Abstract
Given n elements, an integer k ≤ n/2 and a parameter ε ≥ 1/n, we study the problem of selecting an element with rank in (knε, k+nε] using unreliable comparisons where the outcome of each comparison is incorrect independently with a constant error probability, and multiple comparisons between the same pair of elements are independent. In this fault model, the fundamental problems of finding the minimum, selecting the kth smallest element and sorting have been shown to require Θ(n log 1/Q), Θ(n log k/Q) and Θ(n log n/Q) comparisons, respectively, to achieve success probability 1Q [Uriel Feige et al., 1994]. Considering the increasing complexity of modern computing, it is of great interest to develop approximation algorithms that enable a tradeoff between the solution quality and the number of comparisons. In particular, approximation algorithms would even be able to attain a sublinear number of comparisons. Very recently, Leucci and Liu [Stefano Leucci and ChihHung Liu, 2022] proved that the approximate minimum selection problem, which covers the case that k ≤ nε, requires expected Θ(ε^{1} log 1/Q) comparisons, but the general case, i.e., for nε < k ≤ n/2, is still open.
We develop a randomized algorithm that performs expected O(k/n ε^{2} log 1/Q) comparisons to achieve success probability at least 1Q. For k = n ε, the number of comparisons is O(ε^{1} log 1/Q), matching Leucci and Liu’s result [Stefano Leucci and ChihHung Liu, 2022], whereas for k = n/2 (i.e., approximating the median), the number of comparisons is O(ε^{2} log 1/Q). We also prove that even in the absence of comparison faults, any randomized algorithm with success probability at least 1Q performs expected Ω(min{n, k/n ε^{2} log 1/Q}) comparisons. As long as n is large enough, i.e., when n = Ω(k/n ε^{2} log 1/Q), our lower bound demonstrates the optimality of our algorithm, which covers the possible range of attaining a sublinear number of comparisons. Surprisingly, for constant Q, our algorithm performs expected O(k/n ε^{2}) comparisons, matching the best possible approximation algorithm in the absence of computation faults. In contrast, for the exact selection problem, the expected number of comparisons is Θ(n log k) with faults versus Θ(n) without faults. Our results also indicate a clear distinction between approximating the minimum and approximating the kth smallest element, which holds even for the high probability guarantee, e.g., if k = n/2, Q = 1/n and ε = n^{α} for α ∈ (0, 1/2), the asymptotic difference is almost quadratic, i.e., Θ̃(n^α) versus Θ̃(n^{2α}).
BibTeX  Entry
@InProceedings{huang_et_al:LIPIcs.STACS.2023.37,
author = {Huang, Shengyu and Liu, ChihHung and Rutschmann, Daniel},
title = {{Approximate Selection with Unreliable Comparisons in Optimal Expected Time}},
booktitle = {40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)},
pages = {37:137:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959772662},
ISSN = {18688969},
year = {2023},
volume = {254},
editor = {Berenbrink, Petra and Bouyer, Patricia and Dawar, Anuj and Kant\'{e}, Mamadou Moustapha},
publisher = {Schloss Dagstuhl  LeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2023/17689},
URN = {urn:nbn:de:0030drops176898},
doi = {10.4230/LIPIcs.STACS.2023.37},
annote = {Keywords: Approximate Selection, Unreliable Comparisons, Independent Faults}
}
Keywords: 

Approximate Selection, Unreliable Comparisons, Independent Faults 
Collection: 

40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023) 
Issue Date: 

2023 
Date of publication: 

03.03.2023 