Improvements on geometric pattern matching problems

Abstract

We consider the following geometric pattern matching problem: find the minimum Hausdorff distance between two point sets under translation with L1 or L∞ as the underlying metric. Huttenlocher, Kedem, and Sharir have shown that this minimum distance can be found by constructing the upper envelope of certain Voronoi surfaces. Further, they show that if the two sets are each of cardinality n then the complexity of the upper envelope of such surfaces is Ω(n3). We examine the question of whether one can get around this cubic lower bound, and show that under the L1 and L∞ metrics, the time to compute the minimum Hausdorff distance between two point sets is On2 log2n).