Fast Dispersion of Mobile Robots on Arbitrary Graphs

  • Ajay D. Kshemkalyani
  • Anisur Rahaman Molla
  • Gokarna SharmaEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11931)


The dispersion problem on graphs asks \(k\le n\) robots placed initially arbitrarily on the nodes of an n-node anonymous graph to reposition autonomously to reach a configuration in which each robot is on a distinct node of the graph. This problem is of significant interest due to its relationship to other fundamental robot coordination problems, such as exploration, scattering, load balancing, and relocation of self-driven electric cars (robots) to recharge stations (nodes). In this paper, we provide a novel deterministic algorithm for dispersion in arbitrary graphs in a synchronous setting where all robots perform their actions in every time step. Our algorithm has \(O(\min (m,k\varDelta ) \cdot \log k)\) steps runtime using \(O(\log n)\) bits of memory at each robot, where m is the number of edges and \(\varDelta \) is the maximum degree of the graph. This is a significant improvement over the O(mk) steps best previously known algorithm that uses logarithmic memory at each robot. In particular, the runtime of our algorithm is optimal (up to a \(O(\log k)\) factor) in constant-degree arbitrary graphs.


  1. 1.
    Augustine, J., Moses Jr., W.K.: Dispersion of mobile robots: a study of memory-time trade-offs. CoRR, abs/1707.05629, [v4] (2018). (A preliminary version in ICDCN 2018)Google Scholar
  2. 2.
    Bampas, E., Gąsieniec, L., Hanusse, N., Ilcinkas, D., Klasing, R., Kosowski, A.: Euler tour lock-in problem in the rotor-router model. In: Keidar, I. (ed.) DISC 2009. LNCS, vol. 5805, pp. 423–435. Springer, Heidelberg (2009). Scholar
  3. 3.
    Barriere, L., Flocchini, P., Mesa-Barrameda, E., Santoro, N.: Uniform scattering of autonomous mobile robots in a grid. In: IPDPS, pp. 1–8 (2009)Google Scholar
  4. 4.
    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)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms, 3rd edn. The MIT Press, Cambridge (2009)zbMATHGoogle Scholar
  6. 6.
    Cybenko, G.: Dynamic load balancing for distributed memory multiprocessors. J. Parallel Distrib. Comput. 7(2), 279–301 (1989)CrossRefGoogle Scholar
  7. 7.
    Das, S., Flocchini, P., Prencipe, G., Santoro, N., Yamashita, M.: Autonomous mobile robots with lights. Theor. Comput. Sci. 609, 171–184 (2016)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Dereniowski, D., Disser, Y., Kosowski, A., Pajak, D., Uznański, P.: Fast collaborative graph exploration. Inf. Comput. 243(C), 37–49 (2015)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Elor, Y., Bruckstein, A.M.: Uniform multi-agent deployment on a ring. Theor. Comput. Sci. 412(8–10), 783–795 (2011)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Flocchini, P., Prencipe, G., Santoro, N.: Distributed computing by oblivious mobile robots. Synth. Lect. Distrib. Comput. Theory 3(2), 1–185 (2012)CrossRefGoogle Scholar
  11. 11.
    Flocchini, P., Prencipe, G., Santoro, N.: Distributed Computing by Mobile Entities. Theoretical Computer Science and General Issues, vol. 1. Springer, Cham (2019). Scholar
  12. 12.
    Fraigniaud, P., Gasieniec, L., Kowalski, D.R., Pelc, A.: Collective tree exploration. Networks 48(3), 166–177 (2006)MathSciNetCrossRefGoogle Scholar
  13. 13.
    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)MathSciNetCrossRefGoogle Scholar
  14. 14.
    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)Google Scholar
  15. 15.
    Hsiang, T.-R., Arkin, E.M., Bender, M.A., Fekete, S.P., Mitchell, J.S.B.: Algorithms for rapidly dispersing robot swarms in unknown environments. In: Boissonnat, J.-D., Burdick, J., Goldberg, K., Hutchinson, S. (eds.) Algorithmic Foundations of Robotics V. STAR, vol. 7, pp. 77–93. Springer, Heidelberg (2004). Scholar
  16. 16.
    Kshemkalyani, A.D., Ali, F.: Efficient dispersion of mobile robots on graphs. In: ICDCN, pp., 218–227 (2019)Google Scholar
  17. 17.
    Menc, A., Pajak, D., Uznanski, P.: Time and space optimality of rotor-router graph exploration. Inf. Process. Lett. 127, 17–20 (2017)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Molla, A.R., Moses, W.K.: Dispersion of mobile robots: the power of randomness. In: Gopal, T.V., Watada, J. (eds.) TAMC 2019. LNCS, vol. 11436, pp. 481–500. Springer, Cham (2019). Scholar
  19. 19.
    Poudel, P., Sharma, G.: Time-optimal uniform scattering in a grid. In: ICDCN, pp. 228–237 (2019)Google Scholar
  20. 20.
    Shibata, M., Mega, T., Ooshita, F., Kakugawa, H., Masuzawa, T.: Uniform deployment of mobile agents in asynchronous rings. In: PODC, pp. 415–424 (2016)Google Scholar
  21. 21.
    Subramanian, R., Scherson, I.D.: An analysis of diffusive load-balancing. In: SPAA, pp. 220–225 (1994)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Ajay D. Kshemkalyani
    • 1
  • Anisur Rahaman Molla
    • 2
  • Gokarna Sharma
    • 3
    Email author
  1. 1.University of Illinois at ChicagoChicagoUSA
  2. 2.Indian Statistical InstituteKolkataIndia
  3. 3.Kent State UniversityKentUSA

Personalised recommendations