Optimal Lower Bounds for Rank and Select Indexes

Abstract

We develop a new lower bound technique for data structures. We show an optimal $\Omega(n \lg\lg n / \lg n)$ 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 $\Omega((m/t) \lg t)$ 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 the select index given in [Raman et al., SODA’02] and show how to implement both rank and select queries with an index of size $(1 + o(1)) (n \lg\lg n / \lg n) + O(n / \lg n)$ (i.e. we give an explicit constant for storage) in the RAM model with word size $\lg n$.