Sum-of-squares heuristics for bin packing and memory allocation

The sum-of-squares algorithm (SS) was introduced by Csirik, Johnson, Kenyon, Shor, and Weber for online bin packing of integral-sized items into integral-sized bins. First, we show the results of experiments from two new variants of the SS algorithm. The first variant, which runs in time O(nBlogB), appears to have almost identical expected waste as the sum-of-squares algorithm on all the distributions mentioned in the original papers on this topic. The other variant, which runs in O(nlogB) time, performs well on most, but not on all of those distributions. We also apply SS to the online memory-allocation problem. Our experimental comparisons between SS and Best Fit indicate that neither algorithm is consistently better than the other. If the amount of randomness in item sizes is low, SS appears to have lower waste than Best Fit, whereas, if the amount of randomness is high Best Fit appears to have lower waste than SS. Our experiments suggest that in both real and synthetic traces, SS does not seem to have an asymptotic advantage over Best Fit, in contrast with the bin-packing problem.