Static Dictionaries Supporting Rank

A static dictionary is a data structure for storing a subset S of a nite universe U so that membership queries can be answered eeciently. We explore space eecient structures to also nd the rank of an element if found. We rst give a representation of a static dictionary that takes n lg m + O(lg lg m) bits of space and supports membership and rank (of an element present in S) queries in constant time, where n = jSj and m = jU j. Using our structure we also give a representation of a m-ary cardinal tree with n nodes using ndlg me + 2n + o(n) bits of space that supports the tree navigational operations in O(1) time, when m is o(2 lg n= lg lg n). For arbitrary m, we give a structure that takes the same space and supports all the navigational operations, except nding the child labeled i (for any i), in O(1) time. Finding the child labeled i in this structure takes O(lg lg lg m) time.