Noisy sorting without resampling

Mark Braverman, Elchanan Mossel

Research output: Chapter in Book/Report/Conference proceedingConference contribution

96 Scopus citations

Abstract

In this paper we study noisy sorting without re-sampling. In this problem there is an unknown order aπ(1) < ... < aπ(n) where π is a permutation on n elements. The input is the status of (n/2) queries of the form q(ai,aj), for i < j, where q(ai, a j) = +(-) with probability 1/2 + γ if π(i) > π(j)(π(i) < π(j)) for all pairs i ≠ where γ > 0 is a constant. 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 perturbation of a linear order. Note that when γ >1/2 and n is large, it is impossible to recover the original order π. It is known that the weighted feedback arc set problem on tournaments is NP-hard in general. Here we present an algorithm of running time n O-4) and sampling complexity Oγ (n log n) that with high probability solves the noisy sorting without re-sampling problem. We also show that if aσ(i), aσ(2),..., aσ(n) is an optimal solution of the problem then it is "close" to the original order. More formally, with high probability it holds that Σ|σ(i) -π(i)| = Θ(n), max|σ(i) - π(i)| = Θ(log n).

Original languageEnglish (US)
Title of host publicationProceedings of the 19th Annual ACM-SIAM Symposium on Discrete Algorithms
Pages268-276
Number of pages9
StatePublished - Dec 1 2008
Externally publishedYes
Event19th Annual ACM-SIAM Symposium on Discrete Algorithms - San Francisco, CA, United States
Duration: Jan 20 2008Jan 22 2008

Publication series

NameProceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms

Other

Other19th Annual ACM-SIAM Symposium on Discrete Algorithms
CountryUnited States
CitySan Francisco, CA
Period1/20/081/22/08

All Science Journal Classification (ASJC) codes

  • Software
  • Mathematics(all)

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  • Cite this

    Braverman, M., & Mossel, E. (2008). Noisy sorting without resampling. In Proceedings of the 19th Annual ACM-SIAM Symposium on Discrete Algorithms (pp. 268-276). (Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms).