Dijkstra's algorithm

Definition:
An algorithm to find the shortest paths from a single source vertex to all other vertices in a weighted, directed graph. All weights must be nonnegative. The time complexity is O(V²).

See also Bellman-Ford algorithm, Johnson's algorithm, all pairs shortest path.

Note: This is used in Johnson's algorithm. Since Dijkstra's algorithm chooses the closest vertex, it is a greedy algorithm.

Djikstra's algorithm (named after its discover, E.W. Dijkstra) solves the problem of finding the shortest path from a point in a graph (the source) to a destination. It turns out that one can find the shortest paths from a given source to all points in a graph in the same time, hence this problem is sometimes called the single-source shortest paths problem.

The somewhat unexpected result that all the paths can be found as easily as one further demonstrates the value of reading the literature on algorithms!

This problem is related to the spanning tree one. The graph representing all the paths from one vertex to all the others must be a spanning tree - it must include all vertices. There will also be no cycles as a cycle would define more than one path from the selected vertex to at least one other vertex. For a graph,

G = (V,E)

where

V is a set of vertices and
E is a set of edges.

Dijkstra's algorithm keeps two sets of vertices:

S		the set of vertices whose shortest paths from the source have already been determined and
V-S		the remaining vertices.

The other data structures needed are:

d	array of best estimates of shortest path to each vertex
pi	an array of predecessors for each vertex

The basic mode of operation is:

Initialise d and pi,
Set S to empty,
While there are still vertices in V-S,
1. Sort the vertices in V-S according to the current best estimate of their distance from the source,
2. Add u, the closest vertex in V-S, to S,
3. Relax all the vertices still in V-S connected to u

Relaxation

The relaxation process updates the costs of all the vertices, v, connected to a vertex, u, if we could improve the best estimate of the shortest path to v by including (u,v) in the path to v.

The relaxation procedure proceeds as follows:

initialise_single_source( Graph g, Node s )
   for each vertex v in Vertices( g )
       g.d[v] := infinity
       g.pi[v] := nil
   g.d[s] := 0;

This sets up the graph so that each node has no predecessor (pi[v] = nil) and the estimates of the cost (distance) of each node from the source (d[v]) are infinite, except for the source node itself (d[s] = 0).

Note that we have also introduced a further way to store a graph (or part of a graph - as this structure can only store a spanning tree), the predecessor sub-graph - the list of predecessors of each node,

pi[j], 1 <= j <= |V|

The edges in the predecessor sub-graph are (pi[v],v).

The relaxation procedure checks whether the current best estimate of the shortest distance to v (d[v]) can be improved by going through u (i.e. by making u the predecessor of v):

relax( Node u, Node v, double w[][] )
    if d[v] > d[u] + w[u,v] then
       d[v] := d[u] + w[u,v]
       pi[v] := u

The algorithm itself is now:

shortest_paths( Graph g, Node s )
    initialise_single_source( g, s )
    S := { 0 }        /* Make S empty */
    Q := Vertices( g ) /* Put the vertices in a PQ */
    while not Empty(Q)
        u := ExtractCheapest( Q );
        AddNode( S, u ); /* Add u to S */
        for each vertex v in Adjacent( u )
            relax( u, v, w )

Operation of Dijkstra's algorithm

As usual, proof of a greedy algorithm is the trickiest part.

This sequence of diagrams illustrates the operation of Dijkstra's Algorithm.

Initial graph
All nodes have infinite cost except the source

Choose the closest node to s.
As we initialised d[s] to 0, it's s.
Add it to S
Relax all nodes adjacent to s.
Update predecessor (red arrows) for all nodes updated.

Choose the closest node, x
Relax all nodes adjacent to x
Update predecessors for u, v and y.

Now y is the closest, add it to S.
Relax v and adjust its predecessor.

u is now closest, choose it and adjust its neighbour, v.
Finally, add v.
The predecessor list now defines the shortest path from each node to s.