Approximate Selection with Unreliable Comparisons in Optimal Expected Time
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 (k-nε, 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 k-th 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 1-Q [Uriel Feige et al., 1994]. Considering the increasing complexity of modern computing, it is of great interest to develop approximation algorithms that enable a trade-off 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 Chih-Hung 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 1-Q. For k = n ε, the number of comparisons is O(ε^{-1} log 1/Q), matching Leucci and Liu’s result [Stefano Leucci and Chih-Hung 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 1-Q 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 k-th 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}).
Approximate Selection
Unreliable Comparisons
Independent Faults
Theory of computation~Design and analysis of algorithms
37:1-37:23
Regular Paper
https://arxiv.org/abs/2205.01448
The three authors began to investigate this topic when all of them were in ETH Zürich, Switzerland.
Shengyu
Huang
Shengyu Huang
Department of Computer Science, EPFL, Lausanne, Switzerland
Chih-Hung
Liu
Chih-Hung Liu
Department of Electrical Engineering, National Taiwan University, Taipei, Taiwan
https://orcid.org/0000-0001-9683-5982
Yushan Young Fellow Program by Ministry of Education, Taiwan and Research Project 111-2222-E-002-017-MY2 by National Science and Technology Council, Taiwan.
Daniel
Rutschmann
Daniel Rutschmann
Department of Applied Mathematics and Computer Science, Technical University of Denmark, Copenhagen, Denmark
10.4230/LIPIcs.STACS.2023.37
Martin Aigner. Finding the maximum and minimum. Discrete Applied Mathematics, 74(1):1-12, 1997.
Amitava Bagchi. On sorting in the presence of erroneous information. Information Processing Letters, 43(4):213-215, 1992.
Ryan S. Borgstrom and S. Rao Kosaraju. Comparison-based search in the presence of errors. In Proceedings of the Twenty-fifth Symposium on Theory of Computing (STOC93), pages 130-136, 1993.
Mark Braverman, Jieming Mao, and S. Matthew Weinberg. Parallel algorithms for select and partition with noisy comparisons. In Proceedings of the Forty-eighth48th Symposium on Theory of Computing (STOC16), pages 851-862, 2016.
Mark Braverman and Elchanan Mossel. Noisy sorting without resampling. In Proceedings of the Nineteenth Symposium on Discrete Algorithms (SODA08), pages 268-276, 2008.
Xi Chen, Sivakanth Gopi, Jieming Mao, and Jon Schneider. Competitive analysis of the top-k ranking problem. In Proceedings of the Twenty-Eighth Symposium on Discrete Algorithms (SODA17), pages 1245-1264, 2017.
Hyungmin Cho, Larkhoon Leem, and Subhasish Mitra. ERSA: error resilient system architecture for probabilistic applications. IEEE Trans. on CAD of Integrated Circuits and Systems, 31(4):546-558, 2012.
Ferdinando Cicalese. Fault-Tolerant Search Algorithms - Reliable Computation with Unreliable Information. Monographs in Theoretical Computer Science. Springer, 2013.
Uriel Feige, Prabhakar Raghavan, David Peleg, and Eli Upfal. Computing with noisy information. SIAM Journal on Computing, 23(5):1001-1018, 1994.
Irene Finocchi, Fabrizio Grandoni, and Giuseppe F. Italiano. Optimal resilient sorting and searching in the presence of memory faults. Theoretical Computer Science, 410(44):4457-4470, 2009.
Barbara Geissmann, Stefano Leucci, Chih-Hung Liu, and Paolo Penna. Sorting with recurrent comparison errors. In Proceedings of the Twenty-Eighth International Symposium on Algorithms and Computation (ISAAC17), pages 38:1-38:12, 2017.
Barbara Geissmann, Stefano Leucci, Chih-Hung Liu, and Paolo Penna. Optimal sorting with persistent comparison errors. In Proceedings of the Twenty-seventh European Symposium on Algorithms (ESA19), pages 49:1-49:14, 2019.
Barbara Geissmann, Stefano Leucci, Chih-Hung Liu, and Paolo Penna. Optimal dislocation with persistent errors in subquadratic time. Theory Comput. Syst., 64(3):508-521, 2020.
Barbara Geissmann, Matús Mihalák, and Peter Widmayer. Recurring comparison faults: Sorting and finding the minimum. In Proceedings of the Twentieth International Symposium on Fundamentals of Computation Theory (FCT15), pages 227-239, 2015.
Ofer Grossman and Dana Moshkovitz. Amplification and derandomization without slowdown. SIAM Journal on Computing, 49(5):959-998, 2020.
Jie Han and Michael Orshansky. Approximate computing: An emerging paradigm for energy-efficient design. In 18th IEEE European Test Symposium (ETS), pages 1-6, 2013.
Shengyu Huang, Chih-Hung Liu, and Daniel Rutschman. Approximate selection with unreliable comparisons in optimal expected time. CoRR, abs/2205.01448, 2022. URL: https://doi.org/10.48550/arXiv.2205.01448.
https://doi.org/10.48550/arXiv.2205.01448
Claire Kenyon-Mathieu and Warren Schudy. How to rank with few errors. In Proceedings of the Thirty-nineth Symposium on Theory of Computing (STOC07), pages 95-103, 2007.
Christoph M. Kirsch and Hannes Payer. Incorrect systems: it’s not the problem, it’s the solution. In Proceedings of the 49th Design Automation Conference 2012 (DAC), pages 913-917, 2012.
Rolf Klein, Rainer Penninger, Christian Sohler, and David P. Woodruff. Tolerant algorithms. In Proceedings of the Nineteenth European Symposium on Algorithms (ESA11), pages 736 - -747, 2011.
K. B. Lakshmanan, Bala Ravikumar, and K. Ganesan. Coping with erroneous information while sorting. IEEE Transactions on Computers, 40(9):1081-1084, 1991.
Tom Leighton and Yuan Ma. Tight bounds on the size of fault-tolerant merging and sorting networks with destructive faults. SIAM Journal on Computing, 29(1):258-273, 1999.
Stefano Leucci and Chih-Hung Liu. Approximate minimum selection with unreliable comparisons in optimal expected time. Algorithmica, 84(1):60-84, 2022.
Stefano Leucci, Chih-Hung Liu, and Simon Meierhans. Resilient dictionaries for randomly unreliable memory. In Proceedings of the 27th Annual European Symposium on Algorithms, (ESA19), pages 70:1-70:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019.
Philip M. Long. Sorting and searching with a faulty comparison oracle. Technical report, University of California at Santa Cruz, 1992.
Konstantin Makarychev, Yury Makarychev, and Aravindan Vijayaraghavan. Sorting noisy data with partial information. In Proceedings of the Fourth Conference on Innovations in Theoretical Computer Science (ITCS13), pages 515-528, 2013.
M. Mitzenmacher and E. Upfal. Probability and Computing: Randomization and Probabilistic Techniques in Algorithms and Data Analysis. Cambridge University Press, 2 edition, 2017.
Krishna Palem and Avinash Lingamneni. Ten years of building broken chips: The physics and engineering of inexact computing. ACM Transactions on Embedded Computing Systems, 12(2s):87:1-87:23, 2013.
Andrzej Pelc. Searching with known error probability. Theoretical Computer Science, 63(2):185-202, 1989.
Andrzej Pelc. Searching games with errors - fifty years of coping with liars. Theoretical Computer Science, 270(1-2):71-109, 2002.
Bala Ravikumar, K. Ganesan, and K. B. Lakshmanan. On selecting the largest element in spite of erroneous information. In Proceedings of the fourth Symposium on Theoretical Aspects of Computer Science (STACs87), pages 88-99, 1987.
Joseph Sloan, John Sartori, and Rakesh Kumar. On software design for stochastic processors. In Proceedings of the 49th Annual Design Automation Conference 2012 (DAC), pages 918-923, 2012.
Shengyu Huang, Chih-Hung Liu, and Daniel Rutschmann
Creative Commons Attribution 4.0 International license
https://creativecommons.org/licenses/by/4.0/legalcode