Counting Inversions, Offline Orthogonal Range Counting, and Related Problems

Abstract

Previous chapter Next chapter Full AccessProceedings Proceedings of the 2010 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)Counting Inversions, Offline Orthogonal Range Counting, and Related ProblemsTimothy M. Chan and Mihai PătraşcuTimothy M. Chan and Mihai Pătraşcupp.161 - 173Chapter DOI:https://doi.org/10.1137/1.9781611973075.15PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstract We give an -time algorithm for counting the number of inversions in a permutation on n elements. This improves a long-standing previous bound of O(n lg n/lg lg n) that followed from Dietz's data structure [WADS'89], and answers a question of Andersson and Petersson [SODA'95]. As Dietz's result is known to be optimal for the related dynamic rank problem, our result demonstrates a significant improvement in the offline setting. Our new technique is quite simple: we perform a “vertical partitioning” of a trie (akin to van Emde Boas trees), and use ideas from external memory. However, the technique finds numerous applications: for example, we obtain in d dimensions, an algorithm to answer n offline orthogonal range counting queries in time O(n lgd–2+1/d n); an improved construction time for online data structures for orthogonal range counting; an improved update time for the partial sums problem; faster Word RAM algorithms for finding the maximum depth in an arrangement of axis-aligned rectangles, and for the slope selection problem. As a bonus, we also give a simple (1 + ε)-approximation algorithm for counting inversions that runs in linear time, improving the previous O(n lg lg n) bound by Andersson and Petersson. Previous chapter Next chapter RelatedDetails Published:2010ISBN:978-0-89871-701-3eISBN:978-1-61197-307-5 https://doi.org/10.1137/1.9781611973075Book Series Name:ProceedingsBook Code:PR135Book Pages:xviii + 1667