### 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(a_{i},a_{j}), for i < j, where q(a_{i}, 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}(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 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 language | English (US) |
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Title of host publication | Proceedings of the 19th Annual ACM-SIAM Symposium on Discrete Algorithms |

Pages | 268-276 |

Number of pages | 9 |

State | Published - Dec 1 2008 |

Externally published | Yes |

Event | 19th Annual ACM-SIAM Symposium on Discrete Algorithms - San Francisco, CA, United States Duration: Jan 20 2008 → Jan 22 2008 |

### Publication series

Name | Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms |
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### Other

Other | 19th Annual ACM-SIAM Symposium on Discrete Algorithms |
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Country | United States |

City | San Francisco, CA |

Period | 1/20/08 → 1/22/08 |

### All Science Journal Classification (ASJC) codes

- Software
- Mathematics(all)

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## Cite this

*Proceedings of the 19th Annual ACM-SIAM Symposium on Discrete Algorithms*(pp. 268-276). (Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms).