This paper examines the problem of ranking a collection of objects using
pairwise comparisons (rankings of two objects). In general, the ranking of n
objects can be identified by standard sorting methods using nlog2​n
pairwise comparisons. We are interested in natural situations in which
relationships among the objects may allow for ranking using far fewer pairwise
comparisons. Specifically, we assume that the objects can be embedded into a
d-dimensional Euclidean space and that the rankings reflect their relative
distances from a common reference point in Rd. We show that under this
assumption the number of possible rankings grows like n2d and demonstrate
an algorithm that can identify a randomly selected ranking using just slightly
more than dlogn adaptively selected pairwise comparisons, on average. If
instead the comparisons are chosen at random, then almost all pairwise
comparisons must be made in order to identify any ranking. In addition, we
propose a robust, error-tolerant algorithm that only requires that the pairwise
comparisons are probably correct. Experimental studies with synthetic and real
datasets support the conclusions of our theoretical analysis.Comment: 17 pages, an extended version of our NIPS 2011 paper. The new version
revises the argument of the robust section and slightly modifies the result
there to give it more impac