New Results for Adaptive and Approximate Counting of Inversions

Counting inversions is a classic and important problem in databases. The number of inversions, K∗, in a list L = (L(1), L(2), . . . , L(n)) is defined as the number of pairs i < j with L(i) > L(j). In this paper, new results for this problem are presented: 1. In the I/O-model, an adaptive algorithm is presented for calculating K∗. The algorithm performs O( B + N B logM/B( K∗ NB )) I/Os. When K∗ = O(NM), then the algorithm takes only O( B ) I/Os. This algorithm can be modified to match the state of the art for the comparison based model and the RAM model. 2. In the RAM model, a linear-time algorithm is presented to obtain a tight estimate of K∗; specifically a value which lies with high probability in the range [(1 − logN N1/4 )K∗, (1 + logN N1/4 )K∗]. The state of the art linear-time algorithm works for the special case where L is a permutation, i.e., each L(i) is a distinct integer in the range [1, N ]. In this paper, we handle a general case where each L(i) is a real number.