Farthest neighbors and center points in the presence of rectangular obstacles

B. Ben-Moshe, Matthew Katz, J. S.B. Mitchell

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

12 Scopus citations


We study several natural proximity and facility location problems that arise for a set P of n points and a set R of m disjoint rectangular obstacles in the plane, where distances are measured according to the L1 shortest path (geodesic) metric. In particular, we compute, in time O(mn log(m + n)), a data structure of size O(mn) that supports O(log(m + n))-time farthest point queries; we avoid computing the more complicated farthest neighbor Voronoi diagram, whose combinatorial complexity we show to be Θ(mn). We study the center point problem, finding in O(mn log(m + n)) time a center point (and the set of center points) that minimize the maximum distance to sites of P; this result improves the best previous bound by a factor of roughly m. In addition, we give algorithms for approximating the diameter, D, and radius, r, of P, including methods to (i) compute a pair of points a, b ∈ P, such that d(a,b) ≥ (1 - ε)D, in O(n log n + 1/ε(n + m)log m) time; and (ii) compute a po int c′, such that max {d(p,c′) | p ∈ P} ≥ (1 + ε)r, in O(n log(m + n) + (m/ε)log(m + 1/ε)) time. Finally, we show that for all the problems above it is enough to consider only a subset of P. This subset is likely to be much smaller than P, it is computable in O(n log n) time, and using it results in significantly decreased runtime in practice.

Original languageEnglish GB
Title of host publication 17th Annual Symposium on Computational Geometry (SCG'01)
Number of pages8
StatePublished - 1 Jan 2001
Event17th Annual Symposium on Computational Geometry (SCG'01) - Medford, MA, United States
Duration: 3 Jun 20015 Jun 2001


Conference17th Annual Symposium on Computational Geometry (SCG'01)
Country/TerritoryUnited States
CityMedford, MA


  • Approximation algorithms
  • Facility location
  • Farthest neighbors
  • Geodesic paths
  • Shortest paths

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Geometry and Topology
  • Computational Mathematics


Dive into the research topics of 'Farthest neighbors and center points in the presence of rectangular obstacles'. Together they form a unique fingerprint.

Cite this