Counting inversions in lists

In a recent paper, Ajtai et al. [1] give a streaming algorithm to count the number of inversions in a stream Ä ¾ Ñ Ò using two passes and Ç´¯ ½ ÔÒÐÓ Ò´ÐÓ Ñ·ÐÓ Òµµspace. Here, we present a simple randomized streaming algorithm for the same problem that uses one pass and Ç´¯ ¿ ÐÓ ¾ ÒÐÓ Ñµ space. Our algorithm is based on estimating quantiles of the items already seen in the stream, and using that to estimate the number of inversions involving each element. 1 Preliminaries Let Ä be the list of elements appearing as a stream with the ’th element being denoted by Ä . The quantity we want to approximate is ôĵ, the number of inversions in Ä; this is the number of pairs such that Ä Ä . In order to simplify our notation, we restate this in an equivalent form, that of counting non-inversions in the list when the total order, and thus the results of all strict comparisons, has been reversed. Thus, all comparisons between list elements in what fol...

• Type: Conference
• Year: 2003
• Where: SODA
• Authors: Anupam Gupta, Francis Zane
• Keywords: Algorithms, Randomized Streaming Algorithm, SODA 2003, Streaming Algorithm, Total Order