PROMETHEE is not quadratic: An O(qnlog(n)) algorithm

It is generally believed that the preference ranking method PROMETHEE has a quadratic time complexity. In this paper, however, we present an exact algorithm that computes PROMETHEE's net flow scores in time O(qnlog(n)), where q represents the number of criteria and n the number of alternatives. The method is based on first sorting the alternatives after which the unicriterion flow scores of all alternatives can be computed in one scan over the sorted list of alternatives while maintaining a sliding window. This method works with the linear and level criterion preference functions. The algorithm we present is exact and, due to the sub-quadratic time complexity, vastly extends the applicability of the PROMETHEE method. Experiments show that with the new algorithm, PROMETHEE can scale up to millions of tuples.