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DOI: 10.4230/LIPIcs.APPROX-RANDOM.2017.34
URN: urn:nbn:de:0030-drops-75830
URL: http://drops.dagstuhl.de/opus/volltexte/2017/7583/
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Cannon, Sarah ; Levin, David A. ; Stauffer, Alexandre

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

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LIPIcs-APPROX-RANDOM-2017-34.pdf (2 MB)


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.

BibTeX - Entry

@InProceedings{cannon_et_al:LIPIcs:2017:7583,
  author =	{Sarah Cannon and David A. Levin and Alexandre Stauffer},
  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 =	{Klaus Jansen and Jos{\'e} D. P. Rolim and David Williamson and Santosh S. Vempala},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2017/7583},
  URN =		{urn:nbn:de:0030-drops-75830},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2017.34},
  annote =	{Keywords: Random dyadic tilings, spectral gap, rapid mixing}
}

Keywords: Random dyadic tilings, spectral gap, rapid mixing
Seminar: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)
Issue Date: 2017
Date of publication: 31.07.2017


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