4,455 research outputs found

    Minimum Convex Partitions and Maximum Empty Polytopes

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    Let SS be a set of nn points in Rd\mathbb{R}^d. A Steiner convex partition is a tiling of conv(S){\rm conv}(S) with empty convex bodies. For every integer dd, we show that SS admits a Steiner convex partition with at most ⌈(n−1)/d⌉\lceil (n-1)/d\rceil tiles. This bound is the best possible for points in general position in the plane, and it is best possible apart from constant factors in every fixed dimension d≥3d\geq 3. We also give the first constant-factor approximation algorithm for computing a minimum Steiner convex partition of a planar point set in general position. Establishing a tight lower bound for the maximum volume of a tile in a Steiner convex partition of any nn points in the unit cube is equivalent to a famous problem of Danzer and Rogers. It is conjectured that the volume of the largest tile is ω(1/n)\omega(1/n). Here we give a (1−ε)(1-\varepsilon)-approximation algorithm for computing the maximum volume of an empty convex body amidst nn given points in the dd-dimensional unit box [0,1]d[0,1]^d.Comment: 16 pages, 4 figures; revised write-up with some running times improve

    Querying for the Largest Empty Geometric Object in a Desired Location

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    We study new types of geometric query problems defined as follows: given a geometric set PP, preprocess it such that given a query point qq, the location of the largest circle that does not contain any member of PP, but contains qq can be reported efficiently. The geometric sets we consider for PP are boundaries of convex and simple polygons, and point sets. While we primarily focus on circles as the desired shape, we also briefly discuss empty rectangles in the context of point sets.Comment: This version is a significant update of our earlier submission arXiv:1004.0558v1. Apart from new variants studied in Sections 3 and 4, the results have been improved in Section 5.Please note that the change in title and abstract indicate that we have expanded the scope of the problems we stud

    Covering Points by Disjoint Boxes with Outliers

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    For a set of n points in the plane, we consider the axis--aligned (p,k)-Box Covering problem: Find p axis-aligned, pairwise-disjoint boxes that together contain n-k points. In this paper, we consider the boxes to be either squares or rectangles, and we want to minimize the area of the largest box. For general p we show that the problem is NP-hard for both squares and rectangles. For a small, fixed number p, we give algorithms that find the solution in the following running times: For squares we have O(n+k log k) time for p=1, and O(n log n+k^p log^p k time for p = 2,3. For rectangles we get O(n + k^3) for p = 1 and O(n log n+k^{2+p} log^{p-1} k) time for p = 2,3. In all cases, our algorithms use O(n) space.Comment: updated version: - changed problem from 'cover exactly n-k points' to 'cover at least n-k points' to avoid having non-feasible solutions. Results are unchanged. - added Proof to Lemma 11, clarified some sections - corrected typos and small errors - updated affiliations of two author

    Approximation Schemes for Maximum Weight Independent Set of Rectangles

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    In the Maximum Weight Independent Set of Rectangles (MWISR) problem we are given a set of n axis-parallel rectangles in the 2D-plane, and the goal is to select a maximum weight subset of pairwise non-overlapping rectangles. Due to many applications, e.g. in data mining, map labeling and admission control, the problem has received a lot of attention by various research communities. We present the first (1+epsilon)-approximation algorithm for the MWISR problem with quasi-polynomial running time 2^{poly(log n/epsilon)}. In contrast, the best known polynomial time approximation algorithms for the problem achieve superconstant approximation ratios of O(log log n) (unweighted case) and O(log n / log log n) (weighted case). Key to our results is a new geometric dynamic program which recursively subdivides the plane into polygons of bounded complexity. We provide the technical tools that are needed to analyze its performance. In particular, we present a method of partitioning the plane into small and simple areas such that the rectangles of an optimal solution are intersected in a very controlled manner. Together with a novel application of the weighted planar graph separator theorem due to Arora et al. this allows us to upper bound our approximation ratio by (1+epsilon). Our dynamic program is very general and we believe that it will be useful for other settings. In particular, we show that, when parametrized properly, it provides a polynomial time (1+epsilon)-approximation for the special case of the MWISR problem when each rectangle is relatively large in at least one dimension. Key to this analysis is a method to tile the plane in order to approximately describe the topology of these rectangles in an optimal solution. This technique might be a useful insight to design better polynomial time approximation algorithms or even a PTAS for the MWISR problem
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