Noisy Sorting Without Resampling

Abstract

In this paper we study noisy sorting without re-sampling. In this problem there is an unknown order $a_{π(1)} < ... < a_{π(n)}$ where $π$ is a permutation on $n$ elements. The input is the status of $n \choose 2$ queries of the form $q(a_i,x_j)$, where $q(a_i,a_j) = +$ with probability at least $1/2+\ga$ if $π(i) > π(j)$ for all pairs $i \neq j$, where $\ga > 0$ is a constant and $q(a_i,a_j) = -q(a_j,a_i)$ for all $i$ and $j$. It is assumed that the errors are independent. Given the status of the queries the goal is to find the maximum likelihood order. In other words, the goal is find a permutation $σ$ that minimizes the number of pairs $σ(i) > σ(j)$ where $q(σ(i),σ(j)) = -$. The problem so defined is the feedback arc set problem on distributions of inputs, each of which is a tournament obtained as a noisy perturbations of a linear order. Note that when $\ga < 1/2$ and $n$ is large, it is impossible to recover the original order $π$. It is known that the weighted feedback are set problem on tournaments is NP-hard in general. Here we present an algorithm of running time $n^{O(γ^{-4})}$ and sampling complexity $O_γ(n \log n)$ that with high probability solves the noisy sorting without re-sampling problem. We also show that if $a_{σ(1)},a_{σ(2)},...,a_{σ(n)}$ is an optimal solution of the problem then it is ``close'' to the original order. More formally, with high probability it holds that $\sum_i |σ(i) - π(i)| = Θ(n)$ and $\max_i |σ(i) - π(i)| = Θ(\log n)$. Our results are of interest in applications to ranking, such as ranking in sports, or ranking of search items based on comparisons by experts.