Dynamic, First-Fit Packings in Two or More Dimensions

A rectangular storage area or bin , of width w and height h , stores nonoverlapping square objects, of sizes up to k × k , that arrive and depart in an unpredictable sequence. Squares packed at any given time never exceed mw in total area. How large must h be to ensure that there is room for each square when it arrives? This problem generalizes a 1-dimensional packing problem, considered by Robson and others as a model of storage allocation in a computer. All packing algorithms considered here pack the new arrival on the lowest possible level. For all algorithms and all sequences of arrivals and departures, the required bin height h is shown to have an upper bound of the form O ( m log k ). Also, heights greater than a lower bound, also of form O ( m log k ), are actually needed for certain worst-case sequences. These bounds contain multiplicative constants that differ by a factor slightly less than 9. Numerical results show that the factor can be reduced to about 1.7. Similar results hold for packings of cubes, of maximum side k , into a rectangular parallelepiped in space of dimension d ⩾ 3.