Abstract
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 [1] communication cannot last forever. This raises some interesting questions which we address in this paper: Will particle p j ever obtain the initial information of p i ? 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.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Sevryuk, M.: Estimate of the number of collisions of n elastic particles on a line. Theoretical and Mathematical Physics 96(1), 818–826 (1993)
Suzuki, I., Yamashita, M.: Distributed anonymous mobile robots: Formation of geometric patterns. SIAM J. Comput. 28(4), 1347–1363 (1999)
Angluin, D., Aspnes, J., Diamadi, Z., Fischer, M.J., Peralta, R.: Computation in networks of passively mobile finite-state sensors. Dist. Comp. 18(4), 235–253 (2006)
Angluin, D., Aspnes, J., Eisenstat, D.: Stably computable predicates are semilinear. In: PODC, pp. 292–299 (2006)
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)
Flocchini, P., Prencipe, G., Santoro, N., Widmayer, P.: Hard tasks for weak robots: The role of common knowledge in pattern formation by autonomous mobile robots. In: Aggarwal, A.K., Pandu Rangan, C. (eds.) ISAAC 1999. LNCS, vol. 1741, pp. 93–102. Springer, Heidelberg (1999)
Suzuki, I., Yamashita, M.: Distributed anonymous mobile robots: Formation of geometric patterns. SIAM Journal on Computing 28(4), 1347–1363 (1999)
Yamashita, M., Suzuki, I.: Characterizing geometric patterns formable by oblivious anonymous mobile robots. TCS 411(26), 2433–2453 (2010)
Cohen, R., Peleg, D.: Local spreading algorithms for autonomous robot systems. TCS 399(1), 71–82 (2008)
Flocchini, P., Prencipe, G., Santoro, N., Widmayer, P.: Gathering of asynchronous oblivious robots with limited visibility. In: Ferreira, A., Reichel, H. (eds.) STACS 2001. LNCS, vol. 2010, pp. 247–258. Springer, Heidelberg (2001)
Czyzowicz, J., Kranakis, E., Pacheco, E.: Localization for a system of colliding robots. In: Fomin, F.V., Freivalds, R., Kwiatkowska, M., Peleg, D. (eds.) ICALP 2013, Part II. LNCS, vol. 7966, pp. 508–519. Springer, Heidelberg (2013)
Czyzowicz, J., Gąsieniec, L., Kosowski, A., Kranakis, E., Ponce, O.M., Pacheco, E.: Position discovery for a system of bouncing robots. In: Aguilera, M.K. (ed.) DISC 2012. LNCS, vol. 7611, pp. 341–355. Springer, Heidelberg (2012)
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)
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)
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)
Dudek, G., Jenkin, M.: Computational principles of mobile robotics. Cambridge University Press (2010)
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)
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)
Murphy, T.: Dynamics of hard rods in one dimension. Journal of Statistical Physics 74(3), 889–901 (1994)
Tonks, L.: The complete equation of state of one, two and three-dimensional gases of hard elastic spheres. Physical Review 50(10), 955 (1936)
Wylie, J., Yang, R., Zhang, Q.: Periodic orbits of inelastic particles on a ring. Physical Review E 86, 026601(2) (2012)
Cooley, B., Newton, P.: Iterated impact dynamics of n-beads on a ring. SIAM Rev. 47(2), 273–300 (2005)
Murphy, T., Cohen, E.: Maximum number of collisions among identical hard spheres. Journal of Statistical Physics 71(5-6), 1063–1080 (1993)
Friedetzky, T., Gąsieniec, L., Gorry, T., Martin, R.: Observe and remain silent (Communication-less agent location discovery). In: Rovan, B., Sassone, V., Widmayer, P. (eds.) MFCS 2012. LNCS, vol. 7464, pp. 407–418. Springer, Heidelberg (2012)
Chan, T.M.: Remarks on k-level algorithms in the plane. Manuscript, Univ. of Waterloo (1999)
Sharir, M., Agarwal, P.K.: Davenport-Schinzel sequences and their geometric applications. Cambridge University Press (1995)
Erdös, P., Lovász, L., Simmons, A., Straus, E.G.: Dissection graphs of planar point sets. A Survey of Combinatorial Theory, 139–149 (1973)
Lovász, L.: On the number of halving lines. Ann. Univ. Sci. Budapest, Eötvös, Sec. Math. 14, 107–108 (1971)
Dey, T.K.: Improved bounds for planar k-sets and related problems. Discrete & Computational Geometry 19(3), 373–382 (1998)
Tóth, G.: Point sets with many k-sets. Discrete & Computational Geometry 26(2), 187–194 (2001)
Gregory, R.: Classical mechanics. Cambridge University Press (2006)
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)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this paper
Cite this paper
Czyzowicz, J., Kranakis, E., Pacheco, E., Pająk, D. (2015). Information Spreading by Mobile Particles on a Line. In: Scheideler, C. (eds) Structural Information and Communication Complexity. SIROCCO 2015. Lecture Notes in Computer Science(), vol 9439. Springer, Cham. https://doi.org/10.1007/978-3-319-25258-2_20
Download citation
DOI: https://doi.org/10.1007/978-3-319-25258-2_20
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-25257-5
Online ISBN: 978-3-319-25258-2
eBook Packages: Computer ScienceComputer Science (R0)