Computing the distribution of spearman's footrule in O ( n^4 ) time

Abstract

Consider the exact distribution of Spearman's "footrule", D(\\sigma, \\pi) = \\sum_{i=1}^n \\vert \\sigma (i) - \\pi (i)\\vert , where \\sigma and \\pi are chosen independently at random from S_n , the set of all permutations of the first n integers; or, equivalently, where \\sigma is chosen at random and \\pi = (1,2,\\ldots,n) . This can obviously be obtained by complete enumeration of all n ! permutations: that is, in O(n!) time. However, we show that the intrinsically different approach used by Salama and Quade (1990) reduces the computation to the polynomial-time class, specifically, to O(n^4) time.