Average-Case Analysis of Rectangle Packings

Abstract

We study the average-case behavior of algorithms for finding a maximal disjoint subset of a given set of rectangles. In the probability model, a random rectangle is the product of two independent random intervals, each being the interval between two points drawn uniformly at random from [0, 1]. We have proved that the expected cardinality of a maximal disjoint subset of n random rectangles has the tight asymptotic bound Θ (n ^1/2). Although tight bounds for the problem generalized to d > 2 dimensions remain an open problem, we have been able to show that Ω (n ^1/2) and O ((n log^{d − − 1} n)^1/2) are asymptotic lower and upper bounds. In addition, we can prove that Θ (n ^{d/( d + 1)}) is a tight asymptotic bound for the case of random cubes.