Dynamic, first-fit packings in two or more dimensions

A rectangular storage area orbin, of widthwand heighth, stores nonoverlapping square objects, of sizes up tok×k, that arrive and depart in an unpredictable sequence. Squares packed at any given time never exceedmwin total area. How large musthbe 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 heighthis shown to have an upper bound of the formO(mlogk). Also, heights greater than a lower bound, also of formO(mlogk), 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 sidek, into a rectangular parallelepiped in space of dimensiond⩾ 3