A New Weighted Spearman's Footrule as A Measure of Distance between Rankings

TL;DR

This paper introduces a weighted Spearman's footrule metric that emphasizes top-ranked items and provides an efficient approximation method for rank aggregation, improving the measurement of ranking similarities.

Contribution

It proposes a novel weighted Spearman's footrule metric tailored for ranking comparisons, with a polynomial time algorithm for rank aggregation and a 2-approximation guarantee.

Findings

The weighted metric emphasizes top-ranked items effectively.

The algorithm operates in polynomial time.

Achieves a 2-approximation for the weighted Kendall's tau distance.

Abstract

Many applications motivate the distance measure between rankings, such as comparing top-k lists and rank aggregation for voting, and intrigue great interest to researchers. For example, for a search engine, the use of different ranking algorithms may return different ranking lists. The effect of a ranking algorithm can be estimated by computing the distance (similarity) between the result ranking it returns and the appropriate ranking people expect. People may be interested in only the first few items of result ranking, therefore the metric for measuring the distance should emphasize on the items in higher positions. Besides, in an extreme case, if a result ranking is the total reverse of the expected ranking, then it is considered to be the worst ranking with the maximum distance. Therefore, a metric is called for, which can satisfy both of the two intuitions. To address this problem, we present a weighted metric based on the classical Spearman's footrule metric to measure the distance between two permutations of n objects. This metric can be applied in rank aggregation problem with a polynomial time algorithm, and produces a 2-approximation for adopting the weighted Kendall's tau distance proposed by Farnoud et al.