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
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© Springer-Verlag Berlin Heidelberg 2001