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Result-Sensitive Binary Search with Noisy Information

Authors Narthana S. Epa, Junhao Gan , Anthony Wirth



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Author Details

Narthana S. Epa
  • School of Computing and Information Systems, The University of Melbourne, Victoria, Australia
Junhao Gan
  • School of Computing and Information Systems, The University of Melbourne, Victoria, Australia
Anthony Wirth
  • School of Computing and Information Systems, The University of Melbourne, Victoria, Australia

Acknowledgements

We thank our anonymous reviewer for directing us to the work of Karp and Kleinberg [Karp and Kleinberg, 2007].

Cite AsGet BibTex

Narthana S. Epa, Junhao Gan, and Anthony Wirth. Result-Sensitive Binary Search with Noisy Information. In 30th International Symposium on Algorithms and Computation (ISAAC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 149, pp. 60:1-60:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.ISAAC.2019.60

Abstract

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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Sorting and searching
  • Theory of computation → Predecessor queries
Keywords
  • Fault-tolerant search
  • random walks
  • noisy comparisons
  • predecessor queries

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