Online algorithms for 1-space bounded multi dimensional bin packing and hypercube packing

Abstract

In this paper, we study 1-space bounded multi-dimensional bin packing and hypercube packing. A sequence of items arrive over time, each item is a ddimensional hyperbox (in bin packing) or hypercube (in hypercube packing), and the length of each side is no more than 1. These items must be packed without overlapping into d-dimensional hypercubes with unit length on each side. In d-dimensional space, any two dimensions i and j define a space P ij . When an item arrives, we must pack it into an active bin immediately without any knowledge of the future items, and 90 • -rotation on any plane P ij is allowed.The objective is to minimize the total number of bins used for packing all these items in the sequence. In the 1-space bounded variant, there is only one active bin for packing the current item. If the active bin does not have enough space to pack the item, it must be closed and a new active bin is opened. an online algorithm with competitive ratio 4 d is given. Moreover, we consider ddimensional hypercube packing, and give a 2 d+1 -competitive algorithm. These two results are the first study on 1-space bounded multi dimensional bin packing and hypercube packing.

An algorithm for 1-space bounded cube packing

“…The last article also contains an algorithm for packing squares with competitive ratio at most 4.5. In [9], where a 2 d+1 -competitive algorithm was described. The aim of this paper is to improve the upper bound (2 3+1 ) in the 3-dimensional case.…”

Efficient 1-Space Bounded Hypercube Packing Algorithm

“…An algorithm with competitive 2 d+1 that uses only one active bin is presented in the article [31]. In the same paper the authors provide 1-space bounded algorithm for hyperbox packing with competitive ratio 4 d .…”

Drawer algorithms for 1-space bounded multidimensional hyperbox packing

“…A 3-space bounded 3.577-competitive square packing method is given in Grzegorek and Januszewski (2014). The d-dimensional case of one-space bounded hyperbox packing is considered in Zhang et al (2013). The authors give an online algorithm with competitive ratio equal to 4 d .…”

“…We focus on the problem of online packing of d-dimensional hyperboxes into one active bin. The paper contains two algorithms D 1 (d) and D 2 (d): the first method is a 3.5 d -competitive algorithm and for d < 17 works better than the second algorithm having the 12 • 3 d competitive ratio, which is a significant improvement of the ratio 4 d from Zhang et al (2013). Both algorithms are defined inductively from lower dimensions to higher.…”