SSpaths

SSpaths - Chapter 24 Single-Source Shortest Paths Given A...

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Chapter 24: Single-Source Shortest Paths Given: A single source vertex in a weighted , directed graph. Want to compute a shortest path for each possible destination. Similar to BFS. We will assume either no negative-weight edges, or no reachable negative-weight cycles. Algorithm will compute a shortest-path tree . Similar to BFS tree. Many of the slides are from Prof. Plaisted’s resources at University of North Carolina at Chapel Hill
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Outline General Lemmas and Theorems. CLRS now does this last. We’ll still do it first. Bellman-Ford algorithm. DAG algorithm. Dijkstra’s algorithm. We will skip Section 24.4.
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Corollary: Let p = SP from s to v, where p = s u v. Then, δ(s, v) = δ(s, u) + w(u, v). General Results (Relaxation) Lemma 24.1: Let p = ‹v 1 , v 2 , …, v k › be a SP from v 1 to v k . Then, p ij = ‹v i , v i+1 , …, v j › is a SP from v i to v j , where 1 i j k. So, we have the optimal-substructure property . Floyd-Warshall algorithm uses dynamic programming . Dijkstra’s algorithm uses the greedy approach . Let δ(u, v) = weight of SP from u to v. → p' Lemma 24.10: Let s V. For all edges (u,v) E, we have δ(s, v) δ(s, u) + w(u,v).
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Relaxation Initialize(G, s) for each v V[G] do d[v] := ; π [v] := NIL od ; d[s] := 0 Relax(u, v, w) if d[v] > d[u] + w(u, v) then d[v] := d[u] + w(u, v); π [v] := u fi Algorithms keep track of d[v], π [v]. Initialized as follows: These values are changed when an edge (u, v) is relaxed :
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Properties of Relaxation d[v], if not , is the length of some path from s to v. d[v] either stays the same or decreases with time Therefore, if d[v] = δ (s, v) at any time, this holds thereafter Note that d[v] δ (s, v) always After i iterations of relaxing on all (u,v), if the shortest path to v has i edges, then d[v] = δ (s, v).
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Properties of Relaxation Consider any algorithm in which d[v], and π [v] are first initialized by calling Initialize(G, s) [s is the source], and are only changed by calling Relax. We have: Lemma 24.11: ( 2200 v:: d[v] δ (s, v)) is an invariant. Implies d[v] doesn’t change once d[v] =
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SSpaths - Chapter 24 Single-Source Shortest Paths Given A...

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