Optimal Lower Bounds for Rank and Select Indexes

We develop a new lower bound technique for data structures. We show an optimal Ω(n lg lgn/ lgn) space lower bounds for storing an index that allows to implement rank and select queries on a bit vector B provided that B is stored explicitly. These results improve upon [Miltersen, SODA’05]. We show Ω((m/t) lg t) lower bounds for storing rank/select index in the case where B has m 1-bits in it (e.g. low 0-th entropy) and the algorithm is allowed to probe t bits of B. We simplify select index given in [Raman et al., SODA’02] and show how to implement both rank and select queries with an index of size (1+o(1))(n lg lgn/ lgn)+O(n/ lgn) (i.e. we give an explicit constant for storage) in the RAM model with word size lgn.