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DOI: 10.4230/LIPIcs.APPROX-RANDOM.2017.34

Polynomial Mixing of the Edge-Flip Markov Chain for Unbiased Dyadic Tilings

Authors: Sarah Cannon, David A. Levin, and Alexandre Stauffer

Published in: LIPIcs, Volume 81, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)

Abstract

We give the first polynomial upper bound on the mixing time of the edge-flip Markov chain for unbiased dyadic tilings, resolving an open problem originally posed by Janson, Randall, and Spencer in 2002. A dyadic tiling of size n is a tiling of the unit square by n non-overlapping dyadic rectangles, each of area 1/n, where a dyadic rectangle is any rectangle that can be written in the form [a2^{-s}, (a+1)2^{-s}] x [b2^{-t}, (b+1)2^{-t}] for a,b,s,t nonnegative integers. The edge-flip Markov chain selects a random edge of the tiling and replaces it with its perpendicular bisector if doing so yields a valid dyadic tiling. Specifically, we show that the relaxation time of the edge-flip Markov chain for dyadic tilings is at most O(n^{4.09}), which implies that the mixing time is at most O(n^{5.09}). We complement this by showing that the relaxation time is at least Omega(n^{1.38}), improving upon the previously best lower bound of Omega(n*log n) coming from the diameter of the chain.

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Sarah Cannon, David A. Levin, and Alexandre Stauffer. Polynomial Mixing of the Edge-Flip Markov Chain for Unbiased Dyadic Tilings. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 81, pp. 34:1-34:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{cannon_et_al:LIPIcs.APPROX-RANDOM.2017.34,
  author =	{Cannon, Sarah and Levin, David A. and Stauffer, Alexandre},
  title =	{{Polynomial Mixing of the Edge-Flip Markov Chain for Unbiased Dyadic Tilings}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)},
  pages =	{34:1--34:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-044-6},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{81},
  editor =	{Jansen, Klaus and Rolim, Jos\'{e} D. P. and Williamson, David P. and Vempala, Santosh S.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2017.34},
  URN =		{urn:nbn:de:0030-drops-75830},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2017.34},
  annote =	{Keywords: Random dyadic tilings, spectral gap, rapid mixing}
}

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Polynomial Mixing of the Edge-Flip Markov Chain for Unbiased Dyadic Tilings

Abstract

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