# Mittagsseminar (in cooperation with A. Steger, D. Steurer and B. Sudakov)

__Mittagsseminar Talk Information__ | |

**Date and Time**: Tuesday, March 05, 2019, 12:15 pm

**Duration**: 30 minutes

**Location**: OAT S15/S16/S17

**Speaker**: Chih-Hung Liu

## A Nearly Optimal Algorithm for Approximate Minimum Selection with Unreliable Comparisons

We consider the approximate minimum selection problem in presence of independent random comparison faults. This problem asks to select one of the smallest k elements in a linearly ordered collection of n elements by only performing unreliable pairwise comparisons: whenever two elements are compared, there is constant probability that the wrong answer is returned. We design a randomized algorithm that solves this problem with high probability (w.h.p.) for the whole range of values of k using O(log n · ( n/k + log log log n)) expected time. Then, we prove that the expected running time of any algorithm that succeeds w.h.p. must be Omega((n/k) log n), thus implying that our algorithm is optimal, in expectation, for almost all values of k (and it is optimal up to triple-log factors for k = omega( n/log log log n )). These results are quite surprising in the sense that for k between Omega(log n) and c · n, for any constant c < 1, the expected running time must still be Omega((n/k)log n) even in absence of comparison faults.

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