Probabilistic analysis of k-dimensional packing algorithms

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Abstract

In the k-dimensional packing problem, we are given a set I = b1, b2,…, bn of k-dimensional boxes and a k-dimensional box B with unit length in each of the first k − 1 dimensions and unbounded length in the kth dimension. Each box bi is represented by a k-tuplebi = (xi(1),…, xi(k − 1), xi(k)) ϵ (0, 1]k − 1 × (0, ∞), where x(j) denotes its length in the jth dimension, 1 ⩽ jk. We are asked to find a packing of I into B such that each box is packed orthogonally and oriented in all k dimensions and such that the height in the kth dimension of the packing is minimized. The k-dimensional packing problem is known to be NP-hard for each k < 1. In this note, we study the average-case behavior of a class of algorithms, which includes any optimal algorithm and an on-line algorithm. Let A denote an algorithm in this class. Assume that b1, b2,…, bn are independent, identically distributed according to a distribution F(x(1),…, x(k − 1), x(k)) over (0, 1]k −1 × (0, ∞), and the marginal distribution Fk of x(k) satisfies the property that there is a positive number α at which the moment generating function MFk(t) has a finite value Cα < 0. It is shown that for each given s < 0, there is an Ns,F < 0 such that for all nNs,F, Pr(¦ A(b1,…,bn)/n − Γ ¦ < s) < (2 + Cα)exp(−(3)23n13), where Γ = limn → ∞E[A(b1,…, bn)]n and A(b1,…, bn) denotes the height in the kth dimension of the packing of (b1,…, bn) produced by A.

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Research supported in part by the ONR grant N00014-91-J-1383 and in part by the CCIS at UNL.

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