### 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 |
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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 | |

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 |
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Country | USA United States |

City | Berkeley |

Period | 16/08/17 → 18/08/17 |

### Fingerprint

### Keywords

- Random dyadic tilings
- Rapid mixing
- Spectral gap

### ASJC Scopus subject areas

- Software

### Cite this

*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.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*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

}

TY - GEN

T1 - Polynomial mixing of the edge-flip markov chain for unbiased dyadic tilings

AU - Cannon, Sarah

AU - Levin, David A.

AU - Stauffer, Alexandre

PY - 2017/8/1

Y1 - 2017/8/1

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.

KW - Random dyadic tilings

KW - Rapid mixing

KW - Spectral gap

UR - http://www.scopus.com/inward/record.url?scp=85028711102&partnerID=8YFLogxK

UR - http://dx.doi.org/10.4230/LIPIcs.APPROX/RANDOM.2017.34

U2 - 10.4230/LIPIcs.APPROX-RANDOM.2017.34

DO - 10.4230/LIPIcs.APPROX-RANDOM.2017.34

M3 - Conference contribution

BT - Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 20th International Workshop, APPROX 2017 and 21st International Workshop, RANDOM 2017

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

ER -