Approximate Minimum Selection with Unreliable Comparisons
Journal
Algorithmica
Journal Volume
84
Journal Issue
1
Pages
60-84
Date Issued
2022
Author(s)
Leucci S
Abstract
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 a small probability that the wrong comparison outcome is observed. We design a randomized algorithm that solves this problem with a success probability of at least 1 - q for q∈(0,n-kn) and any k∈ [1 , n- 1] using O(nk⌈log1q⌉) comparisons in expectation (if k≥ n or q≥n-kn the problem becomes trivial). Then, we prove that the expected number of comparisons needed by any algorithm that succeeds with probability at least 1 - q must be Ω(nklog1q) whenever q is bounded away from n-kn, thus implying that the expected number of comparisons performed by our algorithm is asymptotically optimal in this range. Moreover, we show that the approximate minimum selection problem can be solved using O((nk+loglog1q)log1q) comparisons in the worst case, which is optimal when q is bounded away from n-kn and k=O(nloglog1q). © 2021, The Author(s).
Subjects
Approximate minimum selection; Independent errors; Unreliable comparisons
Other Subjects
Artificial intelligence; Approximate minimum selection; Asymptotically optimal; Comparison faults; Independent error; K elements; Pair-wise comparison; Randomized Algorithms; Selection problems; Unreliable comparison; Probability
Type
journal article