i++) {		for (j = 0; j < N_NODES; j++) {			/* majority of runtime is actually spent here */			if (i == j) continue;			c = rand() % 100;			if (c < 50) continue;			add_edge(nodes + i, nodes + j, c - 50);		}	} #endif	heap = calloc(sizeof(heap_t), N_NODES + 1);	heap_len = 0; 	calc_all(nodes);	for (i = 0; i < N_NODES; i++) {		show_path(nodes + i);		putchar('\n');	} #if 0	/* real programmers don't free memories (they use Fortran) */	free_edges();	free(heap);	free(nodes);#endif	return 0;}`
output
```a
a-> b(7)
a-> c(9)
a-> c(9) -> d(20)
a-> c(9) -> d(20) -> e(26)
a-> 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.

`#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;}`

## D

Translation of: C++

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;     while (!vertexQueue.empty) {        const u = vertexQueue.front[1];        vertexQueue.removeFront;         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

` -module(dijkstra).-include_lib("eunit/include/eunit.hrl").-export([dijkstrafy/3]). % just hide away recursion so we have a nice interfacedijkstrafy(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

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 algorithmtype 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 typetype 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/heaptype 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
```

Translation of: C++

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.Arrayimport Data.Array.MArrayimport Data.Array.STimport Control.Monad.STimport 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

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 graphend 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 bestMouseend 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

` NB. verbs and adverbparse_table=: ;:@:(LF&= [;._2 -.&CR)mp=: \$:~ :(+/ .*)                       NB. matrix productmin=: <./                               NB. minimumIndex=: (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 definea	 b	 7a	 c	 9a	 f	 14b	 c	 10b	 d	 15c	 d	 11c	 f	 2d	 e	 6e	 f	 9) T=: parse_table INPUTNAMED_LINKS=: _ 2 {. TNODES=: ~. , NAMED_LINKS                NB. vector of boxed namesNUMBERED_LINKS=: NODES i. NAMED_LINKSWEIGHTS=: _ ".&> _ _1 {. TGRAPH=: 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:    TERMINALS0 5    GRAPH0 1  70 2  90 3 141 2 101 4 152 4 112 3  24 5  65 3  9) `

###  J: Alternative Implementation

`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

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) {         return Integer.compare(dist, other.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)

```

## 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"}`

## 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]] */`

## OCaml

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: C++

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 = inttype weight = floattype neighbor = vertex * weightmodule 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

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 6

` 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

I didn't really copy any other code/pseudocode, just followed the basic concept of (update costs) (select lowest cost unvisited) until target reached.
Apart from the one glaring slipup (left in), and the original not coping at all with the "no path" case, it pretty much worked fine first time.
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,Fconstant 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 resend 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 {backtrack(finish,start),cost[finish]} end if (oops, me clobbered start...)        if start=finish then return cost[finish] end if    end whileend function function AFi(integer i)     -- output helper    return 'A'+i-1end function function AFs(sequence s)    -- output helperstring res = ""    for i=1 to length(s) do        res &= AFi(s[i])    end for    return resend function procedure test(sequence testset)integer start,finish,ecostinteger lenstring 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 forend 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.

` <?phpfunction 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

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

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 origintraverse(_). 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

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, queuefrom 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.pushleft(u)            u = previous[u]        s.pushleft(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

` #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

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 prpgram finds 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 jj=1  for length(@abc);     _=substr(@abc,jj,1)                        call value translate(_),jj;    @@.jj=_                        end   /*jj*/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=                     do jv=2  to verts;    call paths verts,jv;      end  /*jv*/sayif bestP==\$.  then do; say 'no path found.';   exit;   end         /*oops-ay. */say 'best path ='  translate(bestP,@abc,123456789)    '          cost ='   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 jj=1  for yy;  _=_||@.jj;  end;           call apath _                     end                else do qq=1  for xx    /*build the vertex paths (recursively)*/                       do kp=1  for ?-1;   if @.kp==!.qq  then iterate qq;   end                     @.?=!.qq;             call .path ?+1                     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

This solution is incorrect. Since the path is directed and f is only a sink, f cannot be in the middle of a path.

Works with: Ruby version 1.9.2+
(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]   endend 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, :epath, 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```

## Scala

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))`

## 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.

`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 pathputs "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
```