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Constant-Distortion Embeddings of Hausdorff Metrics into Constant-Dimensional l_p Spaces

Authors Arturs Backurs, Anastasios Sidiropoulos

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Arturs Backurs
Anastasios Sidiropoulos

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Arturs Backurs and Anastasios Sidiropoulos. Constant-Distortion Embeddings of Hausdorff Metrics into Constant-Dimensional l_p Spaces. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 60, pp. 1:1-1:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016) https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2016.1

Abstract

We show that the Hausdorff metric over constant-size pointsets in constant-dimensional Euclidean space admits an embedding into constant-dimensional l_{infinity} space with constant distortion. More specifically for any s,d>=1, we obtain an embedding of the Hausdorff metric over pointsets of size s in d-dimensional Euclidean space, into  l_{\infinity}^{s^{O(s+d)}} with distortion s^{O(s+d)}. We remark that any metric space M admits an isometric embedding into l_{infinity} with dimension proportional to the size of M. In contrast, we obtain an embedding of a space of infinite size into constant-dimensional l_{infinity}.

We further improve the distortion and dimension trade-offs by considering probabilistic embeddings of the snowflake version of the Hausdorff metric. For the case of pointsets of size s in the real line of bounded resolution, we obtain a probabilistic embedding into l_1^{O(s*log(s()} with distortion O(s).

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Keywords
  • metric embeddings
  • Hausdorff metric
  • distortion
  • dimension

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