Packing random rectangles

E. G. Coffman, George S. Lueker, Joel Spencer, Peter M. Winkler

Research output: Contribution to journalArticle

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 languageEnglish (US)
Pages (from-to)585-599
Number of pages15
JournalProbability Theory and Related Fields
Volume120
Issue number4
StatePublished - Aug 2001

Fingerprint

Rectangle
Packing
Regular hexahedron
Cardinality
Disjoint
Interval
Subset
Open Problems

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

Coffman, E. G., Lueker, G. S., Spencer, J., & Winkler, P. M. (2001). Packing random rectangles. Probability Theory and Related Fields, 120(4), 585-599.

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

In: Probability Theory and Related Fields, Vol. 120, No. 4, 08.2001, p. 585-599.

Research output: Contribution to journalArticle

Coffman, EG, Lueker, GS, Spencer, J & Winkler, PM 2001, 'Packing random rectangles', Probability Theory and Related Fields, vol. 120, no. 4, pp. 585-599.
Coffman EG, Lueker GS, Spencer J, Winkler PM. Packing random rectangles. Probability Theory and Related Fields. 2001 Aug;120(4):585-599.
Coffman, E. G. ; Lueker, George S. ; Spencer, Joel ; Winkler, Peter M. / Packing random rectangles. In: Probability Theory and Related Fields. 2001 ; Vol. 120, No. 4. pp. 585-599.
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