Dijkstra algorithm
- 1. Dijkstra’s Algorithm We consider a weighted connected simple graph G with vertices a = v0, v1, . . . , vn = z and weights w(vi, vj) > 0 where w(vi, vj) = ∞ if {vi, vj} is not an edge. Dijkstra’s algorithm finds the cost of the “cheapest” path between vertices a and z. procedure Dijkstra(G: weighted connected simple graph, with all weights positive) for i = 1 to n L(vi) := ∞ L(a) := 0 S := ∅ while z ∈ S begin u := a vertex not in S with L(u) minimal S := S ∪ {u} for all vertices v ∈ S if L(u) + w(u, v) < L(v) then L(v) := L(u) + w(u, v) end return(L(z)) Example: Use Dijkstra’s algorithm to find the cost of the cheapest path between a and z in the following weighted graph. Describe at each iteration the function L and set S. r. .................................................................................................................................................................................................... r . .................................................................................................................................................................................................... r. ............................................................................................................................................................................................................................................................ r . ............................................................................................................................................................................................................................................................ r. .................................................................................................................................................................................................... r . ..................................................................................................................................................................................................... ........................................................................................................................................................................ . ................................................................................................................................................................................................................................................................................................................... . ........................................................................................................................................................................ a z c e b d 3 4 3 1 2 2 5 16 Solution: The iterations of Dijkstra’s algorithm are described in the following table. S L(a) L(b) L(c) L(d) L(e) L(z) ∅ 0 ∞ ∞ ∞ ∞ ∞ {a} 2 3 ∞ ∞ ∞ {a, b} 3 7 5 ∞ {a, b, c} 7 4 ∞ {a, b, c, e} 5 8 {a, b, c, e, d} 7 {a, b, c, e, d, z} At the last iteration, z ∈ S and L(z) = 7. We conclude that the cheapest path from a to z has a cost of 7. Gilles Cazelais. Typeset with LATEX on December 2, 2006.