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On the bichromatic k-set problem

Published: 03 September 2010 Publication History

Abstract

We study a bichromatic version of the well-known k-set problem: given two sets R and B of points of total size n and an integer k, how many subsets of the form (Rh) ∪ (Bh) can have size exactly k over all halfspaces h? In the dual, the problem is asymptotically equivalent to determining the worst-case combinatorial complexity of the k-level in an arrangement of n halfspaces.
Disproving an earlier conjecture by Linhart [1993], we present the first nontrivial upper bound for all kn in two dimensions: O(nk1/3 + n5/6−ϵ k2/3+2 ϵ+k2) for any fixed ϵ<0. In three dimensions, we obtain the bound O(nk3/2 + n0.5034k2.4932 + k3). Incidentally, this also implies a new upper bound for the original k-set problem in four dimensions: O(n2 k3/2 + n1.5034 k2.4932 + n k3), which improves the best previous result for all kn0.923. Extensions to other cases, such as arrangements of disks, are also discussed.

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cover image ACM Transactions on Algorithms
ACM Transactions on Algorithms  Volume 6, Issue 4
August 2010
308 pages
ISSN:1549-6325
EISSN:1549-6333
DOI:10.1145/1824777
Issue’s Table of Contents
Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Publication History

Published: 03 September 2010
Accepted: 01 July 2009
Received: 01 July 2009
Published in TALG Volume 6, Issue 4

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Author Tags

  1. k-levels
  2. k-sets
  3. Combinatorial geometry
  4. arrangements

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