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Algorithms for p-Faulty Search on a Half-Line

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Abstract

We study p-Faulty Search, a variant of the classic cow-path optimization problem, where a unit speed robot searches the half-line (or 1-ray) for a hidden item. The searcher is probabilistically faulty, and detection of the item with each visitation is an independent Bernoulli trial whose probability of success p is known. The objective is to minimize the worst case expected detection time, relative to the distance of the hidden item to the origin. A variation of the same problem was first proposed by Gal (Search games, Academic Press, New York, 1980). Alpern and Gal (The theory of search games and rendezvous, Springer, Berlin, 2003) proposed a so-called monotone solution for searching the line (2-rays); that is, a trajectory in which the newly searched space increases monotonically in each ray and in each iteration. Moreover, they conjectured that an optimal trajectory for the 2-rays problem must be monotone. We show that an analogous conjecture for the case where the search domain is the half-line cannot be correct. Indeed, we provide a lower bound for all monotone algorithms, which we also match with an upper bound. Our main contribution is the design and analysis of a sequence of refined search strategies, outside the family of monotone algorithms, which we call t-sub-monotone algorithms. Such algorithms induce performance that is strictly decreasing with t, and for all \(p \in (0,1)\). The value of t quantifies, in a certain sense, how much our algorithms deviate from being monotone, demonstrating that monotone algorithms are sub-optimal when searching the half-line.

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Notes

  1. Cyclic and monotone patterns are natural search strategies since they balance the frequency that each direction is explored with the rate that new space is explored.

  2. Alternatively, we could have defined monotone trajectories so as to return to location 1, instead of the origin, since we know that \(d\ge 1\). Our analysis next shows that such a modification would not improve the competitive ratio.

  3. The numerical computation is done in Mathematica over a refined discretization of interval (0, 1) so as to produce smooth results.

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Acknowledgements

The authors would like to thank Huda Chuangpishit, Sophia Park, Bhargav Parsi and Benjamin Reiniger for many fruitful discussions.

Funding

Funding was provided by Canadian Network for Research and Innovation in Machining Technology, Natural Sciences and Engineering Research Council of Canada.

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Authors and Affiliations

  1. Department of Mathematics, Toronto Metropolitan University, 350 Victoria St., Toronto, ON, M5B 2K3, Canada

    Anthony Bonato, Konstantinos Georgiou & Paweł Prałat

  2. Department of Computer Science, University of Toronto, Toronto, ON, M5S 2E4, Canada

    Calum MacRury

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  1. Anthony Bonato
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Correspondence to Konstantinos Georgiou.

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Research supported in part by NSERC.

An extended abstract of this paper appeared in the proceedings of the 14th Latin American Theoretical Informatics Symposium (LATIN’20), São Paulo, Brazil, May 25–29, 2020.

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Bonato, A., Georgiou, K., MacRury, C. et al. Algorithms for p-Faulty Search on a Half-Line. Algorithmica 85, 2485–2514 (2023). https://doi.org/10.1007/s00453-022-01075-y

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