Information Spreading by Mobile Particles on a Line
A set of identical particles is deployed on an infinite line. Each particle moves freely on the line at arbitrary but constant speed. When two particles come into contact they bounce acquiring new velocities according to the law of mechanics for elastic collisions. Each particle initially holds a piece of information. The meeting particles automatically transmit to each other their entire currently possessed information (i.e., the initial one and the one accumulated by means of previous collisions). Due to the fact that the number of collisions in this setting is finite  communication cannot last forever. This raises some interesting questions which we address in this paper: Will particle pj ever obtain the initial information of pi? Are colliding particles able to perform broadcasting, convergecast, or gossiping?
We establish necessary and sufficient conditions for any pair of particles to communicate as well as those needed to achieve gossiping, convergecast, and broadcasting. Although these conditions clearly depend on the initial ordering of the particles along the line, we prove that they are independent of their starting positions. Further, we show how to efficiently decide whether some of the aforementioned communication primitives can take place. Finally we explain how to compute the necessary time to carry out all these communication protocols and we describe a relationship between our problem and an important, longstanding open question in computational geometry.
Keywords and Phrases: Mobile agents, passive mobility, particles, communication, broadcasting, convergecast, gossiping, synchronous systems, elastic collisions.
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- 4.Angluin, D., Aspnes, J., Eisenstat, D.: Stably computable predicates are semilinear. In: PODC, pp. 292–299 (2006)Google Scholar
- 5.Das, S., Flocchini, P., Santoro, N., Yamashita, M.: On the computational power of oblivious robots: forming a series of geometric patterns. In: PODC, pp. 267–276 (2010)Google Scholar
- 13.Michail, O., Spirakis, P.G.: Simple and efficient local codes for distributed stable network construction. In: ACM Symposium on Principles of Distributed Computing, PODC 2014, Paris, France, July 15-18, pp. 76–85 (2014)Google Scholar
- 14.Mertzios, G.B., Nikoletseas, S.E., Raptopoulos, C., Spirakis, P.G.: Determining majority in networks with local interactions and very small local memory. In: Esparza, J., Fraigniaud, P., Husfeldt, T., Koutsoupias, E. (eds.) ICALP 2014, Part I. LNCS, vol. 8572, pp. 871–882. Springer, Heidelberg (2014)Google Scholar
- 15.Cao, Y.U., Fukunaga, A.S., Kahng, A.B., Meng, F.: Cooperative mobile robotics: Antecedents and directions. In: Proceedings of the 1995 IEEE/RSJ International Conference on Intelligent Robots and Systems 1995. ‘Human Robot Interaction and Cooperative Robots, vol. 1, pp. 226–234. IEEE (1995)Google Scholar
- 16.Dudek, G., Jenkin, M.: Computational principles of mobile robotics. Cambridge University Press (2010)Google Scholar
- 17.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)Google Scholar
- 18.Czyzowicz, J., Dobrev, S., An, H.-C., Krizanc, D.: The power of tokens: Rendezvous and symmetry detection for two mobile agents in a ring. In: Geffert, V., Karhumäki, J., Bertoni, A., Preneel, B., Návrat, P., Bieliková, M. (eds.) SOFSEM 2008. LNCS, vol. 4910, pp. 234–246. Springer, Heidelberg (2008)CrossRefGoogle Scholar
- 21.Wylie, J., Yang, R., Zhang, Q.: Periodic orbits of inelastic particles on a ring. Physical Review E 86, 026601(2) (2012)Google Scholar
- 25.Chan, T.M.: Remarks on k-level algorithms in the plane. Manuscript, Univ. of Waterloo (1999)Google Scholar
- 26.Sharir, M., Agarwal, P.K.: Davenport-Schinzel sequences and their geometric applications. Cambridge University Press (1995)Google Scholar
- 27.Erdös, P., Lovász, L., Simmons, A., Straus, E.G.: Dissection graphs of planar point sets. A Survey of Combinatorial Theory, 139–149 (1973)Google Scholar
- 31.Gregory, R.: Classical mechanics. Cambridge University Press (2006)Google Scholar
- 32.Everett, H., Robert, J.M., Van Kreveld, M.: An optimal algorithm for the (≤ k)-levels, with applications to separation and transversal problems. In: Proceedings of the Ninth Annual Symposium on Computational Geometry, ACM, pp. 38–46 (1993)Google Scholar