Low Redundancy in Dictionaries with O(1) Worst Case Lookup Time

A static dictionary is a data structure for storing subsets of a finite universe U , so that membership queries can be answered efficiently. We study this problem in a unit cost RAM model with word size Ω(log |U |), and show that for n-element subsets, constant worst case query time can be obtained using B +O(log log |U |) + o(n) bits of storage, where B = dlog2 (|U | n ) e is the minimum number of bits needed to represent all such subsets. The solution for dense subsets uses B + O( |U | log log |U | log |U | ) bits of storage, and supports constant time rank queries. In a dynamic setting, allowing insertions and deletions, our techniques give an O(B) bit space usage.