Probabilistic Analysis of k-Dimensional Packing Algorithms

In the k-dimensional packing problem, we are given a set Z = (b,, b,, . . , b,} 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-tuple b, = (xj’), . . . , xik-l), xjk’> E (0, ilk-’ X (0, a), where x(I) denotes its length in the jth dimension, 1 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 b,, b,, . . . , 6, are independent, identically distributed according to a distribution F(x(‘), . . . , dk-‘), xck)) over (0, llkP1 X (0, m), and the marginal distribution Fk of x(k) satisfies the property that there is a positive number (Y at which the moment generating function M,k(t) has a finite value C, > 0. It is shown that for each given s > 0, there is an Ns,F > 0 such that for all n 2 Ns,F, Pr( 1 A(b,, . . , b,)/n T 1 > s) < (2 + C,)exp(-(Sa/3) 2/3n1/3), where r = lim ,,,,E[A(b,, . . . , b,)]/n and A(b,, . . , b,) denotes the height in the kth dimension of the packing of (b,, . . . , b,) produced by A.