CodeForces 1389G Directing Edges 边联通分量缩点+树形dp+rerooting technique

最新推荐文章于 2021-10-13 20:45:32 发布

madaidao

最新推荐文章于 2021-10-13 20:45:32 发布

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分类专栏：动态规划图论 rerooting technique

本文链接：https://blog.youkuaiyun.com/madaidao/article/details/109031168

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本文详细介绍了如何在一个无向图中，针对k个特殊顶点，通过定向边以最大化利润。通过Tarjan算法找到边联通分量，将图缩成树，然后利用树形dp和rerooting技术，在O(n)的时间复杂度内求解每个顶点的最大利润。具体策略包括定向边的选择和子树的最优结构分析。

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题目链接：https://codeforces.com/contest/1389/problem/G

G. Directing Edges

time limit per test

4 seconds

memory limit per test

512 megabytes

input

standard input

output

standard output

You are given an undirected connected graph consisting of nn vertices and mm edges. kk vertices of this graph are special.

You have to direct each edge of this graph or leave it undirected. If you leave the ii-th edge undirected, you pay wiwi coins, and if you direct it, you don't have to pay for it.

Let's call a vertex saturated if it is reachable from each special vertex along the edges of the graph (if an edge is undirected, it can be traversed in both directions). After you direct the edges of the graph (possibly leaving some of them undirected), you receive cici coins for each saturated vertex ii. Thus, your total profit can be calculated as ∑i∈Sci−∑j∈Uwj∑i∈Sci−∑j∈Uwj, where SS is the set of saturated vertices, and UU is the set of edges you leave undirected.

For each vertex ii, calculate the maximum possible profit you can get if you have to make the vertex ii saturated.

Input

The first line contains three integers nn, mm and kk (2≤n≤3⋅1052≤n≤3⋅105, n−1≤m≤min(3⋅105,n(n−1)2)n−1≤m≤min(3⋅105,n(n−1)2), 1≤k≤n1≤k≤n).

The second line contains kk pairwise distinct integers v1v1, v2v2, ..., vkvk (1≤vi≤n1≤vi≤n) — the indices of the special vertices.

The third line contains nn integers c1c1, c2c2, ..., cncn (0≤ci≤1090≤ci≤109).

The fourth line contains mm integers w1w1, w2w2, ..., wmwm (0≤wi≤1090≤wi≤109).

Then mm lines follow, the ii-th line contains two