A 2.5 Times Optimal Algorithm for Packing in Two Dimensions

1. Introduction We consider the following two dimensional bin packing problem: Given a set of rectangular pieces, find a way to pack the pieces into a bin of width 1, bounded below but not above, so as to minimize the height to which the pieces fill the bin. The pieces are not allowed to rotate (not even 90 degrees) or overlap. This problem was first considered in [ 13, which includes several applications of the problem. The most interesting one is the allocation of jobs to a shared storage multiprocessor system. la that application each rectangle represents a job whose memory and time requirements are known, and the horizontal dimension of the bin is memory, and the vertical dimension is time. A good way of packing the rectangles into the bin represents a way to run the jobs so that they all get done quickly. It is easy to see that the problem of determining whether a given set of pieces will fit into a given height is "-hard, because we can reduce any instance of the partition problem (problem SPl 1 in [3]) to an instance of this problem. Therefore we have tried to find algorithms that are fast and that give packings whose height is within a small constant factor of the height of an optimal packing. The same approach is taken in [ 11. The authors of [ 11 give an algorithm that packs in 3 h P t where Hopt is the height of an optimal packing. In this note we present an algorithm that packs in 2. 5 h p t and is easier to implement than the one given : I * in [l]. I (1) Stack all the pieces of width greater than 4 on top of one another in the bottom of the bin. Call the height to which they reach ho, and the total area of these pieces Ao. All subsequent packing will occur above ho. height. The pieces will be placed into the bin in this order. Let h l be the height of the tallest of these pieces. bottoms along the line of height ho with the first piece adjacent to the left wall of the bin, and each subsequent piece adjacent to the one just packed. Continue until there is no more room or there are no pieces left to pack. Draw a vertical line that cuts the …