Sparse convex hull coverage

Klimenko, Georgiy; Raichel, Benjamin; Buskirk, Gregory Van

doi:10.1016/j.comgeo.2021.101787

Cited by 7 publications

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“…In the current paper, we select Q so as to minimize the maximum distance of a point in P to CH(Q). [12] instead considered the problem of selecting Q so as to minimize the sum of the distances of points in P to CH(Q). They provided near cubic (or higher) running time algorithms for certain generalized versions of this summed coreset variant.…”

Section: Coresetsmentioning

confidence: 99%

“…Moreover, any cycle C corresponds to the convex hull CH(C). The following lemma is adapted from [12], where Problem 1 was considered but where the cost function was determined by a sum of the distances rather than the maximum distance. To prove the first part of the lemma, we argue that for any point p ∈ P , its contribution to w(C) is at least as large as its contribution to cost(C, P ).…”

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confidence: 99%

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Fast and Exact Convex Hull Simplification

Klimenko,

Raichel

2021

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Given a point set P in the plane, we seek a subset Q ⊆ P , whose convex hull gives a smaller and thus simpler representation of the convex hull of P . Specifically, let cost(Q, P ) denote the Hausdorff distance between the convex hulls CH(Q) and CH(P ). Then given a value ε > 0 we seek the smallest subset Q ⊆ P such that cost(Q, P ) ≤ ε. We also consider the dual version, where given an integer k, we seek the subset Q ⊆ P which minimizes cost(Q, P ), such that |Q| ≤ k. For these problems, when P is in convex position, we respectively give an O(n log 2 n) time algorithm and an O(n log 3 n) time algorithm, where the latter running time holds with high probability. When there is no restriction on P , we show the problem can be reduced to APSP in an unweighted directed graph, yielding an O(n 2.5302 ) time algorithm when minimizing k and an O(min{n 2.5302 , kn 2.376 }) time algorithm when minimizing ε, using prior results for APSP. Finally, we show our near linear algorithms for convex position give 2-approximations for the general case.

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Section: Coresetsmentioning

confidence: 99%

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confidence: 99%