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May 2000 Order of decay of the wasted space for a stochastic packing problem
WanSoo T. Rhee
Ann. Appl. Probab. 10(2): 539-548 (May 2000). DOI: 10.1214/aoap/1019487354


A packing of a collection of rectangles contained in $[0, 1]^2$ is a disjoint subcollection; the wasted space is the measure of the area of the part of $[0, 1]^2$ not covered by the subcollection.A simple packing has the further restriction that each vertical line meets at most one rectangle of the packing. Given a collection of $N$ independent uniformly distributed subrectangles of $[0, 1]$, we proved in a previous work that there exists a number $K$ such that the wasted space $W_N$ in an optimal simple packing of these rectangles satisfies for all $N$

EW_N \leq \frac{K}{\sqrt{N}} \exp K\sqrt{\log N}.

We prove here that

\frac{1}{K\sqrt{N}} \exp \frac{1}{K} \sqrt{N} \leq EW_N.


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WanSoo T. Rhee. "Order of decay of the wasted space for a stochastic packing problem." Ann. Appl. Probab. 10 (2) 539 - 548, May 2000.


Published: May 2000
First available in Project Euclid: 22 April 2002

zbMATH: 1051.60012
MathSciNet: MR1768224
Digital Object Identifier: 10.1214/aoap/1019487354

Primary: 60F05
Secondary: 90B35

Keywords: Rectangle packing , uniform distribution , wasted space

Rights: Copyright © 2000 Institute of Mathematical Statistics


Vol.10 • No. 2 • May 2000
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