Dijkstra Algorithm With Expalna

8/2/2019 Dijkstra Algorithm With Expalna

1/12

Dijkstra's algorithmFrom Wikipedia, the free encyclopediaJump to: navigation , search

Not to be confused with Dykstra's projection algorithm . Dijkstra's algorithm

Dijkstra's algorithm runtime

Class Search algorithm

Data structure Graph

Worst case performance

Graph and tree

search algorithms

A* B*

Beam Bellman Ford

Best-first Bidirectional

Borvka Branch & bound

BFS British Museum

D* DFS
http://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#mw-headhttp://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#mw-headhttp://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#mw-headhttp://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#p-searchhttp://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#p-searchhttp://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#p-searchhttp://en.wikipedia.org/wiki/Dykstra%27s_projection_algorithmhttp://en.wikipedia.org/wiki/Dykstra%27s_projection_algorithmhttp://en.wikipedia.org/wiki/Dykstra%27s_projection_algorithmhttp://en.wikipedia.org/wiki/Search_algorithmhttp://en.wikipedia.org/wiki/Search_algorithmhttp://en.wikipedia.org/wiki/Graph_%28data_structure%29http://en.wikipedia.org/wiki/Graph_%28data_structure%29http://en.wikipedia.org/wiki/Best,_worst_and_average_casehttp://en.wikipedia.org/wiki/Best,_worst_and_average_casehttp://en.wikipedia.org/wiki/Graph_traversalhttp://en.wikipedia.org/wiki/Graph_traversalhttp://en.wikipedia.org/wiki/Tree_traversalhttp://en.wikipedia.org/wiki/Tree_traversalhttp://en.wikipedia.org/wiki/Tree_traversalhttp://en.wikipedia.org/wiki/Tree_traversalhttp://en.wikipedia.org/wiki/Alpha-beta_pruninghttp://en.wikipedia.org/wiki/Alpha-beta_pruninghttp://en.wikipedia.org/wiki/Alpha-beta_pruninghttp://en.wikipedia.org/wiki/Alpha-beta_pruninghttp://en.wikipedia.org/wiki/A*_search_algorithmhttp://en.wikipedia.org/wiki/A*_search_algorithmhttp://en.wikipedia.org/wiki/B*http://en.wikipedia.org/wiki/B*http://en.wikipedia.org/wiki/Beam_searchhttp://en.wikipedia.org/wiki/Beam_searchhttp://en.wikipedia.org/wiki/Bellman%E2%80%93Ford_algorithmhttp://en.wikipedia.org/wiki/Bellman%E2%80%93Ford_algorithmhttp://en.wikipedia.org/wiki/Bellman%E2%80%93Ford_algorithmhttp://en.wikipedia.org/wiki/Bellman%E2%80%93Ford_algorithmhttp://en.wikipedia.org/wiki/Best-first_searchhttp://en.wikipedia.org/wiki/Best-first_searchhttp://en.wikipedia.org/wiki/Bidirectional_searchhttp://en.wikipedia.org/wiki/Bidirectional_searchhttp://en.wikipedia.org/wiki/Bor%C5%AFvka%27s_algorithmhttp://en.wikipedia.org/wiki/Bor%C5%AFvka%27s_algorithmhttp://en.wikipedia.org/wiki/Branch_and_boundhttp://en.wikipedia.org/wiki/Branch_and_boundhttp://en.wikipedia.org/wiki/Breadth-first_searchhttp://en.wikipedia.org/wiki/Breadth-first_searchhttp://en.wikipedia.org/wiki/British_Museum_algorithmhttp://en.wikipedia.org/wiki/British_Museum_algorithmhttp://en.wikipedia.org/wiki/D*http://en.wikipedia.org/wiki/D*http://en.wikipedia.org/wiki/Depth-first_searchhttp://en.wikipedia.org/wiki/Depth-first_searchhttp://en.wikipedia.org/wiki/File:Dijksta_Anim.gifhttp://en.wikipedia.org/wiki/File:Dijksta_Anim.gifhttp://en.wikipedia.org/wiki/Depth-first_searchhttp://en.wikipedia.org/wiki/D*http://en.wikipedia.org/wiki/British_Museum_algorithmhttp://en.wikipedia.org/wiki/Breadth-first_searchhttp://en.wikipedia.org/wiki/Branch_and_boundhttp://en.wikipedia.org/wiki/Bor%C5%AFvka%27s_algorithmhttp://en.wikipedia.org/wiki/Bidirectional_searchhttp://en.wikipedia.org/wiki/Best-first_searchhttp://en.wikipedia.org/wiki/Bellman%E2%80%93Ford_algorithmhttp://en.wikipedia.org/wiki/Beam_searchhttp://en.wikipedia.org/wiki/B*http://en.wikipedia.org/wiki/A*_search_algorithmhttp://en.wikipedia.org/wiki/Alpha-beta_pruninghttp://en.wikipedia.org/wiki/Tree_traversalhttp://en.wikipedia.org/wiki/Tree_traversalhttp://en.wikipedia.org/wiki/Graph_traversalhttp://en.wikipedia.org/wiki/Best,_worst_and_average_casehttp://en.wikipedia.org/wiki/Graph_%28data_structure%29http://en.wikipedia.org/wiki/Search_algorithmhttp://en.wikipedia.org/wiki/Dykstra%27s_projection_algorithmhttp://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#p-searchhttp://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#mw-head


2/12

Depth-limited Dijkstra Edmond

Floyd-Warshall

Hill climbing Iterative deepening

Kruskal Johnson

Lexicographic BFS Prim

Uniform-cost

Listings

Graph algorithms Search algorithms

List of graph algorithms

Related topics

Dynamic programming Graph traversal Tree traversal Search games

v t e

Dijkstra's algorithm , conceived by Dutch computer scientistEdsger Dijkstra in 1956 andpublished in 1959 ,[1][2] is a graph search algorithm that solves the single-source shortest pathproblem for a graph with nonnegative edge path costs, producing a shortest path tree . Thisalgorithm 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 costsof shortest paths from a single vertex to a single destination vertex by stopping the algorithmonce the shortest path to the destination vertex has been determined. For example, if the vertices
http://en.wikipedia.org/wiki/Depth-first_searchhttp://en.wikipedia.org/wiki/Depth-first_searchhttp://en.wikipedia.org/wiki/Depth-limited_searchhttp://en.wikipedia.org/wiki/Depth-limited_searchhttp://en.wikipedia.org/wiki/Edmonds%27_algorithmhttp://en.wikipedia.org/wiki/Edmonds%27_algorithmhttp://en.wikipedia.org/wiki/Floyd%E2%80%93Warshall_algorithmhttp://en.wikipedia.org/wiki/Floyd%E2%80%93Warshall_algorithmhttp://en.wikipedia.org/wiki/Hill_climbinghttp://en.wikipedia.org/wiki/Hill_climbinghttp://en.wikipedia.org/wiki/Iterative_deepening_depth-first_searchhttp://en.wikipedia.org/wiki/Iterative_deepening_depth-first_searchhttp://en.wikipedia.org/wiki/Kruskal%27s_algorithmhttp://en.wikipedia.org/wiki/Kruskal%27s_algorithmhttp://en.wikipedia.org/wiki/Johnson%27s_algorithmhttp://en.wikipedia.org/wiki/Johnson%27s_algorithmhttp://en.wikipedia.org/wiki/Lexicographic_breadth-first_searchhttp://en.wikipedia.org/wiki/Lexicographic_breadth-first_searchhttp://en.wikipedia.org/wiki/Prim%27s_algorithmhttp://en.wikipedia.org/wiki/Prim%27s_algorithmhttp://en.wikipedia.org/wiki/Uniform-cost_searchhttp://en.wikipedia.org/wiki/Uniform-cost_searchhttp://en.wikipedia.org/wiki/Category:Graph_algorithmshttp://en.wikipedia.org/wiki/Category:Graph_algorithmshttp://en.wikipedia.org/wiki/Category:Search_algorithmshttp://en.wikipedia.org/wiki/Category:Search_algorithmshttp://en.wikipedia.org/wiki/List_of_algorithms#Graph_algorithmshttp://en.wikipedia.org/wiki/List_of_algorithms#Graph_algorithmshttp://en.wikipedia.org/wiki/Dynamic_programminghttp://en.wikipedia.org/wiki/Dynamic_programminghttp://en.wikipedia.org/wiki/Graph_traversalhttp://en.wikipedia.org/wiki/Graph_traversalhttp://en.wikipedia.org/wiki/Tree_traversalhttp://en.wikipedia.org/wiki/Tree_traversalhttp://en.wikipedia.org/wiki/Search_gamehttp://en.wikipedia.org/wiki/Search_gamehttp://en.wikipedia.org/wiki/Template:Graph_search_algorithmhttp://en.wikipedia.org/wiki/Template:Graph_search_algorithmhttp://en.wikipedia.org/wiki/Template_talk:Graph_search_algorithmhttp://en.wikipedia.org/wiki/Template_talk:Graph_search_algorithmhttp://en.wikipedia.org/w/index.php?title=Template:Graph_search_algorithm&action=edithttp://en.wikipedia.org/w/index.php?title=Template:Graph_search_algorithm&action=edithttp://en.wikipedia.org/wiki/Computer_scientisthttp://en.wikipedia.org/wiki/Computer_scientisthttp://en.wikipedia.org/wiki/Computer_scientisthttp://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#cite_note-Dijkstra_Interview-0http://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#cite_note-Dijkstra_Interview-0http://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#cite_note-Dijkstra_Interview-0http://en.wikipedia.org/wiki/Graph_search_algorithmhttp://en.wikipedia.org/wiki/Graph_search_algorithmhttp://en.wikipedia.org/wiki/Graph_search_algorithmhttp://en.wikipedia.org/wiki/Shortest_path_problemhttp://en.wikipedia.org/wiki/Shortest_path_problemhttp://en.wikipedia.org/wiki/Shortest_path_problemhttp://en.wikipedia.org/wiki/Shortest_path_problemhttp://en.wikipedia.org/wiki/Graph_%28mathematics%29http://en.wikipedia.org/wiki/Graph_%28mathematics%29http://en.wikipedia.org/wiki/Graph_%28mathematics%29http://en.wikipedia.org/wiki/Edge_%28graph_theory%29http://en.wikipedia.org/wiki/Edge_%28graph_theory%29http://en.wikipedia.org/wiki/Edge_%28graph_theory%29http://en.wikipedia.org/wiki/Shortest_path_treehttp://en.wikipedia.org/wiki/Shortest_path_treehttp://en.wikipedia.org/wiki/Shortest_path_treehttp://en.wikipedia.org/wiki/Routinghttp://en.wikipedia.org/wiki/Routinghttp://en.wikipedia.org/wiki/Routinghttp://en.wikipedia.org/wiki/Vertex_%28graph_theory%29http://en.wikipedia.org/wiki/Vertex_%28graph_theory%29http://en.wikipedia.org/wiki/Vertex_%28graph_theory%29http://en.wikipedia.org/wiki/Vertex_%28graph_theory%29http://en.wikipedia.org/wiki/Routinghttp://en.wikipedia.org/wiki/Shortest_path_treehttp://en.wikipedia.org/wiki/Edge_%28graph_theory%29http://en.wikipedia.org/wiki/Graph_%28mathematics%29http://en.wikipedia.org/wiki/Shortest_path_problemhttp://en.wikipedia.org/wiki/Shortest_path_problemhttp://en.wikipedia.org/wiki/Graph_search_algorithmhttp://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#cite_note-Dijkstra_Interview-0http://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#cite_note-Dijkstra_Interview-0http://en.wikipedia.org/wiki/Computer_scientisthttp://en.wikipedia.org/wiki/Computer_scientisthttp://en.wikipedia.org/w/index.php?title=Template:Graph_search_algorithm&action=edithttp://en.wikipedia.org/wiki/Template_talk:Graph_search_algorithmhttp://en.wikipedia.org/wiki/Template:Graph_search_algorithmhttp://en.wikipedia.org/wiki/Search_gamehttp://en.wikipedia.org/wiki/Tree_traversalhttp://en.wikipedia.org/wiki/Graph_traversalhttp://en.wikipedia.org/wiki/Dynamic_programminghttp://en.wikipedia.org/wiki/List_of_algorithms#Graph_algorithmshttp://en.wikipedia.org/wiki/Category:Search_algorithmshttp://en.wikipedia.org/wiki/Category:Graph_algorithmshttp://en.wikipedia.org/wiki/Uniform-cost_searchhttp://en.wikipedia.org/wiki/Prim%27s_algorithmhttp://en.wikipedia.org/wiki/Lexicographic_breadth-first_searchhttp://en.wikipedia.org/wiki/Johnson%27s_algorithmhttp://en.wikipedia.org/wiki/Kruskal%27s_algorithmhttp://en.wikipedia.org/wiki/Iterative_deepening_depth-first_searchhttp://en.wikipedia.org/wiki/Hill_climbinghttp://en.wikipedia.org/wiki/Floyd%E2%80%93Warshall_algorithmhttp://en.wikipedia.org/wiki/Edmonds%27_algorithmhttp://en.wikipedia.org/wiki/Depth-limited_search


3/12

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 routebetween 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).

Dijkstra's original algorithm does not use a min-priority queue and runs in O(|V |2

). The idea of this algorithm is also given in (Leyzorek et al. 1957 ). The implementation based on a min-priority queue implemented by a Fibonacci heap and running in O(| E | + |V | log | V |) is due to(Fredman & Tarjan 1984 ). This is asymptotically the fastest known single-source shortest-pathalgorithm for arbitrary directed graphs with unbounded nonnegative weights. (For an overviewof earlier shortest path algorithms and later improvements and adaptations, see: Single-sourceshortest-paths algorithms for directed graphs with nonnegative weights .)

Contents

[hide ]

1 Algorithm 2 Description 3 Pseudocode 4 Running time 5 Related problems and algorithms 6 Dynamic programming perspective 7 See also 8 Notes 9 References 10 External links

[edit ] Algorithm

Illustration of Dijkstra's algorithm search for finding path from a start node to a goal node in arobotmotion planning problem. Open nodes represent the "tentative" set. Filled nodes are visited
http://en.wikipedia.org/wiki/Routing_protocolhttp://en.wikipedia.org/wiki/Routing_protocolhttp://en.wikipedia.org/wiki/IS-IShttp://en.wikipedia.org/wiki/IS-IShttp://en.wikipedia.org/wiki/IS-IShttp://en.wikipedia.org/wiki/OSPFhttp://en.wikipedia.org/wiki/OSPFhttp://en.wikipedia.org/wiki/OSPFhttp://en.wikipedia.org/wiki/Min-priority_queuehttp://en.wikipedia.org/wiki/Min-priority_queuehttp://en.wikipedia.org/wiki/Min-priority_queuehttp://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#CITEREFLeyzorekGrayJohnsonLadew1957http://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#CITEREFLeyzorekGrayJohnsonLadew1957http://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#CITEREFLeyzorekGrayJohnsonLadew1957http://en.wikipedia.org/wiki/Fibonacci_heaphttp://en.wikipedia.org/wiki/Fibonacci_heaphttp://en.wikipedia.org/wiki/Fibonacci_heaphttp://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#CITEREFFredmanTarjan1984http://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#CITEREFFredmanTarjan1984http://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#CITEREFFredmanTarjan1984http://en.wikipedia.org/wiki/Asymptotic_computational_complexityhttp://en.wikipedia.org/wiki/Asymptotic_computational_complexityhttp://en.wikipedia.org/wiki/Asymptotic_computational_complexityhttp://en.wikipedia.org/wiki/Single-source_shortest-paths_algorithms_for_directed_graphs_with_nonnegative_weightshttp://en.wikipedia.org/wiki/Single-source_shortest-paths_algorithms_for_directed_graphs_with_nonnegative_weightshttp://en.wikipedia.org/wiki/Single-source_shortest-paths_algorithms_for_directed_graphs_with_nonnegative_weightshttp://en.wikipedia.org/wiki/Single-source_shortest-paths_algorithms_for_directed_graphs_with_nonnegative_weightshttp://en.wikipedia.org/wiki/Dijkstra%27s_algorithmhttp://en.wikipedia.org/wiki/Dijkstra%27s_algorithmhttp://en.wikipedia.org/wiki/Dijkstra%27s_algorithmhttp://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#Algorithmhttp://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#Algorithmhttp://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#Descriptionhttp://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#Descriptionhttp://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#Pseudocodehttp://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#Pseudocodehttp://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#Running_timehttp://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#Running_timehttp://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#Related_problems_and_algorithmshttp://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#Related_problems_and_algorithmshttp://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#Dynamic_programming_perspectivehttp://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#Dynamic_programming_perspectivehttp://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#See_alsohttp://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#See_alsohttp://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#Noteshttp://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#Noteshttp://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#Referenceshttp://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#Referenceshttp://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#External_linkshttp://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#External_linkshttp://en.wikipedia.org/w/index.php?title=Dijkstra%27s_algorithm&action=edit&section=1http://en.wikipedia.org/w/index.php?title=Dijkstra%27s_algorithm&action=edit&section=1http://en.wikipedia.org/wiki/Roboticshttp://en.wikipedia.org/wiki/Roboticshttp://en.wikipedia.org/wiki/File:Dijkstras_progress_animation.gifhttp://en.wikipedia.org/wiki/File:Dijkstras_progress_animation.gifhttp://en.wikipedia.org/wiki/File:Dijkstras_progress_animation.gifhttp://en.wikipedia.org/wiki/File:Dijkstras_progress_animation.gifhttp://en.wikipedia.org/wiki/Roboticshttp://en.wikipedia.org/wiki/Roboticshttp://en.wikipedia.org/w/index.php?title=Dijkstra%27s_algorithm&action=edit&section=1http://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#External_linkshttp://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#Referenceshttp://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#Noteshttp://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#See_alsohttp://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#Dynamic_programming_perspectivehttp://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#Related_problems_and_algorithmshttp://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#Running_timehttp://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#Pseudocodehttp://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#Descriptionhttp://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#Algorithmhttp://en.wikipedia.org/wiki/Dijkstra%27s_algorithmhttp://en.wikipedia.org/wiki/Single-source_shortest-paths_algorithms_for_directed_graphs_with_nonnegative_weightshttp://en.wikipedia.org/wiki/Single-source_shortest-paths_algorithms_for_directed_graphs_with_nonnegative_weightshttp://en.wikipedia.org/wiki/Asymptotic_computational_complexityhttp://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#CITEREFFredmanTarjan1984http://en.wikipedia.org/wiki/Fibonacci_heaphttp://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#CITEREFLeyzorekGrayJohnsonLadew1957http://en.wikipedia.org/wiki/Min-priority_queuehttp://en.wikipedia.org/wiki/OSPFhttp://en.wikipedia.org/wiki/IS-IShttp://en.wikipedia.org/wiki/Routing_protocol


4/12

ones, with color representing the distance: the greener, the further. Nodes in all the differentdirections are explored uniformly, appearing as a more-or-less circular wavefront as Dijkstra'salgorithm uses a heuristic identically equal to 0.

Let the node at which we are starting be called the initial node . Let the distance of node Y be

the distance from the initial node to Y. Dijkstra's algorithm will assign some initial distancevalues and will try to improve them step by step.

1. Assign to every node a tentative distance value: set it to zero for our initial node and toinfinity for all other nodes.

2. Mark all nodes unvisited. Set the initial node as current. Create a set of the unvisitednodes called the unvisited set consisting of all the nodes except the initial node.

3. For the current node, consider all of its unvisited neighbors and calculate their tentative distances. For example, if the current node A is marked with a tentative distance of 6, andthe edge connecting it with a neighbor B has length 2, then the distance to B (through A)will be 6+2=8. If this distance is less than the previously recorded tentative distance of B,

then overwrite that distance. Even though a neighbor has been examined, it is not markedas visited at this time, and it remains in the unvisited set .4. When we are done considering all of the neighbors of the current node, mark the current

node as visited and remove it from the unvisited set . A visited node will never be checkedagain; its distance recorded now is final and minimal.

5. If the destination node has been marked visited (when planning a route between twospecific nodes) or if the smallest tentative distance among the nodes in the unvisited set isinfinity (when planning a complete traversal), then stop. The algorithm has finished.

6. Set the unvisited node marked with the smallest tentative distance as the next "currentnode" and go back to step 3.

[edit ] Description Note: For ease of understanding, this discussion uses the terms intersection , road and map however, formally these terms are vertex , edge and graph , respectively.

Suppose you want to find the shortest path between two intersections on a city map, a startingpoint and a destination. The order is conceptually simple: to start, mark the distance to everyintersection on the map with infinity. This is done not to imply there is an infinite distance, but tonote that that intersection has not yet been visited ; some variants of this method simply leave theintersection unlabeled. Now, at each iteration, select a current intersection. For the first iterationthe current intersection will be the starting point and the distance to it (the intersection's label)

will be zero. For subsequent iterations (after the first) the current intersection will be the closestunvisited intersection to the starting point this will be easy to find.

From the current intersection, update the distance to every unvisited intersection that is directlyconnected to it. This is done by determining the sum of the distance between an unvisitedintersection and the value of the current intersection, and relabeling the unvisited intersectionwith this value if it is less than its current value. In effect, the intersection is relabeled if the pathto it through the current intersection is shorter than the previously known paths. To facilitate
http://en.wikipedia.org/wiki/Wavefronthttp://en.wikipedia.org/wiki/Wavefronthttp://en.wikipedia.org/wiki/Wavefronthttp://en.wikipedia.org/wiki/Consistent_heuristichttp://en.wikipedia.org/wiki/Consistent_heuristichttp://en.wikipedia.org/wiki/Consistent_heuristichttp://en.wikipedia.org/w/index.php?title=Dijkstra%27s_algorithm&action=edit&section=2http://en.wikipedia.org/w/index.php?title=Dijkstra%27s_algorithm&action=edit&section=2http://en.wikipedia.org/wiki/Intersection_%28road%29http://en.wikipedia.org/wiki/Intersection_%28road%29http://en.wikipedia.org/wiki/Intersection_%28road%29http://en.wikipedia.org/wiki/Graph_labelinghttp://en.wikipedia.org/wiki/Graph_labelinghttp://en.wikipedia.org/wiki/Graph_labelinghttp://en.wikipedia.org/wiki/Graph_labelinghttp://en.wikipedia.org/wiki/Intersection_%28road%29http://en.wikipedia.org/w/index.php?title=Dijkstra%27s_algorithm&action=edit&section=2http://en.wikipedia.org/wiki/Consistent_heuristichttp://en.wikipedia.org/wiki/Wavefront


5/12

shortest path identification, in pencil, mark the road with an arrow pointing to the relabeledintersection if you label/relabel it, and erase all others pointing to it. After you have updated thedistances to each neighboring intersection , mark the current intersection as visited and select theunvisited intersection with lowest distance (from the starting point) -- or lowest label as thecurrent intersection. Nodes marked as visited are labeled with the shortest path from the starting

point to it and will not be revisited or returned to.

Continue this process of updating the neighboring intersections with the shortest distances, thenmarking the current intersection as visited and moving onto the closest unvisited intersectionuntil you have marked the destination as visited. Once you have marked the destination as visited(as is the case with any visited intersection) you have determined the shortest path to it, from thestarting point, and can trace your way back, following the arrows in reverse.

Of note is the fact that this algorithm makes no attempt to direct "exploration" towards thedestination as one might expect. Rather, the sole consideration in determining the next "current"intersection is its distance from the starting point. In some sense, this algorithm "expands

outward" from the starting point, iteratively considering every node that is closer in terms of shortest path distance until it reaches the destination. When understood in this way, it is clearhow the algorithm necessarily finds the shortest path, however it may also reveal one of thealgorithm's weaknesses: its relative slowness in some topologies.

[edit ] Pseudocode

In the following algorithm, the code u := vertex in Q with smallest dist[] , searches forthe vertex u in the vertex set Q that has the least dist[ u] value. That vertex is removed from theset Q and returned to the user. dist_between( u, v) calculates the length between the twoneighbor-nodes u and v. The variable alt on line 15 is the length of the path from the root node tothe neighbor node v if it were to go through u. If this path is shorter than the current shortest pathrecorded for v, that current path is replaced with this alt path. The previous array is populatedwith a pointer to the "next-hop" node on the source graph to get the shortest route to the source.

1 function Dijkstra( Graph , source ):2 for each vertex v in Graph : // Initializations 3 dist[ v ] := infinity ; // Unknown distance function

from source to v 4 previous[ v ] := undefined ; // Previous node in optimal

path from source 5 end for ;6 dist[ source ] := 0 ; // Distance from source to

source 7 Q := the set of all nodes in Graph ; // All nodes in the graph are

unoptimized - thus are in Q 8 while Q is not empty: // The main loop 9 u := vertex in Q with smallest distance in dist[] ;

10 if dist[ u ] = infinity:11 break ; // all remaining vertices areinaccessible from source 12 end if ;13 remove u from Q ;
http://en.wikipedia.org/wiki/Neighbourhood_%28graph_theory%29http://en.wikipedia.org/wiki/Neighbourhood_%28graph_theory%29http://en.wikipedia.org/wiki/Neighbourhood_%28graph_theory%29http://en.wikipedia.org/w/index.php?title=Dijkstra%27s_algorithm&action=edit&section=3http://en.wikipedia.org/w/index.php?title=Dijkstra%27s_algorithm&action=edit&section=3http://en.wikipedia.org/w/index.php?title=Dijkstra%27s_algorithm&action=edit&section=3http://en.wikipedia.org/wiki/Neighbourhood_%28graph_theory%29


6/12

14 for each neighbor v of u : // where v has not yet beenremoved from Q. 15 alt := dist[ u ] + dist_between( u , v ) ;16 if alt < dist[ v ]: // Relax (u,v,a) 17 dist[ v ] := alt ;18 previous[ v ] := u ;19 decrease-key v in Q ; // Reorder v in the Queue 20 end if ;21 end for ;22 end while ;23 return dist[] ;24 end Dijkstra.

If we are only interested in a shortest path between vertices source and target , we can terminatethe search at line 13 if u = target . Now we can read the shortest path from source to target byiteration:

1 S := empty sequence2 u := target 3 while previous[ u ] is defined:4 insert u at the beginning of S 5 u := previous[ u ]6 end while ;

Now sequence S is the list of vertices constituting one of the shortest paths from source to target ,or the empty sequence if no path exists.

A more general problem would be to find all the shortest paths between source and target (theremight be several different ones of the same length). Then instead of storing only a single node ineach entry of previous[] we would store all nodes satisfying the relaxation condition. Forexample, if both r and source connect to target and both of them lie on different shortest pathsthrough target (because the edge cost is the same in both cases), then we would add both r andsource to previous[ target ] . When the algorithm completes, previous[] data structure willactually describe a graph that is a subset of the original graph with some edges removed. Its keyproperty will be that if the algorithm was run with some starting node, then every path from thatnode to any other node in the new graph will be the shortest path between those nodes in theoriginal graph, and all paths of that length from the original graph will be present in the newgraph. Then to actually find all these short paths between two given nodes we would use a pathfinding algorithm on the new graph, such as depth-first search.

[edit ] Running time

An upper bound of the running time of Dijkstra's algorithm on a graph with edges and verticescan be expressed as a function of and using Big-O notation .

For any implementation of vertex set the running time is ,

where and are times needed to perform decrease key and extract minimum operationsin set , respectively.
http://en.wikipedia.org/w/index.php?title=Dijkstra%27s_algorithm&action=edit&section=4http://en.wikipedia.org/w/index.php?title=Dijkstra%27s_algorithm&action=edit&section=4http://en.wikipedia.org/wiki/Big_O_notation#Graph_theoryhttp://en.wikipedia.org/wiki/Big_O_notation#Graph_theoryhttp://en.wikipedia.org/wiki/Big_O_notation#Graph_theoryhttp://en.wikipedia.org/wiki/Big_O_notation#Graph_theoryhttp://en.wikipedia.org/w/index.php?title=Dijkstra%27s_algorithm&action=edit&section=4


7/12

The simplest implementation of the Dijkstra's algorithm stores vertices of set in an ordinarylinked list or array, and extract minimum from is simply a linear search through all vertices in

. In this case, the running time is .

For sparse graphs , that is, graphs with far fewer than edges, Dijkstra's algorithm can beimplemented more efficiently by storing the graph in the form of adjacency lists and using abinary heap , pairing heap , or Fibonacci heap as a priority queue to implement extracting

minimum efficiently. With a binary heap, the algorithm requirestime (which is dominated by O( | E | log | V | ), assuming the graph is connected). To avoidO(|V|) look-up in decrease-key step on a vanilla binary heap, it is necessary to maintain asupplementary index mapping each vertex to the heap's index (and keep it up to date as priority

queue changes), making it take only time instead. The Fibonacci heap improves

this to .

Note that for Directed acyclic graphs , it is possible to find shortest paths from a given startingvertex in linear time, by processing the vertices in a topological order, and calculating the pathlength for each vertex to be the minimum length obtained via any of its incoming edges .[3]

[edit ] Related problems and algorithms

The functionality of Dijkstra's original algorithm can be extended with a variety of modifications. For example, sometimes it is desirable to present solutions which are less thanmathematically optimal. To obtain a ranked list of less-than-optimal solutions, the optimalsolution is first calculated. A single edge appearing in the optimal solution is removed from thegraph, and the optimum solution to this new graph is calculated. Each edge of the originalsolution is suppressed in turn and a new shortest-path calculated. The secondary solutions arethen ranked and presented after the first optimal solution.

Dijkstra's algorithm is usually the working principle behind link-state routing protocols , OSPF and IS-IS being the most common ones.

Unlike Dijkstra's algorithm, the Bellman-Ford algorithm can be used on graphs with negativeedge weights, as long as the graph contains no negative cycle reachable from the source vertex s.(The presence of such cycles means there is no shortest path, since the total weight becomeslower each time the cycle is traversed.)

The A* algorithm is a generalization of Dijkstra's algorithm that cuts down on the size of thesubgraph that must be explored, if additional information is available that provides a lowerbound on the "distance" to the target. This approach can be viewed from the perspective of linearprogramming : there is a natural linear program for computing shortest paths , and solutions to itsdual linear program are feasible if and only if they form a consistent heuristic (speaking roughly,since the sign conventions differ from place to place in the literature). This feasible dual / consistent heuristic defines a nonnegative reduced cost and A* is essentially running Dijkstra's
http://en.wikipedia.org/wiki/Sparse_graphhttp://en.wikipedia.org/wiki/Sparse_graphhttp://en.wikipedia.org/wiki/Sparse_graphhttp://en.wikipedia.org/wiki/Adjacency_listhttp://en.wikipedia.org/wiki/Adjacency_listhttp://en.wikipedia.org/wiki/Adjacency_listhttp://en.wikipedia.org/wiki/Binary_heaphttp://en.wikipedia.org/wiki/Binary_heaphttp://en.wikipedia.org/wiki/Pairing_heaphttp://en.wikipedia.org/wiki/Pairing_heaphttp://en.wikipedia.org/wiki/Pairing_heaphttp://en.wikipedia.org/wiki/Fibonacci_heaphttp://en.wikipedia.org/wiki/Fibonacci_heaphttp://en.wikipedia.org/wiki/Fibonacci_heaphttp://en.wikipedia.org/wiki/Priority_queuehttp://en.wikipedia.org/wiki/Priority_queuehttp://en.wikipedia.org/wiki/Priority_queuehttp://en.wikipedia.org/wiki/Fibonacci_heaphttp://en.wikipedia.org/wiki/Fibonacci_heaphttp://en.wikipedia.org/wiki/Fibonacci_heaphttp://en.wikipedia.org/wiki/Directed_acyclic_graphhttp://en.wikipedia.org/wiki/Directed_acyclic_graphhttp://en.wikipedia.org/wiki/Directed_acyclic_graphhttp://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#cite_note-2http://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#cite_note-2http://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#cite_note-2http://en.wikipedia.org/w/index.php?title=Dijkstra%27s_algorithm&action=edit&section=5http://en.wikipedia.org/w/index.php?title=Dijkstra%27s_algorithm&action=edit&section=5http://en.wikipedia.org/wiki/Link-state_routing_protocolhttp://en.wikipedia.org/wiki/Link-state_routing_protocolhttp://en.wikipedia.org/wiki/Link-state_routing_protocolhttp://en.wikipedia.org/wiki/OSPFhttp://en.wikipedia.org/wiki/OSPFhttp://en.wikipedia.org/wiki/OSPFhttp://en.wikipedia.org/wiki/IS-IShttp://en.wikipedia.org/wiki/IS-IShttp://en.wikipedia.org/wiki/IS-IShttp://en.wikipedia.org/wiki/Bellman-Ford_algorithmhttp://en.wikipedia.org/wiki/Bellman-Ford_algorithmhttp://en.wikipedia.org/wiki/Bellman-Ford_algorithmhttp://en.wikipedia.org/wiki/Negative_cyclehttp://en.wikipedia.org/wiki/Negative_cyclehttp://en.wikipedia.org/wiki/Negative_cyclehttp://en.wikipedia.org/wiki/A-star_algorithmhttp://en.wikipedia.org/wiki/A-star_algorithmhttp://en.wikipedia.org/wiki/A-star_algorithmhttp://en.wikipedia.org/wiki/Linear_programminghttp://en.wikipedia.org/wiki/Linear_programminghttp://en.wikipedia.org/wiki/Linear_programminghttp://en.wikipedia.org/wiki/Linear_programminghttp://en.wikipedia.org/wiki/Shortest_path_problem#Linear_programming_formulationhttp://en.wikipedia.org/wiki/Shortest_path_problem#Linear_programming_formulationhttp://en.wikipedia.org/wiki/Shortest_path_problem#Linear_programming_formulationhttp://en.wikipedia.org/wiki/Dual_linear_programhttp://en.wikipedia.org/wiki/Dual_linear_programhttp://en.wikipedia.org/wiki/Consistent_heuristichttp://en.wikipedia.org/wiki/Consistent_heuristichttp://en.wikipedia.org/wiki/Consistent_heuristichttp://en.wikipedia.org/wiki/Reduced_costhttp://en.wikipedia.org/wiki/Reduced_costhttp://en.wikipedia.org/wiki/Reduced_costhttp://en.wikipedia.org/wiki/Reduced_costhttp://en.wikipedia.org/wiki/Consistent_heuristichttp://en.wikipedia.org/wiki/Dual_linear_programhttp://en.wikipedia.org/wiki/Shortest_path_problem#Linear_programming_formulationhttp://en.wikipedia.org/wiki/Linear_programminghttp://en.wikipedia.org/wiki/Linear_programminghttp://en.wikipedia.org/wiki/A-star_algorithmhttp://en.wikipedia.org/wiki/Negative_cyclehttp://en.wikipedia.org/wiki/Bellman-Ford_algorithmhttp://en.wikipedia.org/wiki/IS-IShttp://en.wikipedia.org/wiki/OSPFhttp://en.wikipedia.org/wiki/Link-state_routing_protocolhttp://en.wikipedia.org/w/index.php?title=Dijkstra%27s_algorithm&action=edit&section=5http://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#cite_note-2http://en.wikipedia.org/wiki/Directed_acyclic_graphhttp://en.wikipedia.org/wiki/Fibonacci_heaphttp://en.wikipedia.org/wiki/Priority_queuehttp://en.wikipedia.org/wiki/Fibonacci_heaphttp://en.wikipedia.org/wiki/Pairing_heaphttp://en.wikipedia.org/wiki/Binary_heaphttp://en.wikipedia.org/wiki/Adjacency_listhttp://en.wikipedia.org/wiki/Sparse_graph


8/12

algorithm with these reduced costs. If the dual satisfies the weaker condition of admissibility,then A* is instead more akin to the Bellman-Ford algorithm.

The process that underlies Dijkstra's algorithm is similar to the greedy process used in Prim'salgorithm . Prim's purpose is to find a minimum spanning tree that connects all nodes in the

graph; Dijkstra is concerned with only two nodes. Prim's does not evaluate the total weight of thepath from the starting node, only the individual path.

[edit ] Dynamic programming perspective

From a dynamic programming point of view, Dijkstra's algorithm is a successive approximationscheme that solves the dynamic programming functional equation for the shortest path problemby the Reaching method .[4][5 ][6]

In fact, Dijkstra's explanation of the logic behind the algorithm ,[7] namely

Problem 2. Find the path of minimum total length between two given nodes and .

We use the fact that, if is a node on the minimal path from to , knowledge of the latterimplies the knowledge of the minimal path from to .

is a paraphrasing of Bellman's famous Principle of Optimality in the context of the shortest pathproblem.

[edit ] See also

Computer science portal

Euclidean shortest path Flood fill Longest path problem

[edit ] Notes

1. ^ Dijkstra, Edsger; Thomas J. Misa, Editor (2010-08). "An Interview with Edsger W.Dijkstra". Communications of the ACM 53 (8): 41 47. doi :10.1145/1787234.1787249 . "What is the shortest way to travel from Rotterdam to Groningen? It is the algorithm forthe shortest path which I designed in about 20 minutes. One morning I was shopping withmy young fiance, and tired, we sat down on the caf terrace to drink a cup of coffee andI was just thinking about whether I could do this, and I then designed the algorithm forthe shortest path."

2. ^ Dijkstra 1959 3. ^ http://www.boost.org/doc/libs/1_44_0/libs/graph/doc/dag_shortest_paths.html
http://en.wikipedia.org/wiki/Prim%27s_algorithmhttp://en.wikipedia.org/wiki/Prim%27s_algorithmhttp://en.wikipedia.org/wiki/Prim%27s_algorithmhttp://en.wikipedia.org/wiki/Prim%27s_algorithmhttp://en.wikipedia.org/wiki/Minimum_spanning_treehttp://en.wikipedia.org/wiki/Minimum_spanning_treehttp://en.wikipedia.org/wiki/Minimum_spanning_treehttp://en.wikipedia.org/w/index.php?title=Dijkstra%27s_algorithm&action=edit&section=6http://en.wikipedia.org/w/index.php?title=Dijkstra%27s_algorithm&action=edit&section=6http://en.wikipedia.org/wiki/Dynamic_programminghttp://en.wikipedia.org/wiki/Dynamic_programminghttp://en.wikipedia.org/wiki/Dynamic_programminghttp://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#cite_note-sniedovich_06-3http://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#cite_note-sniedovich_06-3http://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#cite_note-sniedovich_10-5http://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#cite_note-sniedovich_10-5http://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#cite_note-6http://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#cite_note-6http://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#cite_note-6http://en.wikipedia.org/wiki/Richard_Bellmanhttp://en.wikipedia.org/wiki/Richard_Bellmanhttp://en.wikipedia.org/wiki/Richard_Bellmanhttp://en.wikipedia.org/wiki/Principle_of_Optimalityhttp://en.wikipedia.org/wiki/Principle_of_Optimalityhttp://en.wikipedia.org/wiki/Principle_of_Optimalityhttp://en.wikipedia.org/w/index.php?title=Dijkstra%27s_algorithm&action=edit&section=7http://en.wikipedia.org/w/index.php?title=Dijkstra%27s_algorithm&action=edit&section=7http://en.wikipedia.org/wiki/Portal:Computer_sciencehttp://en.wikipedia.org/wiki/Portal:Computer_sciencehttp://en.wikipedia.org/wiki/Euclidean_shortest_pathhttp://en.wikipedia.org/wiki/Euclidean_shortest_pathhttp://en.wikipedia.org/wiki/Flood_fillhttp://en.wikipedia.org/wiki/Flood_fillhttp://en.wikipedia.org/wiki/Longest_path_problemhttp://en.wikipedia.org/wiki/Longest_path_problemhttp://en.wikipedia.org/w/index.php?title=Dijkstra%27s_algorithm&action=edit&section=8http://en.wikipedia.org/w/index.php?title=Dijkstra%27s_algorithm&action=edit&section=8http://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#cite_ref-Dijkstra_Interview_0-0http://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#cite_ref-Dijkstra_Interview_0-0http://en.wikipedia.org/wiki/Digital_object_identifierhttp://en.wikipedia.org/wiki/Digital_object_identifierhttp://dx.doi.org/10.1145%2F1787234.1787249http://dx.doi.org/10.1145%2F1787234.1787249http://dx.doi.org/10.1145%2F1787234.1787249http://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#cite_ref-1http://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#cite_ref-1http://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#cite_ref-1http://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#cite_ref-2http://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#cite_ref-2http://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#cite_ref-2http://en.wikipedia.org/wiki/File:Internet_map_1024.jpghttp://en.wikipedia.org/wiki/File:Internet_map_1024.jpghttp://en.wikipedia.org/wiki/File:Internet_map_1024.jpghttp://en.wikipedia.org/wiki/File:Internet_map_1024.jpghttp://en.wikipedia.org/wiki/File:Internet_map_1024.jpghttp://en.wikipedia.org/wiki/File:Internet_map_1024.jpghttp://en.wikipedia.org/wiki/File:Internet_map_1024.jpghttp://en.wikipedia.org/wiki/File:Internet_map_1024.jpghttp://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#cite_ref-2http://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#cite_ref-2http://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#cite_ref-1http://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#cite_ref-1http://dx.doi.org/10.1145%2F1787234.1787249http://en.wikipedia.org/wiki/Digital_object_identifierhttp://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#cite_ref-Dijkstra_Interview_0-0http://en.wikipedia.org/w/index.php?title=Dijkstra%27s_algorithm&action=edit&section=8http://en.wikipedia.org/wiki/Longest_path_problemhttp://en.wikipedia.org/wiki/Flood_fillhttp://en.wikipedia.org/wiki/Euclidean_shortest_pathhttp://en.wikipedia.org/wiki/Portal:Computer_sciencehttp://en.wikipedia.org/w/index.php?title=Dijkstra%27s_algorithm&action=edit&section=7http://en.wikipedia.org/wiki/Principle_of_Optimalityhttp://en.wikipedia.org/wiki/Richard_Bellmanhttp://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#cite_note-6http://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#cite_note-sniedovich_10-5http://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#cite_note-sniedovich_06-3http://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#cite_note-sniedovich_06-3http://en.wikipedia.org/wiki/Dynamic_programminghttp://en.wikipedia.org/w/index.php?title=Dijkstra%27s_algorithm&action=edit&section=6http://en.wikipedia.org/wiki/Minimum_spanning_treehttp://en.wikipedia.org/wiki/Prim%27s_algorithmhttp://en.wikipedia.org/wiki/Prim%27s_algorithm


9/12

4. ^ Sniedovich, M. (2006). "Dijkstras algorithm revisited: the dynamic programmingconnexion" (PDF ). Journal of Control and Cybernetics 35 (3): 599 620. Online versionof the paper with interactive computational modules.

5. ^ Denardo, E.V. (2003). Dynamic Programming: Models and Applications . Mineola,NY: Dover Publications . ISBN 978-0486428109 .

6.

^ Sniedovich, M. (2010). Dynamic Programming: Foundations and Principles . Francis& Taylor . ISBN 9780824740993 . 7. ^ Dijkstra 1959 , p. 270

[edit ] References

Dijkstra, E. W. (1959). "A note on two problems in connexion with graphs" . Numerische Mathematik 1: 269 271. doi :10.1007/BF01386390 .

Cormen, Thomas H. ; Leiserson, Charles E. ; Rivest, Ronald L. ; Stein, Clifford (2001)."Section 24.3: Dijkstra's algorithm". Introduction to Algorithms (Second ed.). MIT Press and McGraw-Hill . pp. 595 601. ISBN 0-262-03293-7 .

Fredman, Michael Lawrence ; Tarjan, Robert E. (1984). "Fibonacci heaps and their usesin improved network optimization algorithms" . 25th Annual Symposium on Foundationsof Computer Science (IEEE ): 338 346. doi :10.1109/SFCS.1984.715934 .

Fredman, Michael Lawrence ; Tarjan, Robert E. (1987). "Fibonacci heaps and their usesin improved network optimization algorithms" . Journal of the Association for Computing

Machinery 34 (3): 596 615. doi :10.1145/28869.28874 . Zhan, F. Benjamin; Noon, Charles E. (February 1998). "Shortest Path Algorithms: An

Evaluation Using Real Road Networks". Transportation Science 32 (1): 65 73.doi :10.1287/trsc.32.1.65 .

Leyzorek, M.; Gray, R. S.; Johnson, A. A.; Ladew, W. C.; Meaker, Jr., S. R.; Petry, R.M.; Seitz, R. N. (1957). Investigation of Model Techniques First Annual Report 6

June 1956 1 July 1957 A Study of Model Techniques for Communication Systems .Cleveland, Ohio: Case Institute of Technology.

[edit ] External links

Wikimedia Commons has media related to: Dijkstra's algorithm

Oral history interview with Edsger W. Dijkstra , Charles Babbage Institute University of Minnesota, Minneapolis.

Dijkstra's Algorithm in C# Fast Priority Queue Implementation of Dijkstra's Algorithm in C# Applet by Carla Laffra of Pace University Animation of Dijkstra's algorithm Visualization of Dijkstra's Algorithm Shortest Path Problem: Dijkstra's Algorithm Dijkstra's Algorithm Applet Open Source Java Graph package with implementation of Dijkstra's Algorithm
http://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#cite_ref-2http://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#cite_ref-2http://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#cite_ref-sniedovich_06_3-0http://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#cite_ref-sniedovich_06_3-0http://matwbn.icm.edu.pl/ksiazki/cc/cc35/cc3536.pdfhttp://matwbn.icm.edu.pl/ksiazki/cc/cc35/cc3536.pdfhttp://matwbn.icm.edu.pl/ksiazki/cc/cc35/cc3536.pdfhttp://matwbn.icm.edu.pl/ksiazki/cc/cc35/cc3536.pdfhttp://en.wikipedia.org/wiki/PDFhttp://en.wikipedia.org/wiki/PDFhttp://en.wikipedia.org/wiki/PDFhttp://www.ifors.ms.unimelb.edu.au/tutorial/dijkstra_new/index.htmlhttp://www.ifors.ms.unimelb.edu.au/tutorial/dijkstra_new/index.htmlhttp://www.ifors.ms.unimelb.edu.au/tutorial/dijkstra_new/index.htmlhttp://www.ifors.ms.unimelb.edu.au/tutorial/dijkstra_new/index.htmlhttp://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#cite_ref-denardo_03_4-0http://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#cite_ref-denardo_03_4-0http://en.wikipedia.org/wiki/Dover_Publicationshttp://en.wikipedia.org/wiki/Dover_Publicationshttp://en.wikipedia.org/wiki/Dover_Publicationshttp://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://en.wikipedia.org/wiki/Special:BookSources/978-0486428109http://en.wikipedia.org/wiki/Special:BookSources/978-0486428109http://en.wikipedia.org/wiki/Special:BookSources/978-0486428109http://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#cite_ref-sniedovich_10_5-0http://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#cite_ref-sniedovich_10_5-0http://en.wikipedia.org/w/index.php?title=Francis_%26_Taylor&action=edit&redlink=1http://en.wikipedia.org/w/index.php?title=Francis_%26_Taylor&action=edit&redlink=1http://en.wikipedia.org/w/index.php?title=Francis_%26_Taylor&action=edit&redlink=1http://en.wikipedia.org/w/index.php?title=Francis_%26_Taylor&action=edit&redlink=1http://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://en.wikipedia.org/wiki/Special:BookSources/9780824740993http://en.wikipedia.org/wiki/Special:BookSources/9780824740993http://en.wikipedia.org/wiki/Special:BookSources/9780824740993http://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#cite_ref-6http://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#cite_ref-6http://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#cite_ref-6http://en.wikipedia.org/w/index.php?title=Dijkstra%27s_algorithm&action=edit&section=9http://en.wikipedia.org/w/index.php?title=Dijkstra%27s_algorithm&action=edit&section=9http://en.wikipedia.org/wiki/Edsger_W._Dijkstrahttp://en.wikipedia.org/wiki/Edsger_W._Dijkstrahttp://www-m3.ma.tum.de/twiki/pub/MN0506/WebHome/dijkstra.pdfhttp://www-m3.ma.tum.de/twiki/pub/MN0506/WebHome/dijkstra.pdfhttp://www-m3.ma.tum.de/twiki/pub/MN0506/WebHome/dijkstra.pdfhttp://en.wikipedia.org/wiki/Digital_object_identifierhttp://en.wikipedia.org/wiki/Digital_object_identifierhttp://dx.doi.org/10.1007%2FBF01386390http://dx.doi.org/10.1007%2FBF01386390http://dx.doi.org/10.1007%2FBF01386390http://en.wikipedia.org/wiki/Thomas_H._Cormenhttp://en.wikipedia.org/wiki/Thomas_H._Cormenhttp://en.wikipedia.org/wiki/Charles_E._Leisersonhttp://en.wikipedia.org/wiki/Charles_E._Leisersonhttp://en.wikipedia.org/wiki/Charles_E._Leisersonhttp://en.wikipedia.org/wiki/Ronald_L._Rivesthttp://en.wikipedia.org/wiki/Ronald_L._Rivesthttp://en.wikipedia.org/wiki/Ronald_L._Rivesthttp://en.wikipedia.org/wiki/Clifford_Steinhttp://en.wikipedia.org/wiki/Clifford_Steinhttp://en.wikipedia.org/wiki/Clifford_Steinhttp://en.wikipedia.org/wiki/Introduction_to_Algorithmshttp://en.wikipedia.org/wiki/Introduction_to_Algorithmshttp://en.wikipedia.org/wiki/Introduction_to_Algorithmshttp://en.wikipedia.org/wiki/MIT_Presshttp://en.wikipedia.org/wiki/MIT_Presshttp://en.wikipedia.org/wiki/MIT_Presshttp://en.wikipedia.org/wiki/McGraw-Hillhttp://en.wikipedia.org/wiki/McGraw-Hillhttp://en.wikipedia.org/wiki/McGraw-Hillhttp://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://en.wikipedia.org/wiki/Special:BookSources/0-262-03293-7http://en.wikipedia.org/wiki/Special:BookSources/0-262-03293-7http://en.wikipedia.org/wiki/Special:BookSources/0-262-03293-7http://en.wikipedia.org/wiki/Michael_Fredmanhttp://en.wikipedia.org/wiki/Michael_Fredmanhttp://en.wikipedia.org/wiki/Robert_Tarjanhttp://en.wikipedia.org/wiki/Robert_Tarjanhttp://en.wikipedia.org/wiki/Robert_Tarjanhttp://www.computer.org/portal/web/csdl/doi/10.1109/SFCS.1984.715934http://www.computer.org/portal/web/csdl/doi/10.1109/SFCS.1984.715934http://www.computer.org/portal/web/csdl/doi/10.1109/SFCS.1984.715934http://www.computer.org/portal/web/csdl/doi/10.1109/SFCS.1984.715934http://en.wikipedia.org/wiki/IEEEhttp://en.wikipedia.org/wiki/IEEEhttp://en.wikipedia.org/wiki/IEEEhttp://en.wikipedia.org/wiki/Digital_object_identifierhttp://en.wikipedia.org/wiki/Digital_object_identifierhttp://dx.doi.org/10.1109%2FSFCS.1984.715934http://dx.doi.org/10.1109%2FSFCS.1984.715934http://dx.doi.org/10.1109%2FSFCS.1984.715934http://en.wikipedia.org/wiki/Michael_Fredmanhttp://en.wikipedia.org/wiki/Michael_Fredmanhttp://en.wikipedia.org/wiki/Robert_Tarjanhttp://en.wikipedia.org/wiki/Robert_Tarjanhttp://en.wikipedia.org/wiki/Robert_Tarjanhttp://portal.acm.org/citation.cfm?id=28874http://portal.acm.org/citation.cfm?id=28874http://portal.acm.org/citation.cfm?id=28874http://portal.acm.org/citation.cfm?id=28874http://en.wikipedia.org/wiki/Digital_object_identifierhttp://en.wikipedia.org/wiki/Digital_object_identifierhttp://dx.doi.org/10.1145%2F28869.28874http://dx.doi.org/10.1145%2F28869.28874http://dx.doi.org/10.1145%2F28869.28874http://en.wikipedia.org/wiki/Transportation_Sciencehttp://en.wikipedia.org/wiki/Transportation_Sciencehttp://en.wikipedia.org/wiki/Transportation_Sciencehttp://en.wikipedia.org/wiki/Digital_object_identifierhttp://dx.doi.org/10.1287%2Ftrsc.32.1.65http://dx.doi.org/10.1287%2Ftrsc.32.1.65http://dx.doi.org/10.1287%2Ftrsc.32.1.65http://en.wikipedia.org/w/index.php?title=Dijkstra%27s_algorithm&action=edit&section=10http://en.wikipedia.org/w/index.php?title=Dijkstra%27s_algorithm&action=edit&section=10http://commons.wikimedia.org/wiki/Category:Dijkstra%27s_algorithmhttp://commons.wikimedia.org/wiki/Category:Dijkstra%27s_algorithmhttp://commons.wikimedia.org/wiki/Category:Dijkstra%27s_algorithmhttp://www.cbi.umn.edu/oh/display.phtml?id=320http://www.cbi.umn.edu/oh/display.phtml?id=320http://en.wikipedia.org/wiki/Charles_Babbage_Institutehttp://en.wikipedia.org/wiki/Charles_Babbage_Institutehttp://en.wikipedia.org/wiki/Charles_Babbage_Institutehttp://www.codeproject.com/KB/recipes/ShortestPathCalculation.aspxhttp://www.codeproject.com/KB/recipes/ShortestPathCalculation.aspxhttp://www.codeproject.com/KB/recipes/FastHeapDijkstra.aspxhttp://www.codeproject.com/KB/recipes/FastHeapDijkstra.aspxhttp://www.dgp.toronto.edu/people/JamesStewart/270/9798s/Laffra/DijkstraApplet.htmlhttp://www.dgp.toronto.edu/people/JamesStewart/270/9798s/Laffra/DijkstraApplet.htmlhttp://www.cs.sunysb.edu/~skiena/combinatorica/animations/dijkstra.htmlhttp://www.cs.sunysb.edu/~skiena/combinatorica/animations/dijkstra.htmlhttp://students.ceid.upatras.gr/~papagel/english/java_docs/minDijk.htmhttp://students.ceid.upatras.gr/~papagel/english/java_docs/minDijk.htmhttp://www-b2.is.tokushima-u.ac.jp/~ikeda/suuri/dijkstra/Dijkstra.shtmlhttp://www-b2.is.tokushima-u.ac.jp/~ikeda/suuri/dijkstra/Dijkstra.shtmlhttp://www.unf.edu/~wkloster/foundations/DijkstraApplet/DijkstraApplet.htmhttp://www.unf.edu/~wkloster/foundations/DijkstraApplet/DijkstraApplet.htmhttp://code.google.com/p/annas/http://code.google.com/p/annas/http://code.google.com/p/annas/http://www.unf.edu/~wkloster/foundations/DijkstraApplet/DijkstraApplet.htmhttp://www-b2.is.tokushima-u.ac.jp/~ikeda/suuri/dijkstra/Dijkstra.shtmlhttp://students.ceid.upatras.gr/~papagel/english/java_docs/minDijk.htmhttp://www.cs.sunysb.edu/~skiena/combinatorica/animations/dijkstra.htmlhttp://www.dgp.toronto.edu/people/JamesStewart/270/9798s/Laffra/DijkstraApplet.htmlhttp://www.codeproject.com/KB/recipes/FastHeapDijkstra.aspxhttp://www.codeproject.com/KB/recipes/ShortestPathCalculation.aspxhttp://en.wikipedia.org/wiki/Charles_Babbage_Institutehttp://www.cbi.umn.edu/oh/display.phtml?id=320http://commons.wikimedia.org/wiki/Category:Dijkstra%27s_algorithmhttp://en.wikipedia.org/w/index.php?title=Dijkstra%27s_algorithm&action=edit&section=10http://dx.doi.org/10.1287%2Ftrsc.32.1.65http://en.wikipedia.org/wiki/Digital_object_identifierhttp://en.wikipedia.org/wiki/Transportation_Sciencehttp://dx.doi.org/10.1145%2F28869.28874http://en.wikipedia.org/wiki/Digital_object_identifierhttp://portal.acm.org/citation.cfm?id=28874http://portal.acm.org/citation.cfm?id=28874http://en.wikipedia.org/wiki/Robert_Tarjanhttp://en.wikipedia.org/wiki/Michael_Fredmanhttp://dx.doi.org/10.1109%2FSFCS.1984.715934http://en.wikipedia.org/wiki/Digital_object_identifierhttp://en.wikipedia.org/wiki/IEEEhttp://www.computer.org/portal/web/csdl/doi/10.1109/SFCS.1984.715934http://www.computer.org/portal/web/csdl/doi/10.1109/SFCS.1984.715934http://en.wikipedia.org/wiki/Robert_Tarjanhttp://en.wikipedia.org/wiki/Michael_Fredmanhttp://en.wikipedia.org/wiki/Special:BookSources/0-262-03293-7http://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://en.wikipedia.org/wiki/McGraw-Hillhttp://en.wikipedia.org/wiki/MIT_Presshttp://en.wikipedia.org/wiki/Introduction_to_Algorithmshttp://en.wikipedia.org/wiki/Clifford_Steinhttp://en.wikipedia.org/wiki/Ronald_L._Rivesthttp://en.wikipedia.org/wiki/Charles_E._Leisersonhttp://en.wikipedia.org/wiki/Thomas_H._Cormenhttp://dx.doi.org/10.1007%2FBF01386390http://en.wikipedia.org/wiki/Digital_object_identifierhttp://www-m3.ma.tum.de/twiki/pub/MN0506/WebHome/dijkstra.pdfhttp://en.wikipedia.org/wiki/Edsger_W._Dijkstrahttp://en.wikipedia.org/w/index.php?title=Dijkstra%27s_algorithm&action=edit&section=9http://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#cite_ref-6http://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#cite_ref-6http://en.wikipedia.org/wiki/Special:BookSources/9780824740993http://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://en.wikipedia.org/w/index.php?title=Francis_%26_Taylor&action=edit&redlink=1http://en.wikipedia.org/w/index.php?title=Francis_%26_Taylor&action=edit&redlink=1http://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#cite_ref-sniedovich_10_5-0http://en.wikipedia.org/wiki/Special:BookSources/978-0486428109http://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://en.wikipedia.org/wiki/Dover_Publicationshttp://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#cite_ref-denardo_03_4-0http://www.ifors.ms.unimelb.edu.au/tutorial/dijkstra_new/index.htmlhttp://www.ifors.ms.unimelb.edu.au/tutorial/dijkstra_new/index.htmlhttp://en.wikipedia.org/wiki/PDFhttp://matwbn.icm.edu.pl/ksiazki/cc/cc35/cc3536.pdfhttp://matwbn.icm.edu.pl/ksiazki/cc/cc35/cc3536.pdfhttp://en.wikipedia.org/wiki/Dijkstra%27s_algorithm#cite_ref-sniedovich_06_3-0


10/12

Java Implementation of Dijkstra's Algorithm on AlgoWiki QuickGraph, Graph Data Structures and Algorithms for .NET Implementation in Boost C++ library Implementation in T-SQL A Java library for path finding with Dijkstra's Algorithm and example applet

A MATLAB program for Dijkstra's algorithm

View page ratingsRate this pageWhat's this? TrustworthyObjectiveCompleteWell-written

I am highly knowledgeable about this topic (optional)Categories :

1959 in computer science Graph algorithms Search algorithms Routing algorithms Combinatorial optimization

Log in / create account

Article Talk

Read Edit View history

Main page Contents Featured content Current events Random article Donate to Wikipedia

Interaction

Help About Wikipedia Community portal
http://code.google.com/p/annas/http://code.google.com/p/annas/http://algowiki.net/wiki/index.php?title=Dijkstra%27s_algorithmhttp://algowiki.net/wiki/index.php?title=Dijkstra%27s_algorithmhttp://quickgraph.codeplex.com/http://quickgraph.codeplex.com/http://www.boost.org/doc/libs/1_43_0/libs/graph/doc/dijkstra_shortest_paths.htmlhttp://www.boost.org/doc/libs/1_43_0/libs/graph/doc/dijkstra_shortest_paths.htmlhttp://hansolav.net/sql/graphs.htmlhttp://hansolav.net/sql/graphs.htmlhttp://www.stackframe.com/software/PathFinderhttp://www.stackframe.com/software/PathFinderhttp://www.mathworks.com/matlabcentral/fileexchange/20025-advanced-dijkstras-minimum-path-algorithmhttp://www.mathworks.com/matlabcentral/fileexchange/20025-advanced-dijkstras-minimum-path-algorithmhttp://en.wikipedia.org/wiki/Wikipedia:Article%20Feedback%20Toolhttp://en.wikipedia.org/wiki/Wikipedia:Article%20Feedback%20Toolhttp://en.wikipedia.org/wiki/Special:Categorieshttp://en.wikipedia.org/wiki/Special:Categorieshttp://en.wikipedia.org/wiki/Category:1959_in_computer_sciencehttp://en.wikipedia.org/wiki/Category:1959_in_computer_sciencehttp://en.wikipedia.org/wiki/Category:Graph_algorithmshttp://en.wikipedia.org/wiki/Category:Graph_algorithmshttp://en.wikipedia.org/wiki/Category:Search_algorithmshttp://en.wikipedia.org/wiki/Category:Search_algorithmshttp://en.wikipedia.org/wiki/Category:Routing_algorithmshttp://en.wikipedia.org/wiki/Category:Routing_algorithmshttp://en.wikipedia.org/wiki/Category:Combinatorial_optimizationhttp://en.wikipedia.org/wiki/Category:Combinatorial_optimizationhttp://en.wikipedia.org/w/index.php?title=Special:UserLogin&returnto=Dijkstra%27s+algorithm&campaign=ACP3http://en.wikipedia.org/w/index.php?title=Special:UserLogin&returnto=Dijkstra%27s+algorithm&campaign=ACP3http://en.wikipedia.org/wiki/Dijkstra%27s_algorithmhttp://en.wikipedia.org/wiki/Dijkstra%27s_algorithmhttp://en.wikipedia.org/wiki/Talk:Dijkstra%27s_algorithmhttp://en.wikipedia.org/wiki/Talk:Dijkstra%27s_algorithmhttp://en.wikipedia.org/wiki/Dijkstra%27s_algorithmhttp://en.wikipedia.org/wiki/Dijkstra%27s_algorithmhttp://en.wikipedia.org/w/index.php?title=Dijkstra%27s_algorithm&action=edithttp://en.wikipedia.org/w/index.php?title=Dijkstra%27s_algorithm&action=edithttp://en.wikipedia.org/w/index.php?title=Dijkstra%27s_algorithm&action=historyhttp://en.wikipedia.org/w/index.php?title=Dijkstra%27s_algorithm&action=historyhttp://en.wikipedia.org/wiki/Main_Pagehttp://en.wikipedia.org/wiki/Main_Pagehttp://en.wikipedia.org/wiki/Portal:Contentshttp://en.wikipedia.org/wiki/Portal:Contentshttp://en.wikipedia.org/wiki/Portal:Featured_contenthttp://en.wikipedia.org/wiki/Portal:Featured_contenthttp://en.wikipedia.org/wiki/Portal:Current_eventshttp://en.wikipedia.org/wiki/Portal:Current_eventshttp://en.wikipedia.org/wiki/Special:Randomhttp://en.wikipedia.org/wiki/Special:Randomhttp://wikimediafoundation.org/wiki/Special:Landingcheck?landing_page=WMFJA085&language=en&utm_source=donate&utm_medium=sidebar&utm_campaign=20101204SB002http://wikimediafoundation.org/wiki/Special:Landingcheck?landing_page=WMFJA085&language=en&utm_source=donate&utm_medium=sidebar&utm_campaign=20101204SB002http://en.wikipedia.org/wiki/Help:Contentshttp://en.wikipedia.org/wiki/Help:Contentshttp://en.wikipedia.org/wiki/Wikipedia:Abouthttp://en.wikipedia.org/wiki/Wikipedia:Abouthttp://en.wikipedia.org/wiki/Wikipedia:Community_portalhttp://en.wikipedia.org/wiki/Wikipedia:Community_portalhttp://en.wikipedia.org/wiki/Wikipedia:Community_portalhttp://en.wikipedia.org/wiki/Wikipedia:Abouthttp://en.wikipedia.org/wiki/Help:Contentshttp://wikimediafoundation.org/wiki/Special:Landingcheck?landing_page=WMFJA085&language=en&utm_source=donate&utm_medium=sidebar&utm_campaign=20101204SB002http://en.wikipedia.org/wiki/Special:Randomhttp://en.wikipedia.org/wiki/Portal:Current_eventshttp://en.wikipedia.org/wiki/Portal:Featured_contenthttp://en.wikipedia.org/wiki/Portal:Contentshttp://en.wikipedia.org/wiki/Main_Pagehttp://en.wikipedia.org/w/index.php?title=Dijkstra%27s_algorithm&action=historyhttp://en.wikipedia.org/w/index.php?title=Dijkstra%27s_algorithm&action=edithttp://en.wikipedia.org/wiki/Dijkstra%27s_algorithmhttp://en.wikipedia.org/wiki/Talk:Dijkstra%27s_algorithmhttp://en.wikipedia.org/wiki/Dijkstra%27s_algorithmhttp://en.wikipedia.org/w/index.php?title=Special:UserLogin&returnto=Dijkstra%27s+algorithm&campaign=ACP3http://en.wikipedia.org/wiki/Category:Combinatorial_optimizationhttp://en.wikipedia.org/wiki/Category:Routing_algorithmshttp://en.wikipedia.org/wiki/Category:Search_algorithmshttp://en.wikipedia.org/wiki/Category:Graph_algorithmshttp://en.wikipedia.org/wiki/Category:1959_in_computer_sciencehttp://en.wikipedia.org/wiki/Special:Categorieshttp://en.wikipedia.org/wiki/Wikipedia:Article%20Feedback%20Toolhttp://www.mathworks.com/matlabcentral/fileexchange/20025-advanced-dijkstras-minimum-path-algorithmhttp://www.stackframe.com/software/PathFinderhttp://hansolav.net/sql/graphs.htmlhttp://www.boost.org/doc/libs/1_43_0/libs/graph/doc/dijkstra_shortest_paths.htmlhttp://quickgraph.codeplex.com/http://algowiki.net/wiki/index.php?title=Dijkstra%27s_algorithm


11/12

Recent changes Contact Wikipedia

Toolbox

Print/export

Languages

Catal esky Deutsch Eesti Espaol Euskara Franais Hrvatski Bahasa Indonesia Italiano Latvieu Lietuvi Magyar Nederlands orsk okm l Polski Portugus Romn Slovenina Slovenina / Srpski Suomi Svenska Ting Vit
http://en.wikipedia.org/wiki/Wikipedia:Community_portalhttp://en.wikipedia.org/wiki/Wikipedia:Community_portalhttp://en.wikipedia.org/wiki/Special:RecentChangeshttp://en.wikipedia.org/wiki/Special:RecentChangeshttp://en.wikipedia.org/wiki/Wikipedia:Contact_ushttp://en.wikipedia.org/wiki/Wikipedia:Contact_ushttp://bg.wikipedia.org/wiki/%D0%90%D0%BB%D0%B3%D0%BE%D1%80%D0%B8%D1%82%D1%8A%D0%BC_%D0%BD%D0%B0_%D0%94%D0%B5%D0%B9%D0%BA%D1%81%D1%82%D1%80%D0%B0http://bg.wikipedia.org/wiki/%D0%90%D0%BB%D0%B3%D0%BE%D1%80%D0%B8%D1%82%D1%8A%D0%BC_%D0%BD%D0%B0_%D0%94%D0%B5%D0%B9%D0%BA%D1%81%D1%82%D1%80%D0%B0http://ca.wikipedia.org/wiki/Algorisme_de_Dijkstrahttp://ca.wikipedia.org/wiki/Algorisme_de_Dijkstrahttp://cs.wikipedia.org/wiki/Dijkstr%C5%AFv_algoritmushttp://cs.wikipedia.org/wiki/Dijkstr%C5%AFv_algoritmushttp://de.wikipedia.org/wiki/Dijkstra-Algorithmushttp://de.wikipedia.org/wiki/Dijkstra-Algorithmushttp://et.wikipedia.org/wiki/Dijkstra_algoritmhttp://et.wikipedia.org/wiki/Dijkstra_algoritmhttp://el.wikipedia.org/wiki/%CE%91%CE%BB%CE%B3%CF%8C%CF%81%CE%B9%CE%B8%CE%BC%CE%BF%CF%82_%CF%84%CE%BF%CF%85_Dijkstrahttp://el.wikipedia.org/wiki/%CE%91%CE%BB%CE%B3%CF%8C%CF%81%CE%B9%CE%B8%CE%BC%CE%BF%CF%82_%CF%84%CE%BF%CF%85_Dijkstrahttp://es.wikipedia.org/wiki/Algoritmo_de_Dijkstrahttp://es.wikipedia.org/wiki/Algoritmo_de_Dijkstrahttp://eu.wikipedia.org/wiki/Dijkstraren_algoritmohttp://eu.wikipedia.org/wiki/Dijkstraren_algoritmohttp://fa.wikipedia.org/wiki/%D8%A7%D9%84%DA%AF%D9%88%D8%B1%DB%8C%D8%AA%D9%85_%D8%AF%DB%8C%DA%A9%D8%B3%D8%AA%D8%B1%D8%A7http://fa.wikipedia.org/wiki/%D8%A7%D9%84%DA%AF%D9%88%D8%B1%DB%8C%D8%AA%D9%85_%D8%AF%DB%8C%DA%A9%D8%B3%D8%AA%D8%B1%D8%A7http://fr.wikipedia.org/wiki/Algorithme_de_Dijkstrahttp://fr.wikipedia.org/wiki/Algorithme_de_Dijkstrahttp://ko.wikipedia.org/wiki/%EB%8D%B0%EC%9D%B4%ED%81%AC%EC%8A%A4%ED%8A%B8%EB%9D%BC_%EC%95%8C%EA%B3%A0%EB%A6%AC%EC%A6%98http://ko.wikipedia.org/wiki/%EB%8D%B0%EC%9D%B4%ED%81%AC%EC%8A%A4%ED%8A%B8%EB%9D%BC_%EC%95%8C%EA%B3%A0%EB%A6%AC%EC%A6%98http://hy.wikipedia.org/wiki/%D4%B4%D5%A5%D6%84%D5%BD%D5%BF%D6%80%D5%A1%D5%B5%D5%AB_%D5%A1%D5%AC%D5%A3%D5%B8%D6%80%D5%AB%D5%A9%D5%B4http://hy.wikipedia.org/wiki/%D4%B4%D5%A5%D6%84%D5%BD%D5%BF%D6%80%D5%A1%D5%B5%D5%AB_%D5%A1%D5%AC%D5%A3%D5%B8%D6%80%D5%AB%D5%A9%D5%B4http://hr.wikipedia.org/wiki/Dijkstrin_algoritamhttp://hr.wikipedia.org/wiki/Dijkstrin_algoritamhttp://id.wikipedia.org/wiki/Algoritma_Dijkstrahttp://id.wikipedia.org/wiki/Algoritma_Dijkstrahttp://it.wikipedia.org/wiki/Algoritmo_di_Dijkstrahttp://it.wikipedia.org/wiki/Algoritmo_di_Dijkstrahttp://he.wikipedia.org/wiki/%D7%90%D7%9C%D7%92%D7%95%D7%A8%D7%99%D7%AA%D7%9D_%D7%93%D7%99%D7%99%D7%A7%D7%A1%D7%98%D7%A8%D7%94http://he.wikipedia.org/wiki/%D7%90%D7%9C%D7%92%D7%95%D7%A8%D7%99%D7%AA%D7%9D_%D7%93%D7%99%D7%99%D7%A7%D7%A1%D7%98%D7%A8%D7%94http://lv.wikipedia.org/wiki/Deikstras_algoritmshttp://lv.wikipedia.org/wiki/Deikstras_algoritmshttp://lt.wikipedia.org/wiki/Dijkstros_algoritmashttp://lt.wikipedia.org/wiki/Dijkstros_algoritmashttp://hu.wikipedia.org/wiki/Dijkstra-algoritmushttp://hu.wikipedia.org/wiki/Dijkstra-algoritmushttp://nl.wikipedia.org/wiki/Kortstepad-algoritmehttp://nl.wikipedia.org/wiki/Kortstepad-algoritmehttp://ja.wikipedia.org/wiki/%E3%83%80%E3%82%A4%E3%82%AF%E3%82%B9%E3%83%88%E3%83%A9%E6%B3%95http://ja.wikipedia.org/wiki/%E3%83%80%E3%82%A4%E3%82%AF%E3%82%B9%E3%83%88%E3%83%A9%E6%B3%95http://no.wikipedia.org/wiki/Dijkstras_algoritmehttp://pl.wikipedia.org/wiki/Algorytm_Dijkstryhttp://pl.wikipedia.org/wiki/Algorytm_Dijkstryhttp://pt.wikipedia.org/wiki/Algoritmo_de_Dijkstrahttp://pt.wikipedia.org/wiki/Algoritmo_de_Dijkstrahttp://ro.wikipedia.org/wiki/Algoritmul_lui_Dijkstrahttp://ro.wikipedia.org/wiki/Algoritmul_lui_Dijkstrahttp://ru.wikipedia.org/wiki/%D0%90%D0%BB%D0%B3%D0%BE%D1%80%D0%B8%D1%82%D0%BC_%D0%94%D0%B5%D0%B9%D0%BA%D1%81%D1%82%D1%80%D1%8Bhttp://ru.wikipedia.org/wiki/%D0%90%D0%BB%D0%B3%D0%BE%D1%80%D0%B8%D1%82%D0%BC_%D0%94%D0%B5%D0%B9%D0%BA%D1%81%D1%82%D1%80%D1%8Bhttp://sk.wikipedia.org/wiki/Dijkstrov_algoritmushttp://sk.wikipedia.org/wiki/Dijkstrov_algoritmushttp://sl.wikipedia.org/wiki/Dijkstrov_algoritemhttp://sl.wikipedia.org/wiki/Dijkstrov_algoritemhttp://sr.wikipedia.org/wiki/%D0%94%D0%B0%D1%98%D0%BA%D1%81%D1%82%D1%80%D0%B8%D0%BD_%D0%B0%D0%BB%D0%B3%D0%BE%D1%80%D0%B8%D1%82%D0%B0%D0%BChttp://sr.wikipedia.org/wiki/%D0%94%D0%B0%D1%98%D0%BA%D1%81%D1%82%D1%80%D0%B8%D0%BD_%D0%B0%D0%BB%D0%B3%D0%BE%D1%80%D0%B8%D1%82%D0%B0%D0%BChttp://fi.wikipedia.org/wiki/Dijkstran_algoritmihttp://fi.wikipedia.org/wiki/Dijkstran_algoritmihttp://sv.wikipedia.org/wiki/Dijkstras_algoritmhttp://sv.wikipedia.org/wiki/Dijkstras_algoritmhttp://th.wikipedia.org/wiki/%E0%B8%82%E0%B8%B1%E0%B9%89%E0%B8%99%E0%B8%95%E0%B8%AD%E0%B8%99%E0%B8%A7%E0%B8%B4%E0%B8%98%E0%B8%B5%E0%B8%82%E0%B8%AD%E0%B8%87%E0%B9%84%E0%B8%94%E0%B8%84%E0%B9%8C%E0%B8%AA%E0%B8%95%E0%B8%A3%E0%B8%B2http://th.wikipedia.org/wiki/%E0%B8%82%E0%B8%B1%E0%B9%89%E0%B8%99%E0%B8%95%E0%B8%AD%E0%B8%99%E0%B8%A7%E0%B8%B4%E0%B8%98%E0%B8%B5%E0%B8%82%E0%B8%AD%E0%B8%87%E0%B9%84%E0%B8%94%E0%B8%84%E0%B9%8C%E0%B8%AA%E0%B8%95%E0%B8%A3%E0%B8%B2http://uk.wikipedia.org/wiki/%D0%90%D0%BB%D0%B3%D0%BE%D1%80%D0%B8%D1%82%D0%BC_%D0%94%D0%B5%D0%B9%D0%BA%D1%81%D1%82%D1%80%D0%B8http://uk.wikipedia.org/wiki/%D0%90%D0%BB%D0%B3%D0%BE%D1%80%D0%B8%D1%82%D0%BC_%D0%94%D0%B5%D0%B9%D0%BA%D1%81%D1%82%D1%80%D0%B8http://vi.wikipedia.org/wiki/Thu%E1%BA%ADt_to%C3%A1n_Dijkstrahttp://vi.wikipedia.org/wiki/Thu%E1%BA%ADt_to%C3%A1n_Dijkstrahttp://zh.wikipedia.org/wiki/%E8%BF%AA%E7%A7%91%E6%96%AF%E5%BD%BB%E7%AE%97%E6%B3%95http://zh.wikipedia.org/wiki/%E8%BF%AA%E7%A7%91%E6%96%AF%E5%BD%BB%E7%AE%97%E6%B3%95http://zh.wikipedia.org/wiki/%E8%BF%AA%E7%A7%91%E6%96%AF%E5%BD%BB%E7%AE%97%E6%B3%95http://vi.wikipedia.org/wiki/Thu%E1%BA%ADt_to%C3%A1n_Dijkstrahttp://uk.wikipedia.org/wiki/%D0%90%D0%BB%D0%B3%D0%BE%D1%80%D0%B8%D1%82%D0%BC_%D0%94%D0%B5%D0%B9%D0%BA%D1%81%D1%82%D1%80%D0%B8http://th.wikipedia.org/wiki/%E0%B8%82%E0%B8%B1%E0%B9%89%E0%B8%99%E0%B8%95%E0%B8%AD%E0%B8%99%E0%B8%A7%E0%B8%B4%E0%B8%98%E0%B8%B5%E0%B8%82%E0%B8%AD%E0%B8%87%E0%B9%84%E0%B8%94%E0%B8%84%E0%B9%8C%E0%B8%AA%E0%B8%95%E0%B8%A3%E0%B8%B2http://sv.wikipedia.org/wiki/Dijkstras_algoritmhttp://fi.wikipedia.org/wiki/Dijkstran_algoritmihttp://sr.wikipedia.org/wiki/%D0%94%D0%B0%D1%98%D0%BA%D1%81%D1%82%D1%80%D0%B8%D0%BD_%D0%B0%D0%BB%D0%B3%D0%BE%D1%80%D0%B8%D1%82%D0%B0%D0%BChttp://sl.wikipedia.org/wiki/Dijkstrov_algoritemhttp://sk.wikipedia.org/wiki/Dijkstrov_algoritmushttp://ru.wikipedia.org/wiki/%D0%90%D0%BB%D0%B3%D0%BE%D1%80%D0%B8%D1%82%D0%BC_%D0%94%D0%B5%D0%B9%D0%BA%D1%81%D1%82%D1%80%D1%8Bhttp://ro.wikipedia.org/wiki/Algoritmul_lui_Dijkstrahttp://pt.wikipedia.org/wiki/Algoritmo_de_Dijkstrahttp://pl.wikipedia.org/wiki/Algorytm_Dijkstryhttp://no.wikipedia.org/wiki/Dijkstras_algoritmehttp://ja.wikipedia.org/wiki/%E3%83%80%E3%82%A4%E3%82%AF%E3%82%B9%E3%83%88%E3%83%A9%E6%B3%95http://nl.wikipedia.org/wiki/Kortstepad-algoritmehttp://hu.wikipedia.org/wiki/Dijkstra-algoritmushttp://lt.wikipedia.org/wiki/Dijkstros_algoritmashttp://lv.wikipedia.org/wiki/Deikstras_algoritmshttp://he.wikipedia.org/wiki/%D7%90%D7%9C%D7%92%D7%95%D7%A8%D7%99%D7%AA%D7%9D_%D7%93%D7%99%D7%99%D7%A7%D7%A1%D7%98%D7%A8%D7%94http://it.wikipedia.org/wiki/Algoritmo_di_Dijkstrahttp://id.wikipedia.org/wiki/Algoritma_Dijkstrahttp://hr.wikipedia.org/wiki/Dijkstrin_algoritamhttp://hy.wikipedia.org/wiki/%D4%B4%D5%A5%D6%84%D5%BD%D5%BF%D6%80%D5%A1%D5%B5%D5%AB_%D5%A1%D5%AC%D5%A3%D5%B8%D6%80%D5%AB%D5%A9%D5%B4http://ko.wikipedia.org/wiki/%EB%8D%B0%EC%9D%B4%ED%81%AC%EC%8A%A4%ED%8A%B8%EB%9D%BC_%EC%95%8C%EA%B3%A0%EB%A6%AC%EC%A6%98http://fr.wikipedia.org/wiki/Algorithme_de_Dijkstrahttp://fa.wikipedia.org/wiki/%D8%A7%D9%84%DA%AF%D9%88%D8%B1%DB%8C%D8%AA%D9%85_%D8%AF%DB%8C%DA%A9%D8%B3%D8%AA%D8%B1%D8%A7http://eu.wikipedia.org/wiki/Dijkstraren_algoritmohttp://es.wikipedia.org/wiki/Algoritmo_de_Dijkstrahttp://el.wikipedia.org/wiki/%CE%91%CE%BB%CE%B3%CF%8C%CF%81%CE%B9%CE%B8%CE%BC%CE%BF%CF%82_%CF%84%CE%BF%CF%85_Dijkstrahttp://et.wikipedia.org/wiki/Dijkstra_algoritmhttp://de.wikipedia.org/wiki/Dijkstra-Algorithmushttp://cs.wikipedia.org/wiki/Dijkstr%C5%AFv_algoritmushttp://ca.wikipedia.org/wiki/Algorisme_de_Dijkstrahttp://bg.wikipedia.org/wiki/%D0%90%D0%BB%D0%B3%D0%BE%D1%80%D0%B8%D1%82%D1%8A%D0%BC_%D0%BD%D0%B0_%D0%94%D0%B5%D0%B9%D0%BA%D1%81%D1%82%D1%80%D0%B0http://en.wikipedia.org/wiki/Wikipedia:Contact_ushttp://en.wikipedia.org/wiki/Special:RecentChanges


12/12

This page was last modified on 8 March 2012 at 17:06. Text is available under the Creative Commons Attribution-ShareAlike License ;

additional terms may apply. See Terms of use for details.Wikipedia is a registered trademark of the Wikimedia Foundation, Inc. , a non-profitorganization.

Contact us

Privacy policy About Wikipedia Disclaimers Mobile view
http://zh.wikipedia.org/wiki/%E8%BF%AA%E7%A7%91%E6%96%AF%E5%BD%BB%E7%AE%97%E6%B3%95http://zh.wikipedia.org/wiki/%E8%BF%AA%E7%A7%91%E6%96%AF%E5%BD%BB%E7%AE%97%E6%B3%95http://zh.wikipedia.org/wiki/%E8%BF%AA%E7%A7%91%E6%96%AF%E5%BD%BB%E7%AE%97%E6%B3%95http://zh.wikipedia.org/wiki/%E8%BF%AA%E7%A7%91%E6%96%AF%E5%BD%BB%E7%AE%97%E6%B3%95http://zh.wikipedia.org/wiki/%E8%BF%AA%E7%A7%91%E6%96%AF%E5%BD%BB%E7%AE%97%E6%B3%95http://zh.wikipedia.org/wiki/%E8%BF%AA%E7%A7%91%E6%96%AF%E5%BD%BB%E7%AE%97%E6%B3%95http://en.wikipedia.org/wiki/Wikipedia:Text_of_Creative_Commons_Attribution-ShareAlike_3.0_Unported_Licensehttp://en.wikipedia.org/wiki/Wikipedia:Text_of_Creative_Commons_Attribution-ShareAlike_3.0_Unported_Licensehttp://en.wikipedia.org/wiki/Wikipedia:Text_of_Creative_Commons_Attribution-ShareAlike_3.0_Unported_Licensehttp://wikimediafoundation.org/wiki/Terms_of_usehttp://wikimediafoundation.org/wiki/Terms_of_usehttp://wikimediafoundation.org/wiki/Terms_of_usehttp://www.wikimediafoundation.org/http://www.wikimediafoundation.org/http://www.wikimediafoundation.org/http://en.wikipedia.org/wiki/Wikipedia:Contact_ushttp://en.wikipedia.org/wiki/Wikipedia:Contact_ushttp://wikimediafoundation.org/wiki/Privacy_policyhttp://wikimediafoundation.org/wiki/Privacy_policyhttp://en.wikipedia.org/wiki/Wikipedia:Abouthttp://en.wikipedia.org/wiki/Wikipedia:Abouthttp://en.wikipedia.org/wiki/Wikipedia:General_disclaimerhttp://en.wikipedia.org/wiki/Wikipedia:General_disclaimerhttp://en.wikipedia.org/w/index.php?title=Dijkstra%27s_algorithm&useformat=mobilehttp://en.wikipedia.org/w/index.php?title=Dijkstra%27s_algorithm&useformat=mobilehttp://www.mediawiki.org/http://wikimediafoundation.org/http://www.mediawiki.org/http://wikimediafoundation.org/http://en.wikipedia.org/w/index.php?title=Dijkstra%27s_algorithm&useformat=mobilehttp://en.wikipedia.org/wiki/Wikipedia:General_disclaimerhttp://en.wikipedia.org/wiki/Wikipedia:Abouthttp://wikimediafoundation.org/wiki/Privacy_policyhttp://en.wikipedia.org/wiki/Wikipedia:Contact_ushttp://www.wikimediafoundation.org/http://wikimediafoundation.org/wiki/Terms_of_usehttp://en.wikipedia.org/wiki/Wikipedia:Text_of_Creative_Commons_Attribution-ShareAlike_3.0_Unported_License

Documents

Dijkstra Algorithm With Expalna