The.Algorithm.Design.Manual.Springer-Verlag.1998

Dijkstra's Algorithm
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Dijkstra's Algorithm
We can use Dijkstra's algorithm to find the shortest path between any two vertices (,t) in a weighted graph, where each edge has
non-negative edge weight. Although most applications of shortest path involve graphs with positive edge weights, such a
condition is not needed for either Prim's or Kruskal's algorithm to work correctly. The problems that negative edges cause
Dijkstra's algorithm will become apparent once you understand the algorithm.
The principle behind Dijkstra's algorithm is that given the shortest path between and each of a given set of vertices ,
there must exist some other vertex x such that the shortest path from to x must go from to to x, for some .
Specifically, it is the vertex x that minimizes
to j and dist(i,j) is the length of the shortest path between them.
over all , where w(i,j) is the length of the edge from i
This suggests a dynamic programming-like strategy. The shortest path from to itself is trivial unless there are negative weight
edges, so dist(,)=0. Armed with the shortest path to , if (,y) is the lightest edge incident to , then d(,y) = w(,y). As soon as we
decide that we have determined the shortest path to a node x, we search through all the outgoing edges of x to see whether there
is a better path from to some unknown vertex through x:
ShortestPath-Dijkstra(G,s,t)
for i=1 to n,
for each edge (, v), dist[v]=w(, v)
last=
while ( )
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select , the unknown vertex minimizing dist[v]
for each edge ,