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

Sarah Cannon; David A. Levin; Alexandre Stauffer

doi:10.4230/LIPIcs.APPROX-RANDOM.2017.34

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

Sarah Cannon, David A. Levin, Alexandre Stauffer

Research output: Chapter or section in a book/report/conference proceeding › Chapter in a published conference proceeding

2 Citations (SciVal)

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 language	English
Title of host publication	Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 20th International Workshop, APPROX 2017 and 21st International Workshop, RANDOM 2017
Subtitle of host publication	Volume 81
Publisher	Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)	9783959770446
DOIs	https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2017.34
Publication status	Published - 1 Aug 2017
Event	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 Duration: 16 Aug 2017 → 18 Aug 2017

Conference

Conference	20th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2017 and the 21st International Workshop on Randomization and Computation, RANDOM 2017
Country/Territory	USA United States
City	Berkeley
Period	16/08/17 → 18/08/17

Keywords

Random dyadic tilings
Rapid mixing
Spectral gap

ASJC Scopus subject areas

Software

Access to Document

10.4230/LIPIcs.APPROX-RANDOM.2017.34

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 or section in a book/report/conference proceeding › Chapter in a published conference proceeding

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 doi: 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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N2 - 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.

AB - 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.

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

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ASJC Scopus subject areas

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Early Career Fellowship - Mathematical Analysis of Strongly Correlated Processes on Discrete Dynamic Structures

Cite this

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

Abstract

Conference

Keywords

ASJC Scopus subject areas

Access to Document

Other files and links

Fingerprint

Projects

Early Career Fellowship - Mathematical Analysis of Strongly Correlated Processes on Discrete Dynamic Structures

Cite this