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Minimum L_∞ Hausdorff Distance of Point Sets Under Translation: Generalizing Klee’s Measure Problem

Author Timothy M. Chan

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Timothy M. Chan
  • Department of Computer Science, University of Illinois at Urbana-Champaign, IL, USA

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Timothy M. Chan. Minimum L_∞ Hausdorff Distance of Point Sets Under Translation: Generalizing Klee’s Measure Problem. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 24:1-24:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


We present a (combinatorial) algorithm with running time close to O(n^d) for computing the minimum directed L_∞ Hausdorff distance between two sets of n points under translations in any constant dimension d. This substantially improves the best previous time bound near O(n^{5d/4}) by Chew, Dor, Efrat, and Kedem from more than twenty years ago. Our solution is obtained by a new generalization of Chan’s algorithm [FOCS'13] for Klee’s measure problem. To complement this algorithmic result, we also prove a nearly matching conditional lower bound close to Ω(n^d) for combinatorial algorithms, under the Combinatorial k-Clique Hypothesis.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Hausdorff distance
  • geometric optimization
  • Klee’s measure problem
  • fine-grained complexity


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