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

We extend the classic notion of well-separated pair decomposition [P. B. Callahan and S. R. Kosaraju, J. ACM, 42 (1975), pp. 67-90] to the 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(n ^2-2/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.