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On the LSH Distortion of Ulam and Cayley Similarities

Authors Flavio Chierichetti , Mirko Giacchini , Ravi Kumar , Erasmo Tani

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LIPIcs.APPROX-RANDOM.2026.59.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation β†’ Data structures design and analysis
  • Information systems β†’ Similarity measures
  • Theory of computation β†’ Lower bounds and information complexity
  • Theory of computation β†’ Nearest neighbor algorithms
Keywords
  • Locality-sensitive Hashing
  • Ulam metric
  • Cayley Metric
  • Distortion
  • Permutations
  • Representation Theory

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Abstract

Locality-sensitive hashing (LSH) has found widespread use as a fundamental primitive, particularly to accelerate nearest neighbor search. An LSH scheme for a similarity function S:𝒳 Γ— 𝒳 β†’ [0,1] is a distribution over hash functions on 𝒳 with the property that the probability of collision of any two elements x,y ∈ 𝒳 is exactly equal to S(x,y). However, not all similarity functions admit exact LSH schemes. The notion of LSH distortion measures how multiplicatively close a similarity function is to having an LSH scheme. 
In this work, we study the LSH distortion of the Ulam and Cayley similarities, which are popular similarity measures on permutations of n elements. We show that the Ulam similarity admits a sublinear LSH distortion of O(n/√{log n}); we also prove a lower bound of Ω(n^{0.12}) on the best LSH distortion achievable. On the other hand, we show that the LSH distortion of the Cayley similarity is Θ(n).

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Flavio Chierichetti, Mirko Giacchini, Ravi Kumar, and Erasmo Tani. On the LSH Distortion of Ulam and Cayley Similarities. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 59:1-59:21, Schloss Dagstuhl – Leibniz-Zentrum fΓΌr Informatik (2026) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2026.59

Author Details

Flavio Chierichetti
  • Reddit, San Francisco, CA, USA
Mirko Giacchini
  • Department of Computer Science, Sapienza University of Rome, Italy
Ravi Kumar
  • Google Research, Mountain View, CA, USA
Erasmo Tani
  • Department of Computer Science, Sapienza University of Rome, Italy

Funding

  • Chierichetti, Flavio: Work done in part while at Sapienza University of Rome. Supported in part by BiCi - Bertinoro international Center for informatics, by a Google Focused Research Award and by the PRIN project 20229BCXNW (funded by the European Union - Next Generation EU, Mission 4 Component 1 CUP B53D23012910006).

Acknowledgements

The authors wish to thank Alessandro Panconesi for providing useful ideas for the project.

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