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Self-Improving Voronoi Construction for a Hidden Mixture of Product Distributions

Authors Siu-Wing Cheng, Man Ting Wong



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Siu-Wing Cheng
  • Hong Kong University of Science and Technology, Hong Kong, China
Man Ting Wong
  • Hong Kong University of Science and Technology, Hong Kong, China

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Siu-Wing Cheng and Man Ting Wong. Self-Improving Voronoi Construction for a Hidden Mixture of Product Distributions. In 32nd International Symposium on Algorithms and Computation (ISAAC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 212, pp. 8:1-8:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.ISAAC.2021.8

Abstract

We propose a self-improving algorithm for computing Voronoi diagrams under a given convex distance function with constant description complexity. The n input points are drawn from a hidden mixture of product distributions; we are only given an upper bound m = o(√n) on the number of distributions in the mixture, and the property that for each distribution, an input instance is drawn from it with a probability of Ω(1/n). For any ε ∈ (0,1), after spending O(mn log^O(1)(mn) + m^ε n^(1+ε) log(mn)) time in a training phase, our algorithm achieves an O(1/ε n log m + 1/ε n 2^O(log^* n) + 1/ε H) expected running time with probability at least 1 - O(1/n), where H is the entropy of the distribution of the Voronoi diagram output. The expectation is taken over the input distribution and the randomized decisions of the algorithm. For the Euclidean metric, the expected running time improves to O(1/ε n log m + 1/ε H).

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • entropy
  • Voronoi diagram
  • convex distance function
  • hidden mixture of product distributions

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