【机智的图论】牛客2018多校第十场 F. Rikka with Line Graph

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于 2018-08-21 17:33:27 发布

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分类专栏： ACM

本文链接：https://blog.youkuaiyun.com/xuziling_/article/details/81911600

博客介绍了如何解决牛客2018多校第十场 F. Rikka with Line Graph 题目，涉及边权重的无向完全图，通过将边转化为点构建新图并计算每对点的最短路径之和。解题思路包括理解边的最短路计算，使用弗洛伊德算法，并枚举边来计算每个点的最短路径贡献。

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【机智的图论】牛客2018多校第十场 F. Rikka with Line Graph

【题目链接】

题目描述

Line Graph L(G) can be considered as an operator on an undirected graph G just like Complementary Graph and Dual Graph.

Rikka generalizes Line Graph to edge-weighted undirected graphs. For a graph G=(V,E) , L(G) is still an edge-weighted undirected graph which is constructed in the following way:
这里写图片描述

L(G) has |E| vertices and the ith vertex corresponds to the ith edge in G.
There is an edge between i,j in L(G) if and only if edge i and j have at least one common vertices in G. And the edge weight is equal to the sum of the weights of edge i and j in G.

For example, in the following picture, the right graph is the line graph of the left one. Vertex 1,2,3,4 in L(G) correspond to edge (1,2),(1,4),(1,3),(3,4) in G. And if all edges in the left graph have weight 1, the edges in the right graph will have weight 2.

Now, Rikka has an edge-weighted undirected complete graph G with n vertices. And she constructs a graph G’=L(G). It is clear that G’ is connected.

Let d(i,j) be the length of the shortest path between vertex i,j in G’(the length of each edge is equal to its weight),