TY - JOUR

T1 - Getting around a lower bound for the minimum Hausdorff distance

AU - Chew, L. Paul

AU - Kedem, Klara

N1 - Funding Information:
This work was partly supported by US-Israel Binational Science Foundation Grant 94-00279. The first author was also supported by ONR Grant N00014-89-J-1946, by DARPA under contracts N00014-88-K-0591 and N00014-96-1-0699, by the US Army Research Office through the Mathematical Sciences Institute of Corneli University under contract DAAL03-91-C-0027, and by the Cornell Theory Center which receives funding from its Corporate Research Institute, NSF, New York State, DARPA, NIH, and IBM Corporation.

PY - 1998/1/1

Y1 - 1998/1/1

N2 - 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 O(n2 log2 n).

AB - 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 O(n2 log2 n).

KW - Algorithm

KW - Approximate matching

KW - Geometric pattern matching

KW - Optimization

KW - Segment tree

UR - http://www.scopus.com/inward/record.url?scp=0040633364&partnerID=8YFLogxK

U2 - 10.1016/S0925-7721(97)00032-1

DO - 10.1016/S0925-7721(97)00032-1

M3 - Article

AN - SCOPUS:0040633364

SN - 0925-7721

VL - 10

SP - 197

EP - 202

JO - Computational Geometry: Theory and Applications

JF - Computational Geometry: Theory and Applications

IS - 3

ER -