Optimal Encodings for Range Min-Max and Top-k

In this paper we consider various encoding problems for range queries on arrays. In these problems, the goal is that the encoding occupies the information theoretic minimum space required to answer a particular set of range queries. Given an array $A[1..n]$ a range top-$k$ query on an arbitrary range $[i,j] \subseteq [1,n]$ asks us to return the ordered set of indices $\{l_1 ,...,l_k \}$ such that $A[l_m]$ is the $m$-th largest element in $A[i..j]$. We present optimal encodings for range top-$k$ queries, as well as for a new problem which we call range min-max, in which the goal is to return the indices of both the minimum and maximum element in a range

Optimal Encodings for Range Majority Queries

Artículo de publicación ISIWe study the problem of designing a data structure that reports the positions of the distinct τ -majorities within any range of an array A[1, n], without storing A. A τ -majority in a range A[i, j ], for 0 < τ < 1, is an element that occurs more than τ( j − i + 1) times in A[i, j ]. We show that (n log(1/τ ) ) bits are necessary for any data structure just able to count the number of distinct τ -majorities in any range. Then, we design a structure using O(n log(1/τ ) ) bits that returns one position of each τ -majority of A[i, j ] in O((1/τ ) log logw(1/τ ) log n) time, on a RAM machine with word size w (it can output any further position where each τ -majority occurs in O(1) additional time). Finally, we show how to remove a log n factor from the time by adding O(n log log n) bits of space to the structure.Millennium Nucleus Information and Coordination in Networks ICM/FIC, Chile P10-024