Geometric pattern matching in d-dimensional space

L. P. Chew, D. Dor, A. Efrat, K. Kedem

Research output: Contribution to journalArticle

27 Scopus citations

Abstract

We show that, using the L∞ metric, the minimum Hausdorff distance under translation between two point sets of cardinality n in d-dimensional space can be computed in time O(n(4d-2)/3 log2 n) for 3 < d ≤ 8, and in time O(n5d/4 log2 n) for any d > 8. Thus we improve the previous time bound of O(n2d-2 log2 n) due to Chew and Kedem. For d = 3 we obtain a better result of O(n3 log2 n) time by exploiting the fact that the union of n axis-parallel unit cubes can be decomposed into O(n) disjoint axis-parallel boxes. We prove that the number of different translations that achieve the minimum Hausdorff distance in d-space is Θ(n[3d/2]). Furthermore, we present an algorithm which computes the minimum Hausdorff distance under the L2 metric in d-space in time O(n[3d/2]+1+δ), for any δ > 0.

Original languageEnglish (US)
Pages (from-to)257-274
Number of pages18
JournalDiscrete and Computational Geometry
Volume21
Issue number2
DOIs
StatePublished - Mar 1999
Externally publishedYes

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics

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