Pattern Matching for Spatial Point Sets

  • Authors:
  • David E. Cardoze;Leonard J. Schulman

  • Affiliations:
  • -;-

  • Venue:
  • FOCS '98 Proceedings of the 39th Annual Symposium on Foundations of Computer Science
  • Year:
  • 1998

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Abstract

Two sets of points in $d$-dimensional space are given: a {\it data set\/} $D$ consisting of $N$ points, and a {\it pattern set\/} or {\it probe\/} $P$ consisting of $k$ points. We address the problem of determining whether there is a transformation, among a specified group of transformations of the space, carrying $P$ into or near (meaning at a small directed Hausdorff distance of) $D$. The groups we consider are translations and rigid motions. Runtimes of approximately $O(n \log n)$ and $O(n^d \log n)$ respectively are obtained (letting $n=\max\{N,k\}$ and omitting the effects of several secondary parameters). For translations, a runtime of approximately $O(n(ak+1)\log^2 n)$ is obtained for the case that a constant fraction $a