### Abstract

We study the average-case behavior of algorithms for finding a maximal disjoint subset of a given set of rectangles. In the probability model, a random rectangle is the product of two independent random intervals, each being the interval between two points drawn uniformly at random from [0, 1]. We have proved that the expected cardinality of a maximal disjoint subset of n random rectangles has the tight asymptotic bound θ(n^{1/2}). Although tight bounds for the problem generalized to d > 2 dimensions remain an open problem, we have been able to show that ω(n^{1/2}) and O((n log^{d-1} n)^{1/2}) are asymptotic lower and upper bounds. In addition, we can prove that θ(n^{d/(d+1)}) is a tight asymptotic bound for the case of random cubes.

Original language | English (US) |
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Title of host publication | LATIN 2000: Theoretical Informatics - 4th Latin American Symposium, Proceedings |

Pages | 292-297 |

Number of pages | 6 |

Volume | 1776 LNCS |

DOIs | |

State | Published - 2000 |

Event | 4th Latin American Symposium on Theoretical Informatics, LATIN 2000 - Punta del Este, Uruguay Duration: Apr 10 2000 → Apr 14 2000 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 1776 LNCS |

ISSN (Print) | 03029743 |

ISSN (Electronic) | 16113349 |

### Other

Other | 4th Latin American Symposium on Theoretical Informatics, LATIN 2000 |
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Country | Uruguay |

City | Punta del Este |

Period | 4/10/00 → 4/14/00 |

### Fingerprint

### ASJC Scopus subject areas

- Computer Science(all)
- Theoretical Computer Science

### Cite this

*LATIN 2000: Theoretical Informatics - 4th Latin American Symposium, Proceedings*(Vol. 1776 LNCS, pp. 292-297). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 1776 LNCS). https://doi.org/10.1007/10719839_30

**Average-case analysis of rectangle packings.** / Coffman, E. G.; Lueker, George S.; Spencer, Joel; Winkler, Peter M.

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

*LATIN 2000: Theoretical Informatics - 4th Latin American Symposium, Proceedings.*vol. 1776 LNCS, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 1776 LNCS, pp. 292-297, 4th Latin American Symposium on Theoretical Informatics, LATIN 2000, Punta del Este, Uruguay, 4/10/00. https://doi.org/10.1007/10719839_30

}

TY - GEN

T1 - Average-case analysis of rectangle packings

AU - Coffman, E. G.

AU - Lueker, George S.

AU - Spencer, Joel

AU - Winkler, Peter M.

PY - 2000

Y1 - 2000

N2 - We study the average-case behavior of algorithms for finding a maximal disjoint subset of a given set of rectangles. In the probability model, a random rectangle is the product of two independent random intervals, each being the interval between two points drawn uniformly at random from [0, 1]. We have proved that the expected cardinality of a maximal disjoint subset of n random rectangles has the tight asymptotic bound θ(n1/2). Although tight bounds for the problem generalized to d > 2 dimensions remain an open problem, we have been able to show that ω(n1/2) and O((n logd-1 n)1/2) are asymptotic lower and upper bounds. In addition, we can prove that θ(nd/(d+1)) is a tight asymptotic bound for the case of random cubes.

AB - We study the average-case behavior of algorithms for finding a maximal disjoint subset of a given set of rectangles. In the probability model, a random rectangle is the product of two independent random intervals, each being the interval between two points drawn uniformly at random from [0, 1]. We have proved that the expected cardinality of a maximal disjoint subset of n random rectangles has the tight asymptotic bound θ(n1/2). Although tight bounds for the problem generalized to d > 2 dimensions remain an open problem, we have been able to show that ω(n1/2) and O((n logd-1 n)1/2) are asymptotic lower and upper bounds. In addition, we can prove that θ(nd/(d+1)) is a tight asymptotic bound for the case of random cubes.

UR - http://www.scopus.com/inward/record.url?scp=84896754274&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84896754274&partnerID=8YFLogxK

U2 - 10.1007/10719839_30

DO - 10.1007/10719839_30

M3 - Conference contribution

AN - SCOPUS:84896754274

SN - 3540673067

SN - 9783540673064

VL - 1776 LNCS

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 292

EP - 297

BT - LATIN 2000: Theoretical Informatics - 4th Latin American Symposium, Proceedings

ER -