Document

# How to Meet at a Node of Any Connected Graph

## File

LIPIcs.DISC.2022.11.pdf
• Filesize: 1.45 MB
• 16 pages

## Cite As

Subhash Bhagat and Andrzej Pelc. How to Meet at a Node of Any Connected Graph. In 36th International Symposium on Distributed Computing (DISC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 246, pp. 11:1-11:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.DISC.2022.11

## Abstract

Two mobile agents have to meet at the same node of a connected graph with unlabeled nodes. This intensely researched task is known as rendezvous. The adversary assigns the agents different starting nodes in the graph and different integer labels from a set {1,… ,L}. Time is slotted in synchronous rounds. The adversary wakes up the agents in possibly different rounds. After wakeup, the agents move as follows. In each round, an agent can either stay idle or move to an adjacent node. Each agent knows its label but not the label of the other agent, and agents have no a priori information about the graph. They do not know L. They execute the same deterministic algorithm whose parameter is the agent’s label. The time of a rendezvous algorithm is the worst-case number of rounds since the wakeup of the earlier agent till the meeting. In most of the results concerning rendezvous in graphs, the graph is finite and rendezvous relies on the exploration of the entire graph. Thus the time of rendezvous depends on the size of the graph. This approach is inefficient for very large graphs, and cannot be used for infinite graphs. For such graphs it is natural to seek rendezvous algorithms whose time depends on the initial distance D between the agents. In this paper we adopt this approach and consider rendezvous in arbitrary connected graphs with nodes of finite degrees, and whose set of nodes is finite or countably infinite. Our main result is the first deterministic rendezvous algorithm working under this general scenario. For any node v and any positive integer r, let P(v,r) be the number of paths of length r in the graph, starting at node v. For any instance of the rendezvous problem where agents start at nodes v₁ and v₂ at distance D, let P(v₁,v₂,D) = max(P(v₁,D),P(v₂,D)). It is well known that, for example in trees, Ω(D+P(v₁,v₂,D) +log L) is a lower bound on rendezvous time for such an instance. The time of our algorithm, working for arbitrary connected graphs of finite degrees, is polynomial in this lower bound. As an application we solve the problem of approach for synchronous agents in terrains in the plane, in time polynomial in log L and in the initial distance between the agents in the terrain.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Design and analysis of algorithms
• Theory of computation → Distributed algorithms
• Algorithm
• graph
• rendezvous
• mobile agent
• terrain

## Metrics

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

## References

1. Steve Alpern. The rendezvous search problem. SIAM Journal on Control and Optimization, 33(3):673-683, 1995. URL: https://doi.org/10.1137/S0363012993249195.
2. Steve Alpern and Shmuel Gal. The theory of search games and rendezvous, volume 55 of International series in operations research and management science. Kluwer, 2003.
3. Edward J. Anderson and Sándor P. Fekete. Two dimensional rendezvous search. Operations Research, 49(1):107-118, 2001. URL: https://doi.org/10.1287/opre.49.1.107.11191.
4. Evangelos Bampas, Jurek Czyzowicz, Leszek Gasieniec, David Ilcinkas, and Arnaud Labourel. Almost optimal asynchronous rendezvous in infinite multidimensional grids. In Proc. 24th International Symposium on Distributed Computing (DISC 2010), volume 6343, pages 297-311. Springer, 2010. URL: https://doi.org/10.1007/978-3-642-15763-9_28.
5. Vic Baston and Shmuel Gal. Rendezvous on the line when the players' initial distance is given by an unknown probability distribution. SIAM Journal on Control and Optimization, 36(6):1880-1889, 1998. URL: https://doi.org/10.1137/S0363012996314130.
6. Vic Baston and Shmuel Gal. Rendezvous search when marks are left at the starting points. Naval Research Logistics (NRL), 48(8):722-731, 2001. URL: https://doi.org/10.1002/nav.1044.
7. Subhash Bhagat and Andrzej Pelc. Deterministic rendezvous in infinite trees. CoRR, abs/2203.05160, 2022. URL: https://doi.org/10.48550/arXiv.2203.05160.
8. Sébastien Bouchard, Marjorie Bournat, Yoann Dieudonné, Swan Dubois, and Franck Petit. Asynchronous approach in the plane: a deterministic polynomial algorithm. Distributed Computing, 32(4):317-337, 2019. URL: https://doi.org/10.1007/s00446-018-0338-2.
9. Sébastien Bouchard, Yoann Dieudonné, Andrzej Pelc, and Franck Petit. Almost universal anonymous rendezvous in the plane. In Proc. 32nd ACM Symposium on Parallelism in Algorithms and Architectures (SPAA 2020), pages 117-127. ACM, 2020. URL: https://doi.org/10.1145/3350755.3400283.
10. Mark Cieliebak, Paola Flocchini, Giuseppe Prencipe, and Nicola Santoro. Distributed computing by mobile robots: Gathering. SIAM Journal on Computing, 41(4):829-879, 2012. URL: https://doi.org/10.1137/100796534.
11. Andrew Collins, Jurek Czyzowicz, Leszek Gasieniec, Adrian Kosowski, and Russell A. Martin. Synchronous rendezvous for location-aware agents. In Proc. 25th International Symposium on Distributed Computing (DISC 2011), volume 6950, pages 447-459. Springer, 2011. URL: https://doi.org/10.1007/978-3-642-24100-0_42.
12. Jurek Czyzowicz, Leszek Gasieniec, Ryan Killick, and Evangelos Kranakis. Symmetry breaking in the plane: Rendezvous by robots with unknown attributes. In Proc. 2019 ACM Symposium on Principles of Distributed Computing (PODC 2019), pages 4-13. ACM, 2019. URL: https://doi.org/10.1145/3293611.3331608.
13. Jurek Czyzowicz, Andrzej Pelc, and Arnaud Labourel. How to meet asynchronously (almost) everywhere. ACM Trans. Algorithms, 8(4):37:1-37:14, 2012. URL: https://doi.org/10.1145/2344422.2344427.
14. Anders Dessmark, Pierre Fraigniaud, Dariusz R. Kowalski, and Andrzej Pelc. Deterministic rendezvous in graphs. Algorithmica, 46(1):69-96, 2006. URL: https://doi.org/10.1007/s00453-006-0074-2.
15. Yoann Dieudonné and Andrzej Pelc. Deterministic polynomial approach in the plane. Distributed Computing, 28(2):111-129, 2015. URL: https://doi.org/10.1007/s00446-014-0216-5.
16. Yoann Dieudonné and Andrzej Pelc. Anonymous meeting in networks. Algorithmica, 74(2):908-946, 2016. URL: https://doi.org/10.1007/s00453-015-9982-0.
17. Yoann Dieudonné, Andrzej Pelc, and Vincent Villain. How to meet asynchronously at polynomial cost. SIAM Journal on Computing, 44(3):844-867, 2015. URL: https://doi.org/10.1137/130931990.
18. 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. URL: https://doi.org/10.1016/j.tcs.2005.01.001.
19. Dariusz R. Kowalski and Adam Malinowski. How to meet in anonymous network. Theor. Comput. Sci., 399(1-2):141-156, 2008. URL: https://doi.org/10.1016/j.tcs.2008.02.010.
20. Wei Shi Lim and Steve Alpern. Minimax rendezvous on the line. SIAM Journal on Control and Optimization, 34(5):1650-1665, 1996. URL: https://doi.org/10.1137/S036301299427816X.
21. 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. URL: https://doi.org/10.1016/j.tcs.2005.12.016.
22. Avery Miller and Andrzej Pelc. Tradeoffs between cost and information for rendezvous and treasure hunt. J. Parallel Distributed Comput., 83:159-167, 2015. URL: https://doi.org/10.1016/j.jpdc.2015.06.004.
23. Andrzej Pelc. Deterministic rendezvous in networks: A comprehensive survey. Networks, 59(3):331-347, 2012. URL: https://doi.org/10.1002/net.21453.
24. Andrzej Pelc. Deterministic rendezvous algorithms. In Paola Flocchini, Giuseppe Prencipe, and Nicola Santoro, editors, Distributed Computing by Mobile Entities, Current Research in Moving and Computing, volume 11340 of Lecture Notes in Computer Science, pages 423-454. Springer, 2019. URL: https://doi.org/10.1007/978-3-030-11072-7_17.
25. Amnon Ta-Shma and Uri Zwick. Deterministic rendezvous, treasure hunts, and strongly universal exploration sequences. ACM Trans. Algorithms, 10(3):12:1-12:15, 2014. URL: https://doi.org/10.1145/2601068.