The Random Bit Complexity of Mobile Robot Scattering

Abstract : We consider the problem of scattering $n$ robots in a two dimensional continuous space. As this problem is impossible to solve in a deterministic manner, all solutions must be probabilistic. We investigate the amount of randomness (that is, the number of random bits used by the robots) that is required to achieve scattering. We first prove that $n \log n$ random bits are necessary to scatter $n$ robots in any setting. Also, we give a sufficient condition for a scattering algorithm to be random bit optimal. As it turns out that previous solutions for scattering satisfy our condition, they are hence proved random bit optimal for the scattering problem. Then, we investigate the time complexity of scattering when strong multiplicity detection is not available. We prove that such algorithms cannot converge in constant time in the general case and in $o(\log \log n)$ rounds for random bits optimal scattering algorithms. However, we present a family of scattering algorithms that converge as fast as needed without using multiplicity detection. Also, we put forward a specific protocol of this family that is random bit optimal ($O(n \log n)$ random bits are used) and time optimal ($O(\log \log n)$ rounds are used). This improves the time complexity of previous results in the same setting by a $\log n$ factor. Aside from characterizing the random bit complexity of mobile robot scattering, our study also closes the time complexity gap with and without strong multiplicity detection (that is, $O(1)$ time complexity is only achievable when strong multiplicity detection is available, and it is possible to approach a constant value as desired otherwise).
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https://hal.sorbonne-universite.fr/hal-01349017
Contributor : Sébastien Tixeuil <>
Submitted on : Tuesday, July 26, 2016 - 3:35:29 PM
Last modification on : Thursday, October 17, 2019 - 12:36:05 PM

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Quentin Bramas, Sébastien Tixeuil. The Random Bit Complexity of Mobile Robot Scattering. International Journal of Foundations of Computer Science, World Scientific Publishing, 2017, 28 (2), pp.111-133. ⟨10.1142/S0129054117500083⟩. ⟨hal-01349017⟩

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