Threshold data structures and coding theory

Abstract

Data structures for combinatorial objects are traditionally designed to handle objects up to a certain size. We introduce the idea of threshold data structures: representations that allow a richer collection of operations on small objects than large ones. As illustrations of the general concept we discuss threshold data structures for sets and multisets, and show how the former can be applied to cache memory design. Consider threshold representations for subsets of a universe of size n, supporting insertions and deletions at any level, and enumeration of sets whose size does not exceed the threshold t. We derive lower bounds on the space used by any such representation. When t is fixed and n→∞ (the case of interest in memory design), any such representation must use, asymptotically, at least (t+1) log 2n bits of memory. Applying the theory of error-correcting codes, we design a structure, efficiently supporting the required operations, whose space consumption matches the lower bound. Similar results are proved for multisets.