Counting Inversions, Offline Orthogonal Range Counting, and Related Problems
T. Chan, and M. Patrascu.

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 lg d−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.
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