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.
@InProceedings{yu_et_al:LIPIcs.ITCS.2024.100, author = {Yu, Huacheng and Zhan, Wei}, title = {{Sampling, Flowers and Communication}}, booktitle = {15th Innovations in Theoretical Computer Science Conference (ITCS 2024)}, pages = {100:1--100:11}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-309-6}, ISSN = {1868-8969}, year = {2024}, volume = {287}, editor = {Guruswami, Venkatesan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.100}, URN = {urn:nbn:de:0030-drops-196288}, doi = {10.4230/LIPIcs.ITCS.2024.100}, annote = {Keywords: Flower, Sampling, Cell probe, Communcation complexity} }
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