License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2019.63
URN: urn:nbn:de:0030-drops-112787
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Narayanan, Shyam

Pairwise Independent Random Walks Can Be Slightly Unbounded

LIPIcs-APPROX-RANDOM-2019-63.pdf (0.5 MB)


A family of problems that have been studied in the context of various streaming algorithms are generalizations of the fact that the expected maximum distance of a 4-wise independent random walk on a line over n steps is O(sqrt{n}). For small values of k, there exist k-wise independent random walks that can be stored in much less space than storing n random bits, so these properties are often useful for lowering space bounds. In this paper, we show that for all of these examples, 4-wise independence is required by demonstrating a pairwise independent random walk with steps uniform in +/- 1 and expected maximum distance Omega(sqrt{n} lg n) from the origin. We also show that this bound is tight for the first and second moment, i.e. the expected maximum square distance of a 2-wise independent random walk is always O(n lg^2 n). Also, for any even k >= 4, we show that the kth moment of the maximum distance of any k-wise independent random walk is O(n^{k/2}). The previous two results generalize to random walks tracking insertion-only streams, and provide higher moment bounds than currently known. We also prove a generalization of Kolmogorov's maximal inequality by showing an asymptotically equivalent statement that requires only 4-wise independent random variables with bounded second moments, which also generalizes a result of Blasiok.

BibTeX - Entry

  author =	{Shyam Narayanan},
  title =	{{Pairwise Independent Random Walks Can Be Slightly Unbounded}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)},
  pages =	{63:1--63:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-125-2},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{145},
  editor =	{Dimitris Achlioptas and L{\'a}szl{\'o} A. V{\'e}gh},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{},
  URN =		{urn:nbn:de:0030-drops-112787},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2019.63},
  annote =	{Keywords: k-wise Independence, Random Walks, Moments, Chaining}

Keywords: k-wise Independence, Random Walks, Moments, Chaining
Collection: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)
Issue Date: 2019
Date of publication: 17.09.2019

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