Sampling, Flowers and Communication

Authors Huacheng Yu , Wei Zhan



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Author Details

Huacheng Yu
  • Princeton University, NJ, USA
Wei Zhan
  • University of Chicago, IL, USA

Acknowledgements

The authors would like to thank Ran Raz, Emanuele Viola and anonymous reviewers for helpful comments and discussion.

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Huacheng Yu and Wei Zhan. Sampling, Flowers and Communication. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 100:1-100:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ITCS.2024.100

Abstract

Given a distribution over [n]ⁿ such that any k coordinates need k/log^{O(1)}n bits of communication to sample, we prove that any map that samples this distribution from uniform cells requires locality Ω(log(n/k)/log log(n/k)). In particular, we show that for any constant δ > 0, there exists ε = 2^{-Ω(n^{1-δ})} such that Ω(log n/log log n) non-adaptive cell probes on uniform cells are required to: - Sample a uniformly random permutation on n elements with error 1-ε. This provides an exponential improvement on the Ω(log log n) cell probe lower bound by Viola. - Sample an n-vector with each element independently drawn from a random n^{1-δ}-vector, with error 1-ε. This provides the first adaptive vs non-adaptive cell probe separation for sampling. The major technical component in our proof is a new combinatorial theorem about flower with small kernel, i.e. a collection of sets where few elements appear more than once. We show that in a family of n sets, each with size O(log n/log log n), there must be k = poly(n) sets where at most k/log^{O(1)}n elements appear more than once. To show the lower bound on sampling permutation, we also prove a new Ω(k) communication lower bound on sampling uniformly distributed disjoint subsets of [n] of size k, with error 1-2^{-Ω(k²/n)}. This result unifies and subsumes the lower bound for k = Θ(√n) by Ambainis et al., and the lower bound for k = Θ(n) by Göös and Watson.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational complexity and cryptography
Keywords
  • Flower
  • Sampling
  • Cell probe
  • Communcation complexity

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