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- W1998730018 abstract "We give an O(n √lg n)-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/dn);• 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 + e)-approximation algorithm for counting inversions that runs in linear time, improving the previous O (n lg lg n) bound by Andersson and Petersson." @default.
- W1998730018 created "2016-06-24" @default.
- W1998730018 creator A5031221610 @default.
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- W1998730018 date "2010-01-17" @default.
- W1998730018 modified "2023-10-18" @default.
- W1998730018 title "Counting inversions, offline orthogonal range counting, and related problems" @default.
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- W1998730018 doi "https://doi.org/10.5555/1873601.1873616" @default.
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