Probability Theory and Related Fields

, Volume 120, Issue 4, pp 585–599

Packing random rectangles

Authors

  • E.G. Coffman, Jr.
    • Electrical Engineering Department, Columbia University, New York, NY 10027, USA.
  • George S. Lueker
    • Information and Computer Science Department, University of California, Irvine, CA 92697-3425, USA. e-mail: lueker@ics.uci.edu
  • Joel Spencer
    • Mathematics Department, New York University, New York, NY 10003, USA
  • Peter M. Winkler
    • Bell Labs, Lucent Technologies, 700 Mountain Avenue, Murray Hill, NJ 07974, USA

DOI: 10.1007/PL00008793

Cite this article as:
Coffman, Jr., E., Lueker, G., Spencer, J. et al. Probab Theory Relat Fields (2001) 120: 585. doi:10.1007/PL00008793

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 te number Cn of items in a maximum cardinality disjoint subset of n random rectangles satisfies
$$$$
where K is an absolute constant. Although tight bounds for the problem generalized to d > 2 dimensions remain an open problem, we are able to show that, for some absolute constat K,
$$$$
Finally, for a certain distribution of random cubes we show that for some absolute constant K, the number Qn of items in a maximum cardinality disjoint subset of the cubes satisies
$$$$
Mathematics Subject Classification (2000): Primary 52C17; Secondary 05C69, 52C15, 60D05
Key words or phrases:n-dimensional packing – 2-dimensional packing – Intersection graphs – Independent sets – Probabilistic analysis of optimization problems

Copyright information

© Springer-Verlag Berlin Heidelberg 2001