Result-Sensitive Binary Search with Noisy Information
We describe new algorithms for the predecessor problem in the Noisy Comparison Model. In this problem, given a sorted list L of n (distinct) elements and a query q, we seek the predecessor of q in L: denoted by u, the largest element less than or equal to q. In the Noisy Comparison Model, the result of a comparison between two elements is non-deterministic. Moreover, multiple comparisons of the same pair of elements might have different results: each is generated independently, and is correct with probability p > 1/2. Given an overall error tolerance Q, the cost of an algorithm is measured by the total number of noisy comparisons; these must guarantee the predecessor is returned with probability at least 1 - Q. Feige et al. showed that predecessor queries can be answered by a modified binary search with Theta(log (n/Q)) noisy comparisons.
We design result-sensitive algorithms for answering predecessor queries. The query cost is related to the index, k, of the predecessor u in L. Our first algorithm answers predecessor queries with O(log ((log^{*(c)} n)/Q) + log (k/Q)) noisy comparisons, for an arbitrarily large constant c. The function log^{*(c)} n iterates c times the iterated-logarithm function, log^* n. Our second algorithm is a genuinely result-sensitive algorithm whose expected query cost is bounded by O(log (k/Q)), and is guaranteed to terminate after at most O(log((log n)/Q)) noisy comparisons.
Our results strictly improve the state-of-the-art bounds when k is in omega(1) intersected with o(n^epsilon), where epsilon > 0 is some constant. Moreover, we show that our result-sensitive algorithms immediately improve not only predecessor-query algorithms, but also binary-search-like algorithms for solving key applications.
Fault-tolerant search
random walks
noisy comparisons
predecessor queries
Theory of computation~Sorting and searching
Theory of computation~Predecessor queries
60:1-60:15
Regular Paper
We thank our anonymous reviewer for directing us to the work of Karp and Kleinberg [Karp and Kleinberg, 2007].
Narthana S.
Epa
Narthana S. Epa
School of Computing and Information Systems, The University of Melbourne, Victoria, Australia
Junhao
Gan
Junhao Gan
School of Computing and Information Systems, The University of Melbourne, Victoria, Australia
https://orcid.org/0000-0001-9101-1503
Anthony
Wirth
Anthony Wirth
School of Computing and Information Systems, The University of Melbourne, Victoria, Australia
https://orcid.org/0000-0003-3746-6704
Funded by the Melbourne School of Engineering.
10.4230/LIPIcs.ISAAC.2019.60
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Narthana S. Epa, Junhao Gan, and Anthony Wirth
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