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Distance-2-Dispersion: Dispersion with Further Constraints

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Networked Systems (NETYS 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14067))

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Abstract

The aim of the dispersion problem is to place a set of \(k~(\le n)\) mobile robots in the nodes of an unknown graph consisting of n nodes such that in the final configuration each node contains at most one robot, starting from any arbitrary initial configuration of the robots on the graph. In this work, we propose a variant of the dispersion problem, namely Distance-2-Dispersion, in short, D-2-D, where we start with any number of robots, and put an additional constraint that no two adjacent nodes contain robots in the final configuration. However, if a maximal independent set is already formed by the nodes which contain a robot each, then any other unsettled robot, if exists, will not find a node to settle. Hence we allow multiple robots to sit on some nodes only if there is no place to sit. If \(k\ge n\), it is guaranteed that the nodes with robots form a maximal independent set of the underlying network.

The graph \(G=(V, E)\) is a port-labelled graph having n nodes and m edges, where nodes are anonymous. The robots have unique ids in the range [1, L], where \(L \ge k\). Co-located robots can communicate among themselves. We provide an algorithm that solves D-2-D starting from a rooted configuration (i.e., initially all the robots are co-located) and terminates after \(2\varDelta (8m-3n+3)\) synchronous rounds using \(O(\log \varDelta )\) memory per robot without using any global knowledge of the graph parameters m, n and \(\varDelta \), the maximum degree of the graph. We also provide \(\varOmega (m\varDelta )\) lower bound on the number of rounds for the D-2-D problem.

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Notes

  1. 1.

    This is known as the Face-to-Face communication model and has already been considered while studying problems related to mobile robots including exploration [11, 12] and dispersion [1, 13].

  2. 2.

    A configuration where all the robots are initially placed on a single node.

  3. 3.

    We mean \(O(\log \varDelta )\) additional memory, i.e., memory apart from what it requires to store its id.

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Funding

Tanvir Kaur acknowledges the support of the Council of Scientific & Industrial Research (CSIR) during this work (Grant no. 09/1005(0048)/2020-EMR-I). This work was partially supported by the FIST program of the Department of Science and Technology, Government of India, Reference No. SR/FST/MS-I/2018/22(C).

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Authors and Affiliations

  1. Indian Institute of Technology Ropar, Rupnagar, Punjab, India

    Tanvir Kaur & Kaushik Mondal

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  1. Tanvir Kaur
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  2. Kaushik Mondal
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Correspondence to Kaushik Mondal .

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  1. University of Central Florida, Orlando, FL, USA

    David Mohaisen

  2. New York University, New York, NY, USA

    Thomas Wies

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Kaur, T., Mondal, K. (2023). Distance-2-Dispersion: Dispersion with Further Constraints. In: Mohaisen, D., Wies, T. (eds) Networked Systems. NETYS 2023. Lecture Notes in Computer Science, vol 14067. Springer, Cham. https://doi.org/10.1007/978-3-031-37765-5_12

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  • DOI: https://doi.org/10.1007/978-3-031-37765-5_12

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