计算机问题求解一论题3-8 单源最短通路算法 2022年11月2日
计算机问题求解 – 论题3-8 - 单源最短通路算法 2022年11月2日
什么是最短通路问题： In a shortest-paths problem,we are given a weighted,directed graph G=(V,E),with weight function w:ER mapping edges to real-valued weights.The weight w(p)of path p=(vo.v1...v)is the sum of the weights of its constituent edges: 问题1： k w(p）=w(-1,). 输入是什么？输出是 i=1 什么？ We define the shortest-path weight 8(u,v)from u to v by min()ithere isa pathfrom o otherwise A shortest path from vertex u to vertex v is then defined as any path p with weight w(p)=8u,)
什么是最短通路问题?
问题2 单源最短略问题的解必定是一棵树？ A shortest-paths tree rooted at s is a directed subgraph G'=(V',E), where V'C V and E'C E,such that 1.V'is the set of vertices reachable from s in G, 2.G'forms a rooted tree with root s,and 3.for all vV,the unique simple path froms to v in G'is a shortest path froms to v in G. 由.T决定的 Vπ={v∈V:w.π≠NTL}U{s}· 最短路径树： E元={v.π，v)∈E:v∈Vπ-{s}
由.π决定的 最短路径树：
单源最短路问题具有最佳子结构性 Lemma 24.1 (Subpaths of shortest paths are shortest paths) Given a weighted,directed graph G=(V.E)with weight function w:E->R, let p =(vo,v1,....vk)be a shortest path from vertex vo to vertex vk and,for any i and j such that0≤i≤j≤k,let Pij=(i,vi+l,,vi）be the subpath of p from vertex vi to vertex vj.Then,Pij is a shortest path from vi to vj. Proof If we decompose path p into vv,then we have that w(p)=w(Poi)+w(Pij)+w(pik).Now,assume that there is a path p;from vi to with weight w(p(.Then,is a path from v to vk whose weight w(poi)+w(p)+w(pjk)is less than w(p),which contradicts the assumption that p is a shortest path from vo to vk
单源最短路问题具有最佳子结构性
最优子结构： 动态规划必定可行，是否可以“贪心”？ 问题3： 简单的greedy策略不能正确解决最短通路问题！ 为什么？
s 2 6 1 7 v 问题3： 简单的greedy策略不能正确解决最短通路问题! 为什么？ 最优子结构： 动态规划必定可行，是否可以“贪心”？