Polynomial mixing of the edge-flip markov chain for unbiased dyadic tilings

Sarah Cannon, David A. Levin, Alexandre Stauffer

Research output: Chapter in Book/Report/Conference proceedingConference contribution

1 Citation (Scopus)

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 [16]. 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] × [b2-t, (b + 1)2-t] for a, b, s, t 2 Z0. 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(n4.09), which implies that the mixing time is at most O(n5.09). We complement this by showing that the relaxation time is at least (n1.38), improving upon the previously best lower bound of (n log n) coming from the diameter of the chain.

Original languageEnglish
Title of host publicationApproximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 20th International Workshop, APPROX 2017 and 21st International Workshop, RANDOM 2017
Subtitle of host publicationVolume 81
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959770446
DOIs
Publication statusPublished - 1 Aug 2017
Event20th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2017 and the 21st International Workshop on Randomization and Computation, RANDOM 2017 - Berkeley, USA United States
Duration: 16 Aug 201718 Aug 2017

Conference

Conference20th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2017 and the 21st International Workshop on Randomization and Computation, RANDOM 2017
CountryUSA United States
CityBerkeley
Period16/08/1718/08/17

Fingerprint

Markov processes
Polynomials
Relaxation time

Keywords

  • Random dyadic tilings
  • Rapid mixing
  • Spectral gap

ASJC Scopus subject areas

  • Software

Cite this

Cannon, S., Levin, D. A., & Stauffer, A. (2017). Polynomial mixing of the edge-flip markov chain for unbiased dyadic tilings. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 20th International Workshop, APPROX 2017 and 21st International Workshop, RANDOM 2017: Volume 81 [34] Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2017.34

Polynomial mixing of the edge-flip markov chain for unbiased dyadic tilings. / Cannon, Sarah; Levin, David A.; Stauffer, Alexandre.

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 20th International Workshop, APPROX 2017 and 21st International Workshop, RANDOM 2017: Volume 81. Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 2017. 34.

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Cannon, S, Levin, DA & Stauffer, A 2017, Polynomial mixing of the edge-flip markov chain for unbiased dyadic tilings. in Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 20th International Workshop, APPROX 2017 and 21st International Workshop, RANDOM 2017: Volume 81., 34, Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 20th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2017 and the 21st International Workshop on Randomization and Computation, RANDOM 2017, Berkeley, USA United States, 16/08/17. https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2017.34
Cannon S, Levin DA, Stauffer A. Polynomial mixing of the edge-flip markov chain for unbiased dyadic tilings. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 20th International Workshop, APPROX 2017 and 21st International Workshop, RANDOM 2017: Volume 81. Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. 2017. 34 https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2017.34
Cannon, Sarah ; Levin, David A. ; Stauffer, Alexandre. / Polynomial mixing of the edge-flip markov chain for unbiased dyadic tilings. Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 20th International Workshop, APPROX 2017 and 21st International Workshop, RANDOM 2017: Volume 81. Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 2017.
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