### Abstract

A "dyadic rectangle" is a set of the form R = [a2^{-s}, (a + 1)2^{-s}] × [b2^{-t} (b + 1)2^{-t}], where s and t are nonnegative integers. A dyadic tiling is a tiling of the unit square with dyadic rectangles. In this paper we study n-tilings, which consist of 2^{n} nonoverlapping dyadic rectangles, each of area 2^{-n}, whose union is the unit square. We discuss some of the underlying combinatorial structures, provide some efficient methods for uniformly sampling from the set of n-tilings, and study some limiting properties of random tilings.

Original language | English (US) |
---|---|

Pages (from-to) | 225-251 |

Number of pages | 27 |

Journal | Random Structures and Algorithms |

Volume | 21 |

Issue number | 3-4 |

State | Published - Oct 2002 |

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### ASJC Scopus subject areas

- Computer Graphics and Computer-Aided Design
- Software
- Mathematics(all)
- Applied Mathematics

### Cite this

*Random Structures and Algorithms*,

*21*(3-4), 225-251.

**Random Dyadic Tilings of the Unit Square.** / Janson, Svante; Randall, Dana; Spencer, Joel.

Research output: Contribution to journal › Article

*Random Structures and Algorithms*, vol. 21, no. 3-4, pp. 225-251.

}

TY - JOUR

T1 - Random Dyadic Tilings of the Unit Square

AU - Janson, Svante

AU - Randall, Dana

AU - Spencer, Joel

PY - 2002/10

Y1 - 2002/10

N2 - A "dyadic rectangle" is a set of the form R = [a2-s, (a + 1)2-s] × [b2-t (b + 1)2-t], where s and t are nonnegative integers. A dyadic tiling is a tiling of the unit square with dyadic rectangles. In this paper we study n-tilings, which consist of 2n nonoverlapping dyadic rectangles, each of area 2-n, whose union is the unit square. We discuss some of the underlying combinatorial structures, provide some efficient methods for uniformly sampling from the set of n-tilings, and study some limiting properties of random tilings.

AB - A "dyadic rectangle" is a set of the form R = [a2-s, (a + 1)2-s] × [b2-t (b + 1)2-t], where s and t are nonnegative integers. A dyadic tiling is a tiling of the unit square with dyadic rectangles. In this paper we study n-tilings, which consist of 2n nonoverlapping dyadic rectangles, each of area 2-n, whose union is the unit square. We discuss some of the underlying combinatorial structures, provide some efficient methods for uniformly sampling from the set of n-tilings, and study some limiting properties of random tilings.

UR - http://www.scopus.com/inward/record.url?scp=0036441047&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0036441047&partnerID=8YFLogxK

M3 - Article

VL - 21

SP - 225

EP - 251

JO - Random Structures and Algorithms

JF - Random Structures and Algorithms

SN - 1042-9832

IS - 3-4

ER -