Well-separated pair decomposition for the unit-disk graph metric and its applications

  • Authors:
  • Jie Gao;Li Zhang

  • Affiliations:
  • Stanford University, Stanford, CA;Systems Research Center, Hewlett-Packard Labs, Palo Alto, CA

  • Venue:
  • Proceedings of the thirty-fifth annual ACM symposium on Theory of computing
  • Year:
  • 2003

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Abstract

We extend the classic notion of well-separated pair decomposition [10] to the (weighted) unit-disk graph metric: the shortest path distance metric induced by the intersection graph of unit disks. We show that for the unit-disk graph metric of n points in the plane and for any constant c≥1, there exists a c-well-separated pair decomposition with O(n log n) pairs, and the decomposition can be computed in O(n log n) time. We also show that for the unit-ball graph metric in k dimensions where k≥3, there exists a c-well-separated pair decomposition with O(n2-2/k) pairs, and the bound is tight in the worst case. We present the application of the well-separated pair decomposition in obtaining efficient algorithms for approximating the diameter, closest pair, nearest neighbor, center, median, and stretch factor, all under the unit-disk graph metric.