Covering points: Minimizing the maximum depth

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Conference Article

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CCCG 2017 - 29th Canadian Conference on Computational Geometry, Proceedings


We study a variation of the geometric set cover problem. Let P be a set of n points and T be a set of m objects in the plane. We find a cover T0 ⊆ T such that each point is covered by T0 and the depth of T0 is minimum. By depth of a cover T0, we mean the maximum number of objects in T0 which contain a point in P. We prove this problem to be NP-complete in IR2 where the objects are unit squares or unit disks. More precisely, we show that it is NP-complete to decide whether this problem has a cover of depth 1. We present an O((n+m) log n+m log m) time algorithm for the problem where the points are on a real line and objects are unweighted intervals. For weighted intervals, this problem can be solved in O(nm log n) time.

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