Dynamic Range Selection in Linear Space

• Published in: Algorithms and Computation pp. 160 - 169
• Main Authors: He, Meng, Munro, J. Ian, Nicholson, Patrick K.
• Format: Book Chapter
• Language: English
• Published: Berlin, Heidelberg Springer Berlin Heidelberg 2011
• Series: Lecture Notes in Computer Science
• Subjects: Internal Node; Matrix Structure; Query Range; Query Time; Ranking Tree
• ISBN: 3642255906; 9783642255908
• ISSN: 0302-9743; 1611-3349
• DOI: 10.1007/978-3-642-25591-5_18

Given a set S of n points in the plane, we consider the problem of answering range selection queries on S: that is, given an arbitrary x-range Q and an integer k > 0, return the k-th smallest y-coordinate from the set of points that have x-coordinates in Q. We present a linear space data structure that maintains a dynamic set of n points in the plane with real coordinates, and supports range selection queries in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$O((\lg n / \lg \lg n)^2)$\end{document} time, as well as insertions and deletions in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$O((\lg n / \lg \lg n)^2)$\end{document} amortized time. The space usage of this data structure is an \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\Theta(\lg n / \lg \lg n)$\end{document} factor improvement over the previous best result, while maintaining asymptotically matching query and update times. We also present a succinct data structure that supports range selection queries on a dynamic array of n values drawn from a bounded universe.