An Almost Optimal Algorithm for Unbounded Search with Noisy Information
Given a sequence of integers, đź = sâ, sâ,âŠ in ascending order, called the search domain, and an integer t, called the target, the predecessor problem asks for the target index N such that s_N is the largest integer in đź satisfying s_N â€ t. We consider solving the predecessor problem with the least number of queries to a binary comparison oracle. For each query index i, the oracle returns whether s_i â€ t or s_i > t. In particular, we study the predecessor problem under the UnboundedNoisy setting, where (i) the search domain đź is unbounded, i.e., n = |đź| is unknown or infinite, and (ii) the binary comparison oracle is noisy. We denote the former setting by Unbounded and the latter by Noisy. In Noisy, the oracle, for each query, independently returns a wrong answer with a fixed constant probability 0 < p < 1/2. In particular, even for two queries on the same index i, the answers from the oracle may be different. Furthermore, with a noisy oracle, the goal is to correctly return the target index with probability at least 1- Q, where 0 < Q < 1/2 is the failure probability.
Our first result is an algorithm, called NoS, for Noisy that improves the previous result by Ben-Or and Hassidim [FOCS 2008] from an expected query complexity bound to a worst-case bound. We also achieve an expected query complexity bound, whose leading term has an optimal constant factor, matching the lower bound of Ben-Or and Hassidim. Building on NoS, we propose our NoSU algorithm, which correctly solves the predecessor problem in the UnboundedNoisy setting. We prove that the query complexity of NoSU is â_{i = 1}^k (log^{(i)} N) /(1-H(p))+ o(log N) when log Q^{-1} â o(log N), where N is the target index, k = log^* N, the iterated logarithm, and H(p) is the entropy function. This improves the previous bound of O(log (N/Q) / (1-H(p))) by reducing the coefficient of the leading term from a large constant to 1. Moreover, we show that this upper bound can be further improved to (1 - Q) â_{i = 1}^k (log^{(i)} N) /(1-H(p))+ o(log N) in expectation, with the constant in the leading term reduced to 1 - Q. Finally, we show that an information-theoretic lower bound on the expected query cost of the predecessor problem in UnboundedNoisy is at least (1 - Q)(â_{i = 1}^k log^{(i)} N - 2k)/(1-H(p)) - 10. This implies the constant factor in the leading term of our expected upper bound is indeed optimal.
Fault-tolerant search
noisy binary search
query complexity
Theory of computation~Predecessor queries
Theory of computation~Sorting and searching
25:1-25:15
Regular Paper
We thank PrzemysĆaw UznaĆski and Dariusz Dereniowski for the discussions on noisy search, particularly for highlighting the nuances of the binary and ternary comparison models.
Junhao
Gan
Junhao Gan
School of Computing and Information Systems, The University of Melbourne, Australia
https://orcid.org/0000-0001-9101-1503
J.G. is supported in part by Australian Research Council (ARC) Discovery Early Career Researcher Award (DECRA) DE190101118.
Anthony
Wirth
Anthony Wirth
School of Computing and Information Systems, The University of Melbourne, Australia
https://orcid.org/0000-0003-3746-6704
A.W. is supported by the Faculty of Engineering and Information Technology at The University of Melbourne.
Xin
Zhang
Xin Zhang
School of Computing and Information Systems, The University of Melbourne, Australia
https://orcid.org/0000-0001-5411-4968
X. Z.âs research is supported by an Australian Government Research Training Program (RTP) Scholarship.
10.4230/LIPIcs.SWAT.2022.25
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Junhao Gan, Anthony Wirth, and Xin Zhang
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