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## Discussion of Problem 1416. Confidential

O(N^2) solution...
Posted by Dima_Philippov 2 Dec 2011 02:23
What is the O(N^2) solution of this problem?

I wrote simply Kraskal & DFS with assimptoty O(E * Log(E) + V * E) and got Accepted in 1.156... I want to know O(N^2) solution of this problem...
Re: O(N^2) solution...
Posted by die_young 20 Jul 2018 21:41
My 0.093  N square solution:

1) Compute MST with Kruskal in E logE.
2) For each pair of vertices in MST compute dp[u][v] the cost of largest edge on path from u to v. This can be done by calling N depth first searches. DFS runs in O(E) and there is N - 1 edge in a MST. So this step takes (N^2) time.
3) For each edge (u, v) that isn't in a MST try to relax answer with { MST_COST - dp[u][v] + weight(u, v) }. This step is done in O(E).

So, the running time of this solution is O(ElogE + E + N^2) = O(N^2).