throwaway_1234's blog

By throwaway_1234, history, 3 months ago, In English,

I thought of this problem and I was wondering if there is any polynomial time solution?

Given a weighted tree of N nodes, pick K nodes such that sum of distances from each node to the nearest picked node is minimized.

For K = 1, the answer is the centroid of the tree. What about K > 1 ?

 
 
 
 
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3 months ago, # |
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I think you can binary search for the distance possible, and then use a greedy algorithm, where you take nodes as high as you can in the tree.

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3 months ago, # |
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Your problem (on general graphs) is called as "K-median". A quick search reveals that the problem on trees can be solved in O(KN2) time [1].

[1] Tamir, A. An O(pn2) algorithm for the p-median and related problems on tree graphs.

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    3 months ago, # ^ |
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    Thanks!

    The O(pn2) algorithms also solves for a much more general version of the problem (with weighted vertices and custom distance functions) so that's cool!