Abstract
We study a fundamental cooperative message-delivery problem on the plane. Assume n robots which can move in any direction, are placed arbitrarily on the plane. Robots each have their own maximum speed and can communicate with each other face-to-face (i.e., when they are at the same location at the same time). There are also two designated points on the plane, S (the source) and D (the destination). The robots are required to transmit the message from the source to the destination as quickly as possible by face-to-face message passing. We consider both the offline setting where all information (the locations and maximum speeds of the robots) are known in advance and the online setting where each robot knows only its own position and speed along with the positions of S and D.
In the offline case, we discover an important connection between the problem for two-robot systems and the well-known Apollonius circle which we employ to design an optimal algorithm. We also propose a \(\sqrt{2}\) approximation algorithm for systems with any number of robots. In the online setting, we provide an algorithm with competitive ratio \(\frac{1}{7} \left( 5+ 4 \sqrt{2} \right) \) for two-robot systems and show that the same algorithm has a competitive ratio less than 2 for systems with any number of robots. We also show these results are tight for the given algorithm. Finally, we give two lower bounds (employing different arguments) on the competitive ratio of any online algorithm, one of 1.0391 and the other of 1.0405.
E. Kranakis—Research supported in part by NSERC Discovery grant.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Anaya, J., Chalopin, J., Czyzowicz, J., Labourel, A., Pelc, A., Vaxès, Y.: Convergecast and broadcast by power-aware mobile agents. Algorithmica 74(1), 117–155 (2016)
Bärtschi, A., et al.: Energy-efficient delivery by heterogeneous mobile agents. arXiv preprint arXiv:1610.02361 (2016)
Bärtschi, A., Graf, D., Mihalák, M.: Collective fast delivery by energy-efficient agents. In: Potapov, I., Spirakis, P.G., Worrell, J. (eds.) 43rd International Symposium on Mathematical Foundations of Computer Science, MFCS 2018, Liverpool, UK, 27–31 August 2018. LIPIcs, vol. 117, pp. 56:1–56:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)
Bärtschi, A., Graf, D., Penna, P.: Truthful mechanisms for delivery with mobile agents. arXiv preprint arXiv:1702.07665 (2017)
Bärtschi, A., Tschager, T.: Energy-efficient fast delivery by mobile agents. In: Klasing, R., Zeitoun, M. (eds.) FCT 2017. LNCS, vol. 10472, pp. 82–95. Springer, Heidelberg (2017). https://doi.org/10.1007/978-3-662-55751-8_8
Bereg, S., Brunner, A., Caraballo, L.-E., DÃaz-Báñez, J.-M., Lopez, M.A.: On the robustness of a synchronized multi-robot system. J. Comb. Optim. 39(4), 988–1016 (2020). https://doi.org/10.1007/s10878-020-00533-z
Bilò, D., Gualà , L., Leucci, S., Proietti, G., Rossi, M.: New approximation algorithms for the heterogeneous weighted delivery problem. In: Jurdziński, T., Schmid, S. (eds.) SIROCCO 2021. LNCS, vol. 12810, pp. 167–184. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-79527-6_10
Carvalho, I.A., Erlebach, T., Papadopoulos, K.: An efficient algorithm for the fast delivery problem. In: Gąsieniec, L.A., Jansson, J., Levcopoulos, C. (eds.) FCT 2019. LNCS, vol. 11651, pp. 171–184. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-25027-0_12
Carvalho, I.A., Erlebach, T., Papadopoulos, K.: On the fast delivery problem with one or two packages. J. Comput. Syst. Sci. 115, 246–263 (2021)
Chalopin, J., Das, S., Mihal’ák, M., Penna, P., Widmayer, P.: Data delivery by energy-constrained mobile agents. In: Flocchini, P., Gao, J., Kranakis, E., Meyer auf der Heide, F. (eds.) ALGOSENSORS 2013. LNCS, vol. 8243, pp. 111–122. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-642-45346-5_9
Chalopin, J., Godard, E., Métivier, Y., Ossamy, R.: Mobile agent algorithms versus message passing algorithms. In: Shvartsman, M.M.A.A. (ed.) OPODIS 2006. LNCS, vol. 4305, pp. 187–201. Springer, Heidelberg (2006). https://doi.org/10.1007/11945529_14
Chalopin, J., Jacob, R., Mihalák, M., Widmayer, P.: Data delivery by energy-constrained mobile agents on a line. In: Esparza, J., Fraigniaud, P., Husfeldt, T., Koutsoupias, E. (eds.) ICALP 2014. LNCS, vol. 8573, pp. 423–434. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-662-43951-7_36
Chuangpishit, H., Czyzowicz, J., Killick, R., Kranakis, E., Krizanc, D., Morales-Ponce, O.: Optimal rendezvous on a line by location-aware robots in the presence of spies. Discret. Math. Algorithms Appl. (2021, to appear)
Coleman, J., Kranakis, E., Krizanc, D., Morales-Ponce, O.: Message delivery in the plane by robots with different speeds. arXiv preprint arXiv:2109.12185 (2021)
Coleman, J., Kranakis, E., Krizanc, D., Morales-Ponce, O.: The pony express communication problem. In: Proceedings of IWOCA21; also Extended Version as arXiv preprint arXiv:2105.03545 (2021)
Czyzowicz, J., Diks, K., Moussi, J., Rytter, W.: Communication problems for mobile agents exchanging energy. In: Suomela, J. (ed.) SIROCCO 2016. LNCS, vol. 9988, pp. 275–288. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-48314-6_18
Czyzowicz, J., Georgiou, K., Kranakis, E.: Patrolling. In: Flocchini, P., Prencipe, G., Santoro, N. (eds.) Distributed Computing by Mobile Entities. LNCS, vol. 11340, pp. 371–400. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-11072-7_15
Czyzowicz, J., Killick, R., Kranakis, E., Krizanc, D., Morales-Ponce, O.: Gathering in the plane of location-aware robots in the presence of spies. Theor. Comput. Sci. 836, 94–109 (2020)
Das, S., Dereniowski, D., Karousatou, C.: Collaborative exploration by energy-constrained mobile robots. In: Scheideler, C. (ed.) SIROCCO 2014. LNCS, vol. 9439, pp. 357–369. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-25258-2_25
Das, S., Dereniowski, D., Karousatou, C.: Collaborative exploration of trees by energy-constrained mobile robots. Theory Comput. Syst. 62(5), 1223–1240 (2018)
Ogilvy, C.S.: Excursions in Geometry. Dover Publications (1990)
Yiu, P.: Introduction to the Geometry of the Triangle. Version 4.0510, Florida Atlantic University Lecture Notes (2004)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 Springer Nature Switzerland AG
About this paper
Cite this paper
Coleman, J., Kranakis, E., Krizanc, D., Ponce, O.M. (2021). Message Delivery in the Plane by Robots with Different Speeds. In: Johnen, C., Schiller, E.M., Schmid, S. (eds) Stabilization, Safety, and Security of Distributed Systems. SSS 2021. Lecture Notes in Computer Science(), vol 13046. Springer, Cham. https://doi.org/10.1007/978-3-030-91081-5_20
Download citation
DOI: https://doi.org/10.1007/978-3-030-91081-5_20
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-91080-8
Online ISBN: 978-3-030-91081-5
eBook Packages: Computer ScienceComputer Science (R0)