2026-07-22T21:25:29Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:19628 2024-03-06T11:04:31Z ddc:004 open_access

Sampling, Flowers and Communication Yu, Huacheng Zhan, Wei Flower Sampling Cell probe Communcation complexity 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Huacheng Yu and Wei Zhan 2024 Is Part Of LIPIcs, Volume 287, 15th Innovations in Theoretical Computer Science Conference (ITCS 2024) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ITCS.2024.100 urn:nbn:de:0030-drops-196288 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.100 eng https://creativecommons.org/licenses/by/4.0/legalcode