Asympotically optimal encodings for range selection

We consider the problem of preprocessing an array A[1..n] to answer range selection and range top-k queries. Given a query interval [i..j] and a value k, the former query asks for the position of the kth largest value in A[i..j], whereas the latter asks for the positions of all the k largest values in A[i..j]. We consider the encoding version of the problem, where A is not available at query time, and an upper bound kappa on k, the rank that is to be selected, is given at construction time. We obtain data structures with asymptotically optimal size and query time on a RAM model with word size Θ(lg n) : our structures use O(n lg kappa) bits and answer range selection queries in time O(1+ lg k / lg lg n) and range top-k queries in time O(k), for any k ≤ kappa.