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A very very hard problem (need help)
By User0069, history, 18 months ago, In English

Giving an undirected weighted graph of n (n <= 1e5) vertices , n — 1 edges and an positive integer k (k <= 1e9). Count the number of pairs (i , j) that i < j and the weight of the path from i to j is not bigger than k.

Input the first line contain n and k the following n — 1 lines each contains three positive integer u , v , w (u , v <= n , w <= 1e9)

Output the number of pairs (i , j) that the weight of the path from i to j is not bigger than k.

Sample

INPUT
6 8
1 2 2 
2 3 4
2 5 1
4 5 3
5 6 5

OUTPUT
14
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18 months ago, # |
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Centroid decomposition or DSU on tree

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    18 months ago, # ^ |
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    Can you explain more about it

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    18 months ago, # ^ |
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    What do you mean saying DSU on tree? I didn't hear about this technique yet. Please, provide more info.

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    18 months ago, # ^ |
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    Oh wow, if using the centroid decomposition, how to avoid trying every pair(i,j).

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      18 months ago, # ^ |
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      Consider centroid c. We calculate all distances from c to remaing vertices. If distance to vertex v is d, we need to count the number of vertices u with distance to c less or equal to k - d, and path u--v comes through c. We can shrink coordinates and build a segment tree, and the number of such vectices u will be a sum on prefix. If we don't like shrinking coordinates, we can simply use treap instead of segtree, but it can get TLE.

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        18 months ago, # ^ |
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        So basically for centroid c (after centroid decomp) we can precalculate the distances and compress the coords then build a segment tree on it so we can query it quickly? Btw prob a Dynamic ST works?

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          18 months ago, # ^ |
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          Yes, Dynamic ST works, but it requires O(n log n log C) memory. It probaly gets MLE. But you can replace it by classical ST with compressed coordinates.

          Let d be a distance from c to some vertex v. Notice that all coordinates you use in ST can be written as d or k - d. So you need only these values in ST.

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    14 months ago, # ^ |
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    How can you use DSU on tree on arbitrary graph? We can use centroid decomposition on arbitrary graph as explained here, but what about DSU on tree?

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