Dijkstra's algorithm runtime 

Class  Search algorithm 

Data structure  Graph 
Worst case performance  O(  E  +  V  log  V  ) 
Dijkstra's algorithm, conceived by Dutch computer scientist Edsger Dijkstra in 1959,^{[1]} is a graph search algorithm that solves the singlesource shortest path problem for a graph with nonnegative edge path costs, producing a shortest path tree. This algorithm is often used in routing. An equivalent algorithm was developed by Edward F. Moore in 1957.^{[2]}
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 ISIS and OSPF (Open Shortest Path First).
Contents 
Let the node we are starting be called an initial node. Let a distance of a node Y be the distance from the initial node to it. Dijkstra's algorithm will assign some initial distance values and will try to improve them stepbystep.
Suppose you want to find the shortest path between two intersections on a map, a starting point and a destination. To accomplish this, you could highlight the streets (tracing the streets with a marker) in a certain order, until you have a route highlighted from the starting point to the destination. The order is conceptually simple: at each iteration, create a set of intersections consisting of every unmarked intersection that is directly connected to a marked intersection, this will be your set of considered intersections. From that set of considered intersections, find the closest intersection to the destination (this is the "greedy" part, as described above) and highlight it and mark that street to that intersection, draw an arrow with the direction, then repeat. In each stage mark just one new intersection. When you get to the destination, follow the arrows backwards. There will be only one path back against the arrows, the shortest one.
In the following algorithm, the code u := vertex in Q with smallest dist[]
, searches for the vertex u
in the vertex set Q
that has the least dist[u]
value. That vertex is removed from the set Q
and returned to the user. dist_between(u, v)
calculates the length between the two neighbornodes u
and v
. The variable alt
on line 13 is the length of the path from the root node to the neighbor node v
if it were to go through u
. If this path is shorter than the current shortest path recorded for v
, that current path is replaced with this alt
path. The previous
array is populated with a pointer to the "nexthop" 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 dist[source] := 0 // Distance from source to source 6 Q := the set of all nodes in Graph // All nodes in the graph are unoptimized  thus are in Q 7 while Q is not empty: // The main loop 8 u := vertex in Q with smallest dist[] 9 if dist[u] = infinity: 10 break // all remaining vertices are inaccessible from source 11 remove u from Q 12 for each neighbor v of u: // where v has not yet been removed from Q. 13 alt := dist[u] + dist_between(u, v) 14 if alt < dist[v]: // Relax (u,v,a) 15 dist[v] := alt 16 previous[v] := u 17 return dist[]
If we are only interested in a shortest path between vertices source
and target
, we can terminate the search at line 11 if u = target
. Now we can read the shortest path from source
to target
by iteration:
1 S := empty sequence 2 u := target 3 while previous[u] is defined: 4 insert u at the beginning of S 5 u := previous[u]
Now sequence S
is the list of vertices constituting one of the shortest paths from target
to source
, or the empty sequence if no path exists.
A more general problem would be to find all the shortest paths between source
and target
(there might be several different ones of the same length). Then instead of storing only a single node in each entry of previous[]
we would store all nodes satisfying the relaxation condition. For example, if both r
and source
connect to target
and both of them lie on different shortest paths through target
(because the edge cost is the same in both cases), then we would add both r
and source
to previous[target]
. When the algorithm completes, previous[]
data structure will actually describe a graph that is a subset of the original graph with some edges removed. Its key property will be that if the algorithm was run with some starting node, then every path from that node to any other node in the new graph will be the shortest path between those nodes in the original graph, and all paths of that length from the original graph will be present in the new graph. Then to actually find all these short paths between two given nodes we would use a path finding algorithm on the new graph, such as depthfirst search.
An upper bound of the running time of Dijkstra's algorithm on a graph with edges E and vertices V can be expressed as a function of  E  and  V  using the BigO notation.
For any implementation of set Q the running time is , where dk_{Q} and em_{Q} are times needed to perform decrease key and extract minimum operations in set Q, respectively.
The simplest implementation of the Dijkstra's algorithm stores vertices of set Q in an ordinary linked list or array, and extract minimum from Q is simply a linear search through all vertices in Q. In this case, the running time is O(  V  ^{2} +  E  ) = O(  V  ^{2}).
For sparse graphs, that is, graphs with far fewer than O(  V  ^{2}) edges, Dijkstra's algorithm can be implemented more efficiently by storing the graph in the form of adjacency lists and using a binary heap, pairing heap, or Fibonacci heap as a priority queue to implement extracting minimum efficiently. With a binary heap, the algorithm requires O((  E  +  V  )log  V  ) time (which is dominated by O(  E  log  V  ), assuming the graph is connected), and the Fibonacci heap improves this to O(  E  +  V  log  V  ).
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 than mathematically optimal. To obtain a ranked list of lessthanoptimal solutions, the optimal solution is first calculated. A single edge appearing in the optimal solution is removed from the graph, and the optimum solution to this new graph is calculated. Each edge of the original solution is suppressed in turn and a new shortestpath calculated. The secondary solutions are then ranked and presented after the first optimal solution.
Dijkstra's algorithm is usually the working principle behind linkstate routing protocols, OSPF and ISIS being the most common ones.
Unlike Dijkstra's algorithm, the BellmanFord algorithm can be used on graphs with negative edge 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 becomes lower each time the cycle is traversed.)
The A* algorithm is a generalization of Dijkstra's algorithm that cuts down on the size of the subgraph that must be explored, if additional information is available that provides a lower bound on the "distance" to the target. This approach can be viewed from the perspective of linear programming: there is a natural linear program for computing shortest paths, and solutions to its dual 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 algorithm with these reduced costs. If the dual satisfies the weaker condition of admissibility, then A* is instead more akin to the BellmanFord algorithm.
The process that underlies Dijkstra's algorithm is similar to the greedy process used in Prim's algorithm. Prim's purpose is to find a minimum spanning tree for a graph.
From a dynamic programming point of view, Dijkstra's algorithm is a successive approximation scheme that solves the dynamic programming functional equation for the shortest path problem by the Reaching method.^{[3]}^{[4]}
In fact, Dijkstra's explanation of the logic behind the algorithm,^{[5]} namely
Problem 2. Find the path of minimum total length between two given nodes P and Q.
We use the fact that, if R is a node on the minimal path from P to Q, knowledge of the latter implies the knowledge of the minimal path from P to R.
is a paraphrasing of Bellman's famous Principle of Optimality in the context of the shortest path problem.
Class  Search algorithm 

Data structure  Graph 
Worst case performance  $O($ 
Graph search algorithms and Tree search algorithms 

Search

More 
Related 
Dijkstra's algorithm, conceived by Dutch computer scientist Edsger Dijkstra in 1956 and published in 1959,^{[1]}^{[2]} is a graph search algorithm that solves the singlesource shortest path problem for a graph with nonnegative edge path costs, producing a shortest path tree. This algorithm is often used in routing. An equivalent algorithm was developed by Edward F. Moore in 1957.^{[3]}
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 ISIS and OSPF (Open Shortest Path First).
Contents 
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 distance values and will try to improve them step by step.
Suppose you want to find the shortest path between two intersections on a city map, a starting point and a destination. The order is conceptually simple: to start, mark the distance to every intersection on the map with infinity. This is done not to imply there is an infinite distance, but to note that that intersection has not yet been visited. (Some variants of this method simply leave the intersection unlabeled.) Now, at each iteration, select a current intersection. For the first iteration the 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 closest unvisited intersection to the starting point—this will be easy to find.
From the current intersection, update the distance to every unvisited intersection that is directly connected to it. This is done by determining the sum of the distance between an unvisited intersection and the value of the current intersection, and relabeling the unvisited intersection with this value if it is less than its current value. In effect, the intersection is relabeled if the path to it through the current intersection is shorter than the previously known paths. To facilitate shortest path identification, in pencil, mark the road with an arrow pointing to the relabeled intersection if you label/relabel it, and erase all others pointing to it. After you have updated the distances to each neighboring intersection, mark the current intersection as visited and select the unvisited intersection with lowest distance (from the starting point)  or lowest label—as the current 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, then marking the current intersection as visited and moving onto the closest unvisited intersection until 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 the starting point, and can trace your way back, following the arrows in reverse.
In the accompanying animated graphic, the starting and destination intersections are colored in light pink and blue and labelled a and b respectively. The visited intersections are colored in red, and the current intersection in a pale blue.
Of note is the fact that this algorithm makes no attempt to direct "exploration" towards the destination 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 clear how the algorithm necessarily finds the shortest path, however it may also reveal one of the algorithm's weaknesses: its relative slowness in some topologies.
In the following algorithm, the code u := vertex in Q with smallest dist[]
, searches for the vertex u
in the vertex set Q
that has the least dist[u]
value. That vertex is removed from the set Q
and returned to the user. dist_between(u, v)
calculates the length between the two neighbornodes u
and v
. The variable alt
on line 15 is the length of the path from the root node to the neighbor node v
if it were to go through u
. If this path is shorter than the current shortest path recorded for v
, that current path is replaced with this alt
path. The previous
array is populated with a pointer to the "nexthop" 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 dist[] ; 10 if dist[u] = infinity: 11 break ; // all remaining vertices are inaccessible from source 12 fi ; 13 remove u from Q ; 14 for each neighbor v of u: // where v has not yet been removed 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 fi ; 20 end for ; 21 end while ; 22 return dist[] ; 23 end Dijkstra.
If we are only interested in a shortest path between vertices source
and target
, we can terminate the search at line 13 if u = target
.
Now we can read the shortest path from source
to target
by iteration:
1 S := empty sequence 2 u := target 3 while previous[u] is defined: 4 insert u at the beginning of S 5 u := previous[u]
Now sequence S
is the list of vertices constituting one of the shortest paths from target
to source
, or the empty sequence if no path exists.
A more general problem would be to find all the shortest paths between source
and target
(there might be several different ones of the same length). Then instead of storing only a single node in each entry of previous[]
we would store all nodes satisfying the relaxation condition. For example, if both r
and source
connect to target
and both of them lie on different shortest paths through target
(because the edge cost is the same in both cases), then we would add both r
and source
to previous[target]
. When the algorithm completes, previous[]
data structure will actually describe a graph that is a subset of the original graph with some edges removed. Its key property will be that if the algorithm was run with some starting node, then every path from that node to any other node in the new graph will be the shortest path between those nodes in the original graph, and all paths of that length from the original graph will be present in the new graph. Then to actually find all these short paths between two given nodes we would use a path finding algorithm on the new graph, such as depthfirst search.
An upper bound of the running time of Dijkstra's algorithm on a graph with edges $E$ and vertices $V$ can be expressed as a function of $E$ and $V$ using the BigO notation.
For any implementation of set $Q$ the running time is $O(E\; \backslash cdot\; dk\_Q\; +\; V\; \backslash cdot\; em\_Q)$, where $dk\_Q$ and $em\_Q$ are times needed to perform decrease key and extract minimum operations in set $Q$, respectively.
The simplest implementation of the Dijkstra's algorithm stores vertices of set $Q$ in an ordinary linked list or array, and extract minimum from $Q$ is simply a linear search through all vertices in $Q$. In this case, the running time is $O(V^2\; +\; E)\; =\; O(V^2)$.
For sparse graphs, that is, graphs with far fewer than $O(V^2)$ edges, Dijkstra's algorithm can be implemented more efficiently by storing the graph in the form of adjacency lists and using a binary heap, pairing heap, or Fibonacci heap as a priority queue to implement extracting minimum efficiently. With a binary heap, the algorithm requires $O((E\; +\; V)\; \backslash log\; V)$ time (which is dominated by $O(E\; log\; V)$, assuming the graph is connected), and the Fibonacci heap improves this to $O(E\; +\; V\; \backslash logV)$.
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 than mathematically optimal. To obtain a ranked list of lessthanoptimal solutions, the optimal solution is first calculated. A single edge appearing in the optimal solution is removed from the graph, and the optimum solution to this new graph is calculated. Each edge of the original solution is suppressed in turn and a new shortestpath calculated. The secondary solutions are then ranked and presented after the first optimal solution.
Dijkstra's algorithm is usually the working principle behind linkstate routing protocols, OSPF and ISIS being the most common ones.
Unlike Dijkstra's algorithm, the BellmanFord algorithm can be used on graphs with negative edge 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 becomes lower each time the cycle is traversed.)
The A* algorithm is a generalization of Dijkstra's algorithm that cuts down on the size of the subgraph that must be explored, if additional information is available that provides a lower bound on the "distance" to the target. This approach can be viewed from the perspective of linear programming: there is a natural linear program for computing shortest paths, and solutions to its dual 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 algorithm with these reduced costs. If the dual satisfies the weaker condition of admissibility, then A* is instead more akin to the BellmanFord algorithm.
The process that underlies Dijkstra's algorithm is similar to the greedy process used in Prim's algorithm. Prim's purpose is to find a minimum spanning tree for a graph.
From a dynamic programming point of view, Dijkstra's algorithm is a successive approximation scheme that solves the dynamic programming functional equation for the shortest path problem by 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 $P$ and $Q$.
We use the fact that, if $R$ is a node on the minimal path from $P$ to $Q$, knowledge of the latter implies the knowledge of the minimal path from $P$ to $R$.
is a paraphrasing of Bellman's famous Principle of Optimality in the context of the shortest path problem.
