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Maximum Overlap of Convex Polytopes under Translation

  • Hee-Kap Ahn
  • Siu-Wing Cheng
  • Iris Reinbacher
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6507)

Abstract

We study the problem of maximizing the overlap of two convex polytopes under translation in \({\mathbb R}^d\) for some constant d ≥ 3. Let n be the number of bounding hyperplanes of the polytopes. We present an algorithm that, for any ε> 0, finds an overlap at least the optimum minus ε and reports a translation realizing it. The running time is \(O(n^{{\lfloor d/2 \rfloor}+1} \log^d n)\) with probability at least 1 − n − O(1), which can be improved to O(nlog3.5 n) in \({\mathbb R}^3\). The time complexity analysis depends on a bounded incidence condition that we enforce with probability one by randomly perturbing the input polytopes. This causes an additive error ε, which can be made arbitrarily small by decreasing the perturbation magnitude. Our algorithm in fact computes the maximum overlap of the perturbed polytopes. All bounds and their big-O constants are independent of ε.

Keywords

Convex Polygon Convex Polytopes Steep Ascent Face Pair Time Complexity Analysis 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Hee-Kap Ahn
    • 1
  • Siu-Wing Cheng
    • 2
  • Iris Reinbacher
    • 1
  1. 1.Department of Computer Science and EngineeringPOSTECHKorea
  2. 2.Department of Computer Science and EngineeringHKUSTHong Kong

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