Optimal Encodings for Range Top-k, Selection, and Min-Max

We consider encoding problems for range queries on arrays. In these problems the goal is to store a structure capable of recovering the answer to all queries that occupies the information theoretic minimum space possible, to within lower order terms. As input, we are given an array A[1..n], and a fixed parameter k is an element of [1, n]. A range top-k query on an arbitrary range [i, j] subset of [1, n] asks us to return the ordered set of indices {l1, ..., l(k)} such that A[l(m)] is the m-th largest element in A[i..j], for 1 <= m <= k. A range selection query for an arbitrary range [i, j] subset of [1, n] and query parameter k' is an element of [1, k] asks us to return the index of the k'-th largest element in A[i..j]. We completely resolve the space complexity of both of these heavily studied problems-to within lower order terms-for all k = o(n). Previously, the constant factor in the space complexity was known only for k = 1. We also resolve the space complexity of another problem, that we call range min-max, in which the goal is to return the indices of both the minimum and maximum elements in a range.