Dynamic Elias-Fano Representation

We show that it is possible to store a dynamic ordered set S(n, u) of n integers drawn from a bounded universe of size u in space close to the information-theoretic lower bound and yet preserve the asymptotic time optimality of the operations. Our results leverage on the EliasFano representation of S(n, u) which takes EF(S(n, u)) = ndlog u ne+ 2n bits of space and can be shown to be less than half a bit per element away from the information-theoretic minimum. Considering a RAM model with memory words of Θ(log u) bits, we focus on the case in which the integers of S are drawn from a polynomial universe of size u = n , for any γ = Θ(1). We represent S(n, u) with EF(S(n, u)) + o(n) bits of space and: 1. support static predecessor/successor queries in O(min{1 + log u n , log logn}); 2. make S grow in an append-only fashion by spending O(1) per inserted element; 3. support random access in O(logn/ log logn) worstcase, insertions/deletions in O(logn/ log logn) amortized and predecessor/successor queries in O(min{1 + log u n , log logn}) worst-case time. These time bounds are optimal. 1998 ACM Subject Classification E.1 Data Structures, E.4 Coding and Information Theory, F.2.2 Nonnumerical Algorithms and Problems