Computing the penetration depth of two convex polytopes in 3d

Pankaj K. Agarwal, Leonidas J. Guibas, Sariel Har-Peled, Alexander Rabinovitch, Micha Sharir

Research output: Chapter in Book/Report/Conference proceedingConference contribution


Let A and B be two convex polytopes in R3 with m and n facets, respectively. The penetration depth of A and B, denoted as π(A, B), is the minimum distance by which A has to be translated so that A and B do not intersect. We present a randomized algorithm that computes π(A,B) in O(m3/4+ɛn3/4+ɛ + m1+ɛ + n1+ɛ) expected time, for any constant ɛ > 0. It also computes a vector t such that ||t|| = π(A,B) and int(A + t) ⊓ B = θ. We show that if the Minkowski sum B ⊗ (—A) has K facets, then the expected running time of our algorithm is O (K1/2+ɛm1/4n1/4 + m1+ɛ + n1+ɛ), for any ɛ > 0. We also present an approximation algorithm for computing π(A,B). For any δ > 0, we can compute, in time O(m + n + (log2(m + n))/δ), a vector t such that ||t|| < (1 + δ)π(A, B) and int(A +t) ⊓ B = θ. Our result also gives a δ-approximation algorithm for computing the width of A in time O(n + (log2 n)/δ), which is simpler and slightly faster than the recent algorithm by Chan.

Original languageEnglish (US)
Title of host publicationAlgorithm Theory - SWAT 2000 - 7th Scandinavian Workshop on Algorithm Theory, 2000, Proceedings
EditorsMagnús M. Halldórsson
Number of pages11
ISBN (Print)3540676902, 9783540676904
StatePublished - 2000
Externally publishedYes
Event7th Scandinavian Workshop on Algorithm Theory, SWAT 2000 - Bergen, Norway
Duration: Jul 5 2000Jul 7 2000

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349


Other7th Scandinavian Workshop on Algorithm Theory, SWAT 2000

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science(all)


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