Document

# Asynchronous Approach in the Plane: A Deterministic Polynomial Algorithm

## File

LIPIcs.DISC.2017.8.pdf
• Filesize: 441 kB
• 16 pages

## Cite As

Sébastien Bouchard, Marjorie Bournat, Yoann Dieudonné, Swan Dubois, and Franck Petit. Asynchronous Approach in the Plane: A Deterministic Polynomial Algorithm. In 31st International Symposium on Distributed Computing (DISC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 91, pp. 8:1-8:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
https://doi.org/10.4230/LIPIcs.DISC.2017.8

## Abstract

In this paper we study the task of approach of two mobile agents having the same limited range of vision and moving asynchronously in the plane. This task consists in getting them in finite time within each other's range of vision. The agents execute the same deterministic algorithm and are assumed to have a compass showing the cardinal directions as well as a unit measure. On the other hand, they do not share any global coordinates system (like GPS), cannot communicate and have distinct labels. Each agent knows its label but does not know the label of the other agent or the initial position of the other agent relative to its own. The route of an agent is a sequence of segments that are subsequently traversed in order to achieve approach. For each agent, the computation of its route depends only on its algorithm and its label. An adversary chooses the initial positions of both agents in the plane and controls the way each of them moves along every segment of the routes, in particular by arbitrarily varying the speeds of the agents. Roughly speaking, the goal of the adversary is to prevent the agents from solving the task, or at least to ensure that the agents have covered as much distance as possible before seeing each other. A deterministic approach algorithm is a deterministic algorithm that always allows two agents with any distinct labels to solve the task of approach regardless of the choices and the behavior of the adversary. The cost of a complete execution of an approach algorithm is the length of both parts of route travelled by the agents until approach is completed. Let Delta and l be the initial distance separating the agents and the length of (the binary representation of) the shortest label, respectively. Assuming that Delta and l are unknown to both agents, does there exist a deterministic approach algorithm whose cost is polynomial in Delta and l? Actually the problem of approach in the plane reduces to the network problem of rendezvous in an infinite oriented grid, which consists in ensuring that both agents end up meeting at the same time at a node or on an edge of the grid. By designing such a rendezvous algorithm with appropriate properties, as we do in this paper, we provide a positive answer to the above question. Our result turns out to be an important step forward from a computational point of view, as the other algorithms allowing to solve the same problem either have an exponential cost in the initial separating distance and in the labels of the agents, or require each agent to know its starting position in a global system of coordinates, or only work under a much less powerful adversary.
##### Keywords
• mobile agents
• asynchronous rendezvous
• plane
• infinite grid
• deterministic algorithm
• polynomial cost

## Metrics

• Access Statistics
• Total Accesses (updated on a weekly basis)
0

## References

1. Noa Agmon and David Peleg. Fault-tolerant gathering algorithms for autonomous mobile robots. SIAM J. Comput., 36(1):56-82, 2006.
2. Steve Alpern and Shmuel Gal. The theory of search games and rendezvous. Int. Series in Operations Research and Management Science, Kluwer Academic Publishers, 2003.
3. Evangelos Bampas, Jurek Czyzowicz, Leszek Gasieniec, David Ilcinkas, and Arnaud Labourel. Almost optimal asynchronous rendezvous in infinite multidimensional grids. In Distributed Computing, 24th International Symposium, DISC 2010, Proceedings, pages 297-311, 2010.
4. Sébastien Bouchard, Yoann Dieudonné, and Bertrand Ducourthial. Byzantine gathering in networks. Distributed Computing, 29(6):435-457, 2016.
5. Jérémie Chalopin, Yoann Dieudonné, Arnaud Labourel, and Andrzej Pelc. Rendezvous in networks in spite of delay faults. Distributed Computing, 29(3):187-205, 2016.
6. Mark Cieliebak, Paola Flocchini, Giuseppe Prencipe, and Nicola Santoro. Distributed computing by mobile robots: Gathering. SIAM J. Comput., 41(4):829-879, 2012.
7. Reuven Cohen and David Peleg. Convergence properties of the gravitational algorithm in asynchronous robot systems. SIAM J. Comput., 34(6):1516-1528, 2005.
8. Reuven Cohen and David Peleg. Convergence of autonomous mobile robots with inaccurate sensors and movements. SIAM J. Comput., 38(1):276-302, 2008.
9. Andrew Collins, Jurek Czyzowicz, Leszek Gasieniec, and Arnaud Labourel. Tell me where I am so I can meet you sooner. In Automata, Languages and Programming, 37th International Colloquium, ICALP 2010, Proceedings, Part II, pages 502-514, 2010.
10. Jurek Czyzowicz, Adrian Kosowski, and Andrzej Pelc. How to meet when you forget: log-space rendezvous in arbitrary graphs. Distributed Computing, 25(2):165-178, 2012.
11. Jurek Czyzowicz, Andrzej Pelc, and Arnaud Labourel. How to meet asynchronously (almost) everywhere. ACM Transactions on Algorithms, 8(4):37, 2012.
12. Gianlorenzo D'Angelo, Gabriele Di Stefano, and Alfredo Navarra. Gathering on rings under the look-compute-move model. Distributed Computing, 27(4):255-285, 2014.
13. Shantanu Das, Dariusz Dereniowski, Adrian Kosowski, and Przemyslaw Uznanski. Rendezvous of distance-aware mobile agents in unknown graphs. In Structural Information and Communication Complexity - 21st International Colloquium, SIROCCO 2014, Proceedings, pages 295-310, 2014.
14. Shantanu Das, Flaminia L. Luccio, and Euripides Markou. Mobile agents rendezvous in spite of a malicious agent. In Algorithms for Sensor Systems - 11th International Symposium on Algorithms and Experiments for Wireless Sensor Networks, ALGOSENSORS 2015, Revised Selected Papers, pages 211-224, 2015.
15. Xavier Défago, Maria Gradinariu, Stéphane Messika, and Philippe Raipin Parvédy. Fault-tolerant and self-stabilizing mobile robots gathering. In Distributed Computing, 20th International Symposium, DISC 2006, Proceedings, pages 46-60, 2006.
16. Anders Dessmark, Pierre Fraigniaud, Dariusz R. Kowalski, and Andrzej Pelc. Deterministic rendezvous in graphs. Algorithmica, 46(1):69-96, 2006.
17. Yoann Dieudonné and Andrzej Pelc. Deterministic polynomial approach in the plane. Distributed Computing, 28(2):111-129, 2015.
18. Yoann Dieudonné and Andrzej Pelc. Anonymous meeting in networks. Algorithmica, 74(2):908-946, 2016.
19. Yoann Dieudonné, Andrzej Pelc, and David Peleg. Gathering despite mischief. ACM Transactions on Algorithms, 11(1):1, 2014.
20. Yoann Dieudonné, Andrzej Pelc, and Vincent Villain. How to meet asynchronously at polynomial cost. SIAM J. Comput., 44(3):844-867, 2015.
21. Yoann Dieudonné and Franck Petit. Self-stabilizing gathering with strong multiplicity detection. Theor. Comput. Sci., 428:47-57, 2012.
22. Paola Flocchini, Giuseppe Prencipe, Nicola Santoro, and Peter Widmayer. Gathering of asynchronous robots with limited visibility. Theor. Comput. Sci., 337(1-3):147-168, 2005.
23. Pierre Fraigniaud and Andrzej Pelc. Deterministic rendezvous in trees with little memory. In Distributed Computing, 22nd International Symposium, DISC 2008, Proceedings, pages 242-256, 2008.
24. Pierre Fraigniaud and Andrzej Pelc. Delays induce an exponential memory gap for rendezvous in trees. ACM Transactions on Algorithms, 9(2):17, 2013.
25. Taisuke Izumi, Samia Souissi, Yoshiaki Katayama, Nobuhiro Inuzuka, Xavier Défago, Koichi Wada, and Masafumi Yamashita. The gathering problem for two oblivious robots with unreliable compasses. SIAM J. Comput., 41(1):26-46, 2012.
26. Dariusz R. Kowalski and Adam Malinowski. How to meet in anonymous network. Theor. Comput. Sci., 399(1-2):141-156, 2008.
27. Evangelos Kranakis, Danny Krizanc, and Sergio Rajsbaum. Mobile agent rendezvous: A survey. In Structural Information and Communication Complexity, 13th International Colloquium, SIROCCO 2006, Proceedings, pages 1-9, 2006.
28. Gianluca De Marco, Luisa Gargano, Evangelos Kranakis, Danny Krizanc, Andrzej Pelc, and Ugo Vaccaro. Asynchronous deterministic rendezvous in graphs. Theor. Comput. Sci., 355(3):315-326, 2006.
29. Avery Miller and Andrzej Pelc. Fast rendezvous with advice. Theor. Comput. Sci., 608:190-198, 2015.
30. Avery Miller and Andrzej Pelc. Time versus cost tradeoffs for deterministic rendezvous in networks. Distributed Computing, 29(1):51-64, 2016.
31. Linda Pagli, Giuseppe Prencipe, and Giovanni Viglietta. Getting close without touching: near-gathering for autonomous mobile robots. Distributed Computing, 28(5):333-349, 2015.
32. Thomas Schelling. The Strategy of Conflict. Oxford University Press, Oxford, 1960.
33. Ichiro Suzuki and Masafumi Yamashita. Distributed anonymous mobile robots: Formation of geometric patterns. SIAM J. Comput., 28(4):1347-1363, 1999.
34. Amnon Ta-Shma and Uri Zwick. Deterministic rendezvous, treasure hunts, and strongly universal exploration sequences. ACM Transactions on Algorithms, 10(3):12, 2014.
X

Feedback for Dagstuhl Publishing