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Noisy Sorting Without Resampling (0707.1051v1)

Published 6 Jul 2007 in cs.DS

Abstract: In this paper we study noisy sorting without re-sampling. In this problem there is an unknown order $a_{\pi(1)} < ... < a_{\pi(n)}$ where $\pi$ 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 $\pi(i) > \pi(j)$ 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 $\sigma$ that minimizes the number of pairs $\sigma(i) > \sigma(j)$ where $q(\sigma(i),\sigma(j)) = -$. 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 $\pi$. 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 $n{O(\gamma{-4})}$ and sampling complexity $O_{\gamma}(n \log n)$ that with high probability solves the noisy sorting without re-sampling problem. We also show that if $a_{\sigma(1)},a_{\sigma(2)},...,a_{\sigma(n)}$ is an optimal solution of the problem then it is ``close'' to the original order. More formally, with high probability it holds that $\sum_i |\sigma(i) - \pi(i)| = \Theta(n)$ and $\max_i |\sigma(i) - \pi(i)| = \Theta(\log n)$. Our results are of interest in applications to ranking, such as ranking in sports, or ranking of search items based on comparisons by experts.

Citations (204)

Summary

  • The paper presents a randomized algorithm that resolves noisy sorting without resampling, delivering high-probability near-optimal order recovery.
  • It achieves efficient performance with polynomial time complexity and requires only O(n log n) comparisons while bounding permutation deviations.
  • The research offers practical insights applicable to sports ranking and expert evaluations, providing a scalable solution for noise-affected pairwise comparisons.

Overview of Noisy Sorting Without Resampling

The paper "Noisy sorting without resampling" by Mark Braverman and Elchanan Mossel addresses the challenging problem of ranking or ordering items when the input comparisons are subject to noise. This is a significant issue in real-world scenarios such as sports rankings and expert evaluation of items, where repeated comparisons are impractical or impossible.

Problem Definition and Importance

The central focus of this research is on the so-called "Noisy Sorting Without Resampling" (NSWR) problem. Here, the task is to determine an order σ\sigma for a set of nn items, given noisy pairwise comparisons, that maximizes the likelihood of reflecting the true order π\pi. The problem involves a permutation π\pi on nn elements such that aπ(1)<aπ(2)<<aπ(n)a_{\pi(1)} < a_{\pi(2)} < \ldots < a_{\pi(n)}. The comparisons are modeled as noisy, with a probability greater than 12\frac{1}{2} representing correct outcomes. Unlike previous models that allow for repeated sampling, this research posits a single comparison for each pair, highlighting its practical implications where resampling is not feasible.

Key Results

The authors devise a randomized algorithm with time complexity nO(γ4)n^{O(\gamma^{-4})} and sampling complexity Oγ(nlogn)O_{\gamma}(n \log n), which capably resolves the NSWR problem with high probability. A notable outcome of their work is the demonstration that any optimal permutation σ\sigma obtained is "close" to the true permutation π\pi: it deviates from the true order by a sum distance of Θ(n)\Theta(n) and by a maximum distance of Θ(logn)\Theta(\log n) for individual elements.

  • Time and Sampling Efficiency: The algorithm operates in polynomial time concerning the number of items, and it requires O(nlogn)O(n \log n) comparisons, making it efficient and scalable.
  • Theoretical Boundaries: The authors provide rigorous proofs demonstrating that deviations in permutation discrepancies are restrained to linear and logarithmic scales, respectively.

Implications and Future Directions

The implications of this work extend into several domains where accurate ranking under uncertain comparisons is critical. This includes sports analytics, information retrieval, and decision-making processes reliant on expert evaluations. Given the NP-hard nature of the general feedback arc set problem, the proposed approach offers a practically implementable solution that balances computational feasibility and accuracy.

Future research could delve into extending this framework to more complex scenarios, such as those involving dynamic changes in the item set or exploring variations with even higher levels of noise. Further exploration could involve hybrid models that integrate information theory to optimize the noise handling capabilities of the sorting algorithm.

Conclusion

This paper provides a robust algorithmic solution to the NSWR problem by tightly bounding the permissible noise and computational resources, and it lays a solid groundwork for subsequent studies in noisy sorting without resampling. The blend of practical applications and theoretical advancements found in this paper equips it with the potential for extensive utility across diverse fields requiring efficient ranking methodologies.