10,531 research outputs found

    An Improved Implementation and Inalysis of the Diaz and O'Rourke Algorithm for Finding the Simpson Point of a Convex Polygon

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    This paper focuses on the well-known Diaz and O'Rourke [M. Diaz and J. O'Rourke, Algorithms for computing the center of area of a convex polygon, Visual Comput. 10 (1994), 432–442.] iterative search algorithm to find the Simpson Point of a market, described by a convex polygon. In their paper, they observed that their algorithm did not appear to converge pointwise, and therefore, modified it to do so. We first present an enhancement of their algorithm that improves its time complexity from O(log2?) to O(n log 1/?). This is then followed by a proof of pointwise convergence and derivation of explicit bounds on convergence rates of our algorithm. It is also shown that with an appropriate interpretation, our convergence results extend to all similar iterative search algorithms to find the Simpson Point – a class that includes the original unmodified Diaz–O'Rourke algorithm. Finally, we explore how our algorithm and its convergence guarantees might be modified to find the Simpson Point when the demand distribution is non-uniform

    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

    Approximating the Maximum Overlap of Polygons under Translation

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    Let PP and QQ be two simple polygons in the plane of total complexity nn, each of which can be decomposed into at most kk convex parts. We present an (1−ε)(1-\varepsilon)-approximation algorithm, for finding the translation of QQ, which maximizes its area of overlap with PP. Our algorithm runs in O(cn)O(c n) time, where cc is a constant that depends only on kk and ε\varepsilon. This suggest that for polygons that are "close" to being convex, the problem can be solved (approximately), in near linear time

    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
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