, Volume 120, Issue 4, pp 585-599

Packing random rectangles

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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 C n 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 Q n of items in a maximum cardinality disjoint subset of the cubes satisies

Received: 1 September 1999 / Revised version: 3 November 2000 / Published online: 14 June 2001