Asymptotically Optimal Encodings of Range Data Structures for Selection and Top- k Queries

Abstract

Given an array A [1, n ] of elements with a total order, we consider the problem of building a data structure that solves two queries: ( a ) selection queries receive a range [ i , j ] and an integer k and return the position of the k th largest element in A [ i , j ]; ( b ) top- k queries receive [ i , j ] and k and return the positions of the k largest elements in A [ i , j ]. These problems can be solved in optimal time, O (1+lg k /lg lg n ) and O ( k ), respectively, using linear-space data structures. We provide the first study of the encoding data structures for the above problems, where A cannot be accessed at query time. Several applications are interested in the relative order of the entries of A , and their positions, rather their actual values, and thus we do not need to keep A at query time. In those cases, encodings save storage space: we first show that any encoding answering such queries requires n lg k - O ( n + k lg k ) bits of space; then, we design encodings using O ( n lg k ) bits, that is, asymptotically optimal up to constant factors, while preserving optimal query time.