Abstract
Given any arbitrary initial configuration of \(k\le n\) robots positioned on the nodes of an n-node anonymous graph, the problem of dispersion is to autonomously reposition the robots such that each node will contain at most one robot. This problem gained significant interest due to its resemblance with several fundamental problems such as exploration, scattering, load balancing, relocation of electric cars to charging stations, etc. The objective is to solve dispersion simultaneously minimizing (or providing a trade-off between) time and memory requirement at each robot. The literature mainly dealt with dispersion on undirected anonymous graphs. In this paper, we initiate the study of dispersion on directed anonymous graphs. We first show that it may not always be possible to solve dispersion when the directed graph is not strongly connected. We then establish some lower bounds on both time and memory requirements at each robot for solving dispersion on a strongly connected directed graph. Finally, we provide two deterministic algorithms solving dispersion on any strongly connected directed graph. Let D be the graph diameter and \(\varDelta _{out}\) be its maximum out-degree. The first algorithm solves dispersion in \(O(k^2\cdot \varDelta _{out})\) time with \(O(\log (k+\varDelta _{out}))\) bits at each robot. The second algorithm solves dispersion in \(O(k\cdot D)\) time with \(O(k\cdot \log (k+\varDelta _{out}))\) bits at each robot, provided that robots in the 1-hop neighborhood can communicate. Both algorithms extend to handle crash faults.
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References
Albers, S., Henzinger, M.R.: Exploring unknown environments. In: STOC, pp. 416–425. ACM (1997)
Augustine, J., Moses Jr., W.K.: Dispersion of mobile robots: a study of memory-time trade-offs. In: ICDCN, pp. 1:1–1:10 (2018)
Bampas, E., Gasieniec, L., Hanusse, N., Ilcinkas, D., Klasing, R., Kosowski, A.: Euler tour lock-in problem in the rotor-router model: I choose pointers and you choose port numbers. In: DISC, pp. 423–435 (2009)
Barriere, L., Flocchini, P., Mesa-Barrameda, E., Santoro, N.: Uniform scattering of autonomous mobile robots in a grid. In: IPDPS, pp. 1–8 (2009)
Bender, M.A., Fernández, A., Ron, D., Sahai, A., Vadhan, S.P.: The power of a pebble: exploring and mapping directed graphs. In: STOC, pp. 269–278 (1998)
Bender, M.A., Slonim, D.K.: The power of team exploration: two robots can learn unlabeled directed graphs. In: FOCS, pp. 75–85 (1994)
Cohen, R., Fraigniaud, P., Ilcinkas, D., Korman, A., Peleg, D.: Label-guided graph exploration by a finite automaton. ACM Trans. Algorithms 4(4), 42:1–42:18 (2008)
Cybenko, G.: Dynamic load balancing for distributed memory multiprocessors. J. Parallel Distrib. Comput. 7(2), 279–301 (1989)
Dereniowski, D., Disser, Y., Kosowski, A., Pajak, D., Uznański, P.: Fast collaborative graph exploration. Inf. Comput. 243(C), 37–49 (2015)
Elor, Y., Bruckstein, A.M.: Uniform multi-agent deployment on a ring. Theor. Comput. Sci. 412(8–10), 783–795 (2011)
Flocchini, P., Prencipe, G., Santoro, N.: Distributed Computing by Oblivious Mobile Robots. Morgan & Claypool Publishers, Synthesis Lectures on Distributed Computing Theory (2012)
Flocchini, P., Prencipe, G., Santoro, N.: Distributed Computing by Mobile Entities, Theoretical Computer Science and General Issues, vol. 1. Springer (2019). https://doi.org/10.1007/978-3-030-11072-7
Foerster, K., Wattenhofer, R.: Lower and upper competitive bounds for online directed graph exploration. Theor. Comput. Sci. 655, 15–29 (2016)
Fraigniaud, P., Gasieniec, L., Kowalski, D.R., Pelc, A.: Collective tree exploration. Networks 48(3), 166–177 (2006)
Fraigniaud, P., Ilcinkas, D., Peer, G., Pelc, A., Peleg, D.: Graph exploration by a finite automaton. Theor. Comput. Sci. 345(2–3), 331–344 (2005)
Hsiang, T.R., Arkin, E.M., Bender, M.A., Fekete, S., Mitchell, J.S.B.: Online dispersion algorithms for swarms of robots. In: SoCG, pp. 382–383 (2003)
Hsiang, T., Arkin, E.M., Bender, M.A., Fekete, S.P., Mitchell, J.S.B.: Algorithms for rapidly dispersing robot swarms in unknown environments. In: WAFR, pp. 77–94 (2002)
Kshemkalyani, A.D., Ali, F.: Fast graph exploration by a mobile robot. In: AIKE, pp. 115–118 (2018)
Kshemkalyani, A.D., Ali, F.: Efficient dispersion of mobile robots on graphs. In: ICDCN, pp. 218–227 (2019)
Kshemkalyani, A.D., Molla, A.R., Sharma, G.: Fast dispersion of mobile robots on arbitrary graphs. In: ALGOSENSORS, pp. 23–40 (2019)
Kshemkalyani, A.D., Molla, A.R., Sharma, G.: Dispersion of mobile robots in the global communication model. In: ICDCN, pp. 12:1–12:10 (2020)
Kshemkalyani, A.D., Molla, A.R., Sharma, G.: Dispersion of mobile robots on grids. In: WALCOM, pp. 183–197 (2020)
Kshemkalyani, A.D., Molla, A.R., Sharma, G.: Efficient dispersion of mobile robots on dynamic graphs. In: ICDCS, pp. 732–742 (2020)
Kshemkalyani, A.D., Sharma, G.: Near-optimal dispersion on arbitrary anonymous graphs. CoRR (2021)
Menc, A., Pajak, D., Uznanski, P.: Time and space optimality of rotor-router graph exploration. Inf. Process. Lett. 127, 17–20 (2017)
Molla, A.R., Jr., W.K.M.: Dispersion of mobile robots: The power of randomness. In: TAMC, pp. 481–500 (2019)
Molla, A.R., Mondal, K., Jr., W.K.M.: Efficient dispersion on an anonymous ring in the presence of weak byzantine robots. In: ALGOSENSORS, pp. 154–169 (2020)
Molla, A.R., Mondal, K., Jr., W.K.M.: Byzantine dispersion on graphs. In: IPDPS, pp. 942–951. IEEE (2021)
Pattanayak, D., Sharma, G., Mandal, P.S.: Dispersion of mobile robots tolerating faults. In: ICDCN, pp. 133–138 (2021)
Poudel, P., Sharma, G.: Time-optimal uniform scattering in a grid. In: ICDCN, pp. 228–237 (2019)
Shibata, M., Mega, T., Ooshita, F., Kakugawa, H., Masuzawa, T.: Uniform deployment of mobile agents in asynchronous rings. In: PODC, pp. 415–424 (2016)
Shintaku, T., Sudo, Y., Kakugawa, H., Masuzawa, T.: Efficient dispersion of mobile agents without global knowledge. In: SSS, pp. 280–294 (2020)
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Italiano, G.F., Pattanayak, D., Sharma, G. (2022). Dispersion of Mobile Robots on Directed Anonymous Graphs. In: Parter, M. (eds) Structural Information and Communication Complexity. SIROCCO 2022. Lecture Notes in Computer Science, vol 13298. Springer, Cham. https://doi.org/10.1007/978-3-031-09993-9_11
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