CISC 365 Lecture Notes - Lecture 3: Prims, Ion, Nyishi (Tribe)

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We started with a sketch of the proof that dijkstra"s algorithm correctly nds the least-weight paths from a to all other vertices in the graph. Claim: after each iteration of the main while-loop, the algorithm has correctly identi ed the least-weight path from a to one more vertex. Base case: after the rst iteration the algorithm has moved a to the set r, with a computed distance of 0 and a path containing no edges. This is correct because it costs nothing to travel from a to a. Inductive step: suppose that after the ith iteration, for some i 1, the algorithm has correctly computed the distance and least-weight paths to all vertices in r. If there is no following iteration, the claim is true. Then that preferable path would have to start at a, continue through some vertices in r, then jump to one of the vertices, then continue on to x.

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