A Space Efficient Metadata Structure for Ranking Subset Sums

The top-k variation of the Subset Sum problem can be successfully used in solving some popular problems in Recommendation Systems as well as other domains. Given a set of n real numbers, we generate the k best subsets so that the sums of their elements are minimized, where k is a positive integer chosen by the user. Our solution methodology is based on constructing a metadata structure G for a given n. The metadata structure G is constructed as a layered directed acyclic graph where in each node an n-bit vector is kept from which a suitable subset can be retrieved. The explicit construction of the whole graph is never needed; only an implicit traversal is carried out in an output-sensitive manner to reduce the total time and space requirement. We then improve the efficiency of our algorithm by reporting each subset incrementally, doing away with the storage of the bit vector in each node. We have implemented our algorithms and compared one of the variations with an existing algorithm, which illustrates the superiority of our algorithm by a constant factor both with respect to time and space complexity.