算法导论习题解-第23章最小生成树

最新推荐文章于 2022-09-25 11:44:48 发布

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最新推荐文章于 2022-09-25 11:44:48 发布 · 1w 阅读

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#算法导论

本文详述了《算法导论》中关于最小生成树的若干习题，包括最小边的性质、反例展示、环路上最大边的讨论以及不同情况下的最小生成树算法。内容涵盖教授Sabatier的错误猜想、轻量边不构成MST的反例、唯一轻量边的MST唯一性证明，并探讨了次优生成树、稀疏图的MST算法优化等主题。

参考资料

23.1节的答案见 http://blog.youkuaiyun.com/anye3000/article/details/12091125

23.2节的答案见 https://sites.google.com/site/clrssolutions/home/chapter23

#23.1-1

设(u,v)是连通图G中权重最小的边，证明(u,v)是某棵最小生成树的边。

解：将顶点集V割为{u}和{V-u}

#23.1-2 反例

Professor Sabatier conjectures the following converse of Theorem 23.1. Let G = (V, E) be a connected, undirected graph with a real-valued weight function w defined on E. Let A be a subset of E that is included in some minimum spanning tree for G, let (S, V - S) be any cut of G that respects A, and let (u, v) be a safe edge for A crossing (S, V - S). Then, (u, v) is a light edge for the cut. Show that the professor's conjecture is incorrect by giving a counterexample.

解：令