On-Line Algorithms for 2-Space Bounded Cube and Hypercube Packing

We consider the problem of packing d-dimensional cubes into the minimum number of unit cubes with 2-space bounded, as the generalization of the classic bin packing problem. Given a sequence of items, each of which is a d-dimensional (d ≥ 3) hypercube with side length not greater than 1 and an infinite number of d-dimensional (d ≥ 3) hypercube bins with unit length on each side, we want to pack all items in the sequence into a minimum number of bins. The constraint is that only two bins are active at anytime during the packing process. Each item should be orthogonally packed without overlapping with others. Items are given in an on-line manner which means each item comes without knowing any information about the subsequent items. We extend the technique of brick partitioning in paper [1] for square packing and obtain two results: a three dimensional box partitioning scheme for cube packing and a d-dimensional hyperbox partitioning scheme for hypercube packing. We give a 5.43-competitive algorithm for cube packing and a 32/21 · 2^d-competitive algorithm for hypercube packing. To the best of our knowledge these are the first known results on 2-space bounded cube and hypercube packing.