Counting Inversions, Offline Orthogonal Range Counting, and Related Problems

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/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 m...

• Type: Conference
• Year: 2010
• Where: SODA
• Authors: Timothy M. Chan, Mihai Patrascu
• Keywords: Discrete Algorithms, Lg N)-time Algorithm, Lg N/, Orthogonal Range Counting, SODA 2010