eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2021-11-29
26:1
26:17
10.4230/LIPIcs.FSTTCS.2021.26
article
Fast and Exact Convex Hull Simplification
Klimenko, Georgiy
1
Raichel, Benjamin
1
Department of Computer Science, University of Texas at Dallas, Richardson, TX, USA
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 log²n) time algorithm and an O(n log³n) 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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol213-fsttcs2021/LIPIcs.FSTTCS.2021.26/LIPIcs.FSTTCS.2021.26.pdf
Convex hull
coreset
exact algorithm