Math @ Duke

Publications [#235464] of Pankaj K. Agarwal
Papers Published
 Agarwal, PK; Nevo, E; Pach, J; Pinchasi, R; Sharir, M; Smorodinsky, S, Lenses in arrangements of pseudocircles and their applications,
Journal of the ACM, vol. 51 no. 2
(2004),
pp. 139186, ISSN 00045411 [doi]
(last updated on 2018/03/23)
Abstract: A collection of simple closed Jordan curves in the plane is called a family of pseudocircles if any two of its members intersect at most twice. A closed curve composed of two subarcs of distinct pseudocircles is said to be an empty lens if the closed Jordan region that it bounds does not intersect any other member of the family. We establish a linear upper bound on the number of empty lenses in an arrangement of n pseudocircles with the property that any two curves intersect precisely twice. We use this bound to show that any collection of n xmonotone pseudocircles can be cut into O(n 8/5) arcs so that any two intersect at most once; this improves a previous bound of O(n 5/3) due to Tamaki and Tbkuyama. If, in addition, the given collection admits an algebraic representation by three real parameters that satisfies some simple conditions, then the number of cuts can be further reduced to O(n 3/2(log n) O(αs(n))), where α(n) is the inverse Ackermann function, and s is a constant that depends on the the representation of the pseudocircles. For arbitrary collections of pseudocircles, any two of which intersect exactly twice, the number of necessary cuts reduces still further to O(n 4/3). As applications, we obtain improved bounds for the number of incidences, the complexity of a single level, and the complexity of many faces in arrangements of circles, of pairwise intersecting pseudocircles, of arbitrary xmonotone pseudocircles, of parabolas, and of homothetic copies of any fixed simply shaped convex curve. We also obtain a variant of the GallaiSylvester theorem for arrangements of pairwise intersecting pseudocircles, and a new lower bound on the number of distinct distances under any wellbehaved norm.


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