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Evacuating from \(\ell _p\) Unit Disks in the Wireless Model

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Algorithms for Sensor Systems (ALGOSENSORS 2021)

Part of the book series: Lecture Notes in Computer Science ((LNCCN,volume 12961))

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Abstract

The search-type problem of evacuating 2 robots in the wireless model from the (Euclidean) unit disk was first introduced and studied by Czyzowicz et al. [DISC’2014]. Since then, the problem has seen a long list of follow-up results pertaining to variations as well as to upper and lower bound improvements. All established results in the area study this 2-dimensional search-type problem in the Euclidean metric space where the search space, i.e. the unit disk, enjoys significant (metric) symmetries.

We initiate and study the problem of evacuating 2 robots in the wireless model from \(\ell _p\) unit disks, \(p \in [1,\infty )\), where in particular robots’ moves are measured in the underlying metric space. To the best of our knowledge, this is the first study of a search-type problem with mobile agents in more general metric spaces. The problem is particularly challenging since even the circumference of the \(\ell _p\) unit disks have been the subject of technical studies. In our main result, and after identifying and utilizing the very few symmetries of \(\ell _p\) unit disks, we design optimal evacuation algorithms that vary with p. Our main technical contributions are two-fold. First, in our upper bound results, we provide (nearly) closed formulae for the worst case cost of our algorithms. Second, and most importantly, our lower bounds’ arguments reduce to a novel observation in convex geometry which analyzes trade-offs between arc and chord lengths of \(\ell _p\) unit disks as the endpoints of the arcs (chords) change position around the perimeter of the disk, which we believe is interesting in its own right. Part of our argument pertaining to the latter property relies on a computer assisted numerical verification that can be done for non-extreme values of p.

K. Georgiou—Research supported in part by NSERC.

J. Lucier—Research supported by a NSERC USRA.

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Notes

  1. 1.

    An underlying assumption is also that robots can distinguish points (x, y) by their coordinates, and they can move between them at will. As a byproduct, robots have a sense of orientation. This specification was not mentioned explicitly before for the Euclidean space, since all arguments were invariant under rotations (which is not the case any more). However, even in the \(\ell _2\) case this specification was silently assumed by fixing the cost of the optimal offline algorithm to 1 (a searcher that knows the location of the exit goes directly there), hence all previous results were performing competitive analysis by just doing worst case analysis.

  2. 2.

    For arbitrary algorithms one should define the cost as the supremum over all exit placements. Since in Algorithm Wireless-Search\(_p\)(\(\phi \)) the searched space remains contiguous and its boundaries keep expanding with time, the maximum always exists.

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Authors and Affiliations

  1. Department of Mathematics, Ryerson University, Toronto, ON, M5B 2K3, Canada

    Konstantinos Georgiou, Sean Leizerovich, Jesse Lucier & Somnath Kundu

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  1. Konstantinos Georgiou
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  2. Sean Leizerovich
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  3. Jesse Lucier
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  4. Somnath Kundu
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Correspondence to Konstantinos Georgiou .

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Editors and Affiliations

  1. University of Liverpool, Liverpool, UK

    Leszek Gąsieniec

  2. CNRS and University of Bordeaux, Talence, France

    Ralf Klasing

  3. King's College London, London, UK

    Tomasz Radzik

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Georgiou, K., Leizerovich, S., Lucier, J., Kundu, S. (2021). Evacuating from \(\ell _p\) Unit Disks in the Wireless Model. In: Gąsieniec, L., Klasing, R., Radzik, T. (eds) Algorithms for Sensor Systems. ALGOSENSORS 2021. Lecture Notes in Computer Science(), vol 12961. Springer, Cham. https://doi.org/10.1007/978-3-030-89240-1_6

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