Krydom: 暁の水平线に胜利を刻むのです

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@krydom6月前

04/20
07:53
kruskal

[CF 472D] Design Tutorial: Inverse the Problem

♦♦♦♦♦♦   Description   ♦♦♦♦♦♦

There is an easy way to obtain a new task from an old one called "Inverse the problem": we give an output of the original task, and ask to generate an input, such that solution to the original problem will produce the output we provided. The hard task of Topcoder Open 2014 Round 2C, InverseRMQ, is a good example.

Now let's create a task this way. We will use the task: you are given a tree, please calculate the distance between any pair of its nodes. Yes, it is very easy, but the inverse version is a bit harder: you are given an n × n distance matrix. Determine if it is the distance matrix of a weighted tree (all weights must be positive integers).

♦♦♦♦♦♦   Input   ♦♦♦♦♦♦

The first line contains an integer n (1 ≤ n ≤ 2000) — the number of nodes in that graph.

Then next n lines each contains n integers di, j (0 ≤ di, j ≤ 109) — the distance between node i and node j.

♦♦♦♦♦♦   Output   ♦♦♦♦♦♦

If there exists such a tree, output "YES", otherwise output "NO".

♦♦♦♦♦♦   Sample Input   ♦♦♦♦♦♦

3
0 2 7
2 0 9
7 9 0

♦♦♦♦♦♦   Sample Output   ♦♦♦♦♦♦

YES

♦♦♦♦♦♦   Hint   ♦♦♦♦♦♦

the required tree exists. It has one edge between nodes 1 and 2 with weight 2, another edge between nodes 1 and 3 with weight 7.

♦♦♦♦♦♦   题解  ♦♦♦♦♦♦

我们考虑kruskal的过程,如果当前处理到的边连接着处于不同联通块的两个点,那么这条边一定在最后生成的树中,因为假设这条边不选,那么这两个点之后的连接方案一定是 到x1联通块内某个点的距离+一条权值大于等于这条边的边+到x2联通块内某个点的距离, 很显然这个距离是大于当前处理的边的

 

[CF 472D] Design Tutorial: Inverse the Problem