Abstract
We consider the following geometric pattern matching problem: find the minimum Hausdorff distance between two point sets under translation with L 1 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 L 1 and L ∞ metrics, the time to compute the minimum Hausdorff distance between two point sets is On2 log2 n).
The first author was supported by AFOSR Grant AFOSR-91-0328, ONR Grant N00014-89-J-1946, NSF Grant IRI-9006137, and DARPA under ONR contract N00014-88-K-0591. The second author was supported by the Eshkol grant 04601-90 from the Israeli Ministry of Science and Technology.
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© 1992 Springer-Verlag Berlin Heidelberg
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Chew, L.P., Kedem, K. (1992). Improvements on geometric pattern matching problems. In: Nurmi, O., Ukkonen, E. (eds) Algorithm Theory — SWAT '92. SWAT 1992. Lecture Notes in Computer Science, vol 621. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-55706-7_28
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DOI: https://doi.org/10.1007/3-540-55706-7_28
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