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Dijkstra's algorithm

Dijkstra's algorithm, named after its inventor the Dutch computer scientist Edsger Dijkstra, solves a shortest path problem for a directed and connected graph G(V,E) which has nonnegative (>=0) edge weights. The set V is the set of all vertices in the graph G. The set E is the set of ordered pairs which represent connected vertexes in the graph (if (u,v) belongs to E then there is a connection from vertex u to vertex v).

Assume that the function w: V x V -> [0, ∞] describes the cost w(x,y) of moving from vertex x to vertex y (non-negative cost). (We can define the cost to be infinite for pairs of vertices that are not connected by an edge.) The cost of a path between two vertices is the sum of costs of the edges in that path. The cost of an edge can be thought of as (a generalisation of) the distance between those two vertices. For a given pair of vertices s,t in V, the algorithm finds the path from s to t with lowest cost (i.e. the shortest path).

The algorithm works by constructing a subgraph S of such that the distance of any vertex v' (in S) from s is known to be a minimum within G. Initially S is simply the single vertex s, and the distance of s from itself is known to be zero. Edges are added to S at each stage by (a) identifying all the edges ei = (vi1,vi2) in G-S such that vi1 is in S and vi2 is in G, and then (b) choosing the edge ej = (vj1,vj2) in G-S which gives the minimum distance of its vertex vj2 (in G) from s from all edges ei. The algorithm terminates either when S becomes a spanning tree of G, or when all the vertices of interest are within S.

The procedure for adding an edge ej to S maintains the property that the distances of all the vertices within S from s are known to be minimum.

A few subroutines for use with Dijkstra's algorithm:

Initialize-Single-Source(G,s)

 1 for each vertex v in V[G]
 2    do d[v] := infinite
 3       previous[v] := 0
 4 d[s] := 0

Relax(u,v,w)

 1 if d[v] > d[u] + w(u,v)
 2    then d[v] := d[u] + w(u,v)
 3         previous[v] := u

v = Extract-Min(Q) searches for the vertex v in the vertex set Q that has the least d[v] value. That vertex is removed from the set Q and then returned.

The algorithm:

Dijkstra(G,w,s)

 1 Initialize-Single-Source(G,s)
 2 S := empty set
 3 Q := set of all vertexes
 4 while Q is not an empty set
 5       do u := Extract-Min(Q)
 6          S := S union {u}
 7          for each vertex v which is a neighbour of u
 8              do Relax(u,v,w)

Dijkstra's algorithm can be implemented efficiently by storing the graph in the form of adjacency lists and using a heap as priority queue to implement the Extract-Min function. If the graph has m edges and n vertices, then the algorithm's time requirements are Θ(m + n log n) (see Big O notation), assuming that comparisons of edge weights take constant time.

If we are only interested in a shortest path between vertexes s and t, we can terminate the search at line 5 if u == t.

Now we can read the shortest path from s to t by iteration:

 1 S = empty sequence
 2 u := t
 3 S = u + S  /* insert u to the beginning of S */
 4 if u == s
 5    end
 6 u = previous[u]
 7 goto 3

Now sequence S has the shortest path from s to t.

OSPF Open shortest path first is a well known real world implementation used in internet routing.

A related problem is the traveling salesman problem, which is the problem of finding the shortest path that goes through every vertex exactly once, and returns to the start. That problem is NP-hard, so it can't be solved by Dijkstra's algorithm, nor by any other known, polynomial-time algorithm.

See Also

References

  • Cormen, Leiserson, Rivest: Introduction to Algorithms



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