Inverse Range Selection Queries

Abstract

On a given sequence $X=\langle x_1x_2\ldots x_n \rangle$, the range selection queries denoted by Q(i, j, k) return the $k^{th}$-smallest element on $\langle x_ix_{i+1}\ldots x_j \rangle$. The problem has received significant attention in recent years and many solutions aiming to achieve this task with a cost lower than dynamically sorting the elements on the queried range have been proposed. The reverse problem interestingly has not yet received that much attention, although there exists practical usage scenarios especially in the time–series analysis. This study investigates the inverse range selection query $\bar{Q}(\upsilon ,k)$ that aims to detect all possible intervals on X such that the $k^{th}$-smallest element is less than or equal to $\upsilon$. We present the basic solution first and then discuss how that basic solution can be implemented with different data structures previously proposed for regular range selection queries.