### Abstract

A random rectangle is the product of two independent random intervals, each being the interval between two random points drawn independently and uniformly from [0, 1]. We prove that the number C
_{n} of items in a maximum cardinality disjoint subset of n random rectangles satisfies n
^{1/2}/K ≤ EC
_{n} ≤ Kn
^{1/2}, where K is an absolute constant. Although tight bounds for the problem generalized to d > 2 dimensions remain an open problem, we are able to show that, for some absolute constant K, n
^{1/2}/K ≤ EC
_{n} ≤ K (n log
^{d-1} n)
^{1/2}. Finally, for a certain distribution of random cubes we show that for some absolute constant K, the number Q
_{n} of items in a maximum cardinality disjoint subset of the cubes satisfies n
^{d/(d+1)}/K ≤ EQ
_{n} ≤ Kn
^{d/(d+1)}.

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

Pages (from-to) | 585-599 |

Number of pages | 15 |

Journal | Probability Theory and Related Fields |

Volume | 120 |

Issue number | 4 |

State | Published - Aug 2001 |

### Fingerprint

### Keywords

- 2-dimensional packing
- Independent sets
- Intersection graphs
- n-dimensional packing
- Probabilistic analysis of optimization problems

### ASJC Scopus subject areas

- Mathematics(all)
- Analysis
- Statistics and Probability

### Cite this

*Probability Theory and Related Fields*,

*120*(4), 585-599.

**Packing random rectangles.** / Coffman, E. G.; Lueker, George S.; Spencer, Joel; Winkler, Peter M.

Research output: Contribution to journal › Article

*Probability Theory and Related Fields*, vol. 120, no. 4, pp. 585-599.

}

TY - JOUR

T1 - Packing random rectangles

AU - Coffman, E. G.

AU - Lueker, George S.

AU - Spencer, Joel

AU - Winkler, Peter M.

PY - 2001/8

Y1 - 2001/8

N2 - A random rectangle is the product of two independent random intervals, each being the interval between two random points drawn independently and uniformly from [0, 1]. We prove that the number C n of items in a maximum cardinality disjoint subset of n random rectangles satisfies n 1/2/K ≤ EC n ≤ Kn 1/2, where K is an absolute constant. Although tight bounds for the problem generalized to d > 2 dimensions remain an open problem, we are able to show that, for some absolute constant K, n 1/2/K ≤ EC n ≤ K (n log d-1 n) 1/2. Finally, for a certain distribution of random cubes we show that for some absolute constant K, the number Q n of items in a maximum cardinality disjoint subset of the cubes satisfies n d/(d+1)/K ≤ EQ n ≤ Kn d/(d+1).

AB - A random rectangle is the product of two independent random intervals, each being the interval between two random points drawn independently and uniformly from [0, 1]. We prove that the number C n of items in a maximum cardinality disjoint subset of n random rectangles satisfies n 1/2/K ≤ EC n ≤ Kn 1/2, where K is an absolute constant. Although tight bounds for the problem generalized to d > 2 dimensions remain an open problem, we are able to show that, for some absolute constant K, n 1/2/K ≤ EC n ≤ K (n log d-1 n) 1/2. Finally, for a certain distribution of random cubes we show that for some absolute constant K, the number Q n of items in a maximum cardinality disjoint subset of the cubes satisfies n d/(d+1)/K ≤ EQ n ≤ Kn d/(d+1).

KW - 2-dimensional packing

KW - Independent sets

KW - Intersection graphs

KW - n-dimensional packing

KW - Probabilistic analysis of optimization problems

UR - http://www.scopus.com/inward/record.url?scp=0035614406&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0035614406&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0035614406

VL - 120

SP - 585

EP - 599

JO - Probability Theory and Related Fields

JF - Probability Theory and Related Fields

SN - 0178-8051

IS - 4

ER -