Article

Journal of Combinatorial Optimization

, Volume 26, Issue 2, pp 223-236

First online:

Open Access This content is freely available online to anyone, anywhere at any time.

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

  • Yong ZhangAffiliated withDepartment of Computer Science, The University of Hong KongShenzhen Institutes of Advanced Technology, Chinese Academy of Sciences Email author 
  • , Francis Y. L. ChinAffiliated withDepartment of Computer Science, The University of Hong Kong
  • , Hing-Fung TingAffiliated withDepartment of Computer Science, The University of Hong Kong
  • , Xin HanAffiliated withSchool of Software, Dalian University of Technology

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 d-dimensional 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. For d-dimensional bin packing, an online algorithm with competitive ratio 4 d is given. Moreover, we consider d-dimensional 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.

Keywords

Online algorithms Bin packing 1-space bounded Multi dimensional