The Simultaneous Communication of Disjointness with Applications to Data Streams

Ontology type: schema:Chapter

Chapter Info

DATE

2015-06-20

AUTHORS ABSTRACT

We study k-party number-in-hand set disjointness in the simultaneous message-passing model, and show that even if each element i∈[n]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$i\in [n]$$ \end{document} is guaranteed to either belong to all k parties or to at most O(1) parties in expectation (and to at most O(logn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$O(\log n)$$ \end{document} parties with high probability), then Ω(nmin(log1/δ,logk)/k)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\varOmega (n \min (\log 1/\delta , \log k) / k )$$ \end{document} communication is required by any δ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\delta$$ \end{document}-error communication protocol for this problem (assuming k=Ω(logn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$k = \varOmega (\log n)$$ \end{document}).We use the strong promise of our lower bound, together with a recent characterization of turnstile streaming algorithms as linear sketches, to obtain new lower bounds for the well-studied problem in data streams of approximating the frequency moments. We obtain a space lower bound of Ω(n1-2/pε-2logMlog1/δ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\varOmega (n^{1-2/p} \varepsilon ^{-2} \log M \log 1/\delta )$$ \end{document} bits for any algorithm giving a (1+ε)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$(1+\varepsilon )$$ \end{document}-approximation to the p-th moment ∑i=1n|xi|p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\sum _{i=1}^n |x_i|^p$$ \end{document} of an n-dimensional vector x∈{±M}n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$x\in \{\pm M\}^n$$ \end{document} with probability 1-δ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$1-\delta$$ \end{document}, for any δ≥2-o(n1/p)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\delta \ge 2^{-o(n^{1/p})}$$ \end{document}. Our lower bound improves upon a prior Ω(n1-2/pε-2logM)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\varOmega (n^{1-2/p} \varepsilon ^{-2} \log M)$$ \end{document} lower bound which did not capture the dependence on δ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\delta$$ \end{document}, and our bound is optimal whenever ε≤1/poly(logn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\varepsilon \le 1/\text {poly}(\log n)$$ \end{document}. This is the first example of a lower bound in data streams which uses a characterization in terms of linear sketches to obtain stronger lower bounds than obtainable via the one-way communication model; indeed, our set disjointness lower bound provably cannot hold in the one-way model. More... »

PAGES

1082-1093

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Automata, Languages, and Programming

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978-3-662-47671-0
978-3-662-47672-7

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https://app.dimensions.ai/details/publication/pub.1016316224

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Our lower bound improves upon a prior Ω(n1-2/pε-2logM)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\varOmega (n^{1-2/p} \varepsilon ^{-2} \log M)$$ \end{document} lower bound which did not capture the dependence on δ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\delta$$ \end{document}, and our bound is optimal whenever ε≤1/poly(logn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\varepsilon \le 1/\text {poly}(\log n)$$ \end{document}. This is the first example of a lower bound in data streams which uses a characterization in terms of linear sketches to obtain stronger lower bounds than obtainable via the one-way communication model; indeed, our set disjointness lower bound provably cannot hold in the one-way model.
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