Tables 2.3 - Cycles Shortest paths can’t contain cycles Already ruled out negative-weight cycles Positive-weight we can get a shorter path by

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Unformatted text preview: Cycles Shortest paths can’t contain cycles: Already ruled out negative-weight cycles. Positive-weight: we can get a shorter path by omitting the cycle. Zero-weight: no reason to use them won’t use them. CSE 2011 Prof. J. Elder - 176 - assume that our solutions Last Updated: 4/1/10 10:16 AM Shortest-Path Example: Single-Source CSE 2011 Prof. J. Elder - 177 - Last Updated: 4/1/10 10:16 AM Output of a single-source shortest-path algorithm For each vertex v in V: d[v] = (s, v). Initially, d[v]= . Reduce as algorithm progresses. But always maintain d[v] (s, v). Call d[v] a shortest-path estimate. [v] = predecessor of v on a shortest path from s. If no predecessor, [v] = NIL. induces a tree — shortest-path tree. CSE 2011 Prof. J. Elder - 178 - Last Updated: 4/1/10 10:16 AM Initialization All shortest-paths algorithms start with the same initialization: INIT-SINGLE-SOURCE(V, s) for each v in V do d[v] [v] d[s] NIL 0 CSE 2011 Prof. J. Elder - 179 - Last Updated: 4/1/10 10:16 AM Relaxing an edge Can we improve shortest-path estimate for v by first going to u and then following edge (u,v)? RELAX(u, v, w) if d[v] > d[u] + w(u, v) then d[v] [v] CSE 2011 Prof. J. Elder d[u] + w(u, v) u - 180 - Last Updated: 4/1/10 10:16 AM ...
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This note was uploaded on 02/14/2012 for the course CSE 2011Z taught by Professor Elder during the Fall '11 term at York University.

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