# A Recursive Approach to Multi-robot Exploration of Trees

• Christian Ortolf
• Christian Schindelhauer
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8576)

## Abstract

The multi-robot exploration problem is to explore an unknown graph of size n and depth d with k robots starting from the same node. For known graphs a traversal of all nodes takes at most $${\cal O}(d + n/k)$$ steps. The ratio between the time until cooperating robots explore an unknown graph and the optimal traversal of a known graph is called the competitive exploration time ratio.

It is known that for any algorithm this ratio is at least $$\Omega\left((\log k)/\log \log k\right)$$. For k ≤ n robots the best algorithm known so far achieves a competitive time ratio of $${\cal O}\left({k}/{\log k} \right)$$.

Here, we improve this bound for trees with bounded depth or a minimum number of robots. Starting from a simple $${\cal O}(d)$$-competitive algorithm, called Yo-yo, we recursively improve it by the Yo-star algorithm, which for any 0 < α < 1 transforms a g(d,k)-competitive algorithm into a $${\cal O}( (g(d^{\alpha},k) \log k + d^{1-\alpha})(\log k + \log n))$$-competitive algorithm. So, we achieve a competitive bound of $${\cal O}\left(2^{{\cal O}(\sqrt{(\log d)(\log\log k)})}(\log k)(\log k+ \log n)\right)$$. This improves the best known bounds for trees of depth d, whenever the number of robots is at least $$k=2^{\omega(\sqrt{(\log d)(\log \log d)})}$$ and $$n=2^{O(2^{\sqrt{\log d}})}$$.

### Keywords

competitive analysis robot collective graph exploration

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### References

1. 1.
Albers, S., Henzinger, M.R.: Exploring Unknown Environments. SIAM Journal on Computing 29(4), 1164 (2000)
2. 2.
Albers, S., Kursawe, K., Schuierer, S.: Exploring unknown environments with obstacles. In: Proceedings of the Tenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 1999, pp. 842–843. Society for Industrial and Applied Mathematics, Philadelphia (1999)Google Scholar
3. 3.
Bender, M.A.: The power of team exploration: Two robots can learn unlabeled directed graphs. In: Proceedings of the Thirty Fifth Annual Symposium on Foundations of Computer Science, pp. 75–85 (1994)Google Scholar
4. 4.
Bender, M.A., Fernández, A., Ron, D., Sahai, A., Vadhan, S.: The power of a pebble: Exploring and mapping directed graphs. Information and Computation 176(1), 1–21 (2002)
5. 5.
Brass, P., Cabrera-Mora, F., Gasparri, A., Xiao, J.: Multirobot tree and graph exploration. IEEE Transactions on Robotics 27(4), 707–717 (2011)
6. 6.
Deng, X., Papadimitriou, C.: Exploring an unknown graph. In: Proceedings of the 31st Annual Symposium on Foundations of Computer Science, vol. 1, pp. 355–361 (October 1990)Google Scholar
7. 7.
Dereniowski, D., Disser, Y., Kosowski, A., Pająk, D., Uznański, P.: Fast collaborative graph exploration. In: Fomin, F.V., Freivalds, R., Kwiatkowska, M., Peleg, D. (eds.) ICALP 2013, Part II. LNCS, vol. 7966, pp. 520–532. Springer, Heidelberg (2013)
8. 8.
Dessmark, A., Pelc, A.: Optimal graph exploration without good maps. Theor. Comput. Sci. 326, 343–362 (2004)
9. 9.
Dynia, M., Kutyłowski, J., Meyer auf der Heide, F., Schindelhauer, C.: Smart robot teams exploring sparse trees. In: Královič, R., Urzyczyn, P. (eds.) MFCS 2006. LNCS, vol. 4162, pp. 327–338. Springer, Heidelberg (2006)
10. 10.
Dynia, M., Łopuszański, J., Schindelhauer, C.: Why robots need maps. In: Prencipe, G., Zaks, S. (eds.) SIROCCO 2007. LNCS, vol. 4474, pp. 41–50. Springer, Heidelberg (2007)
11. 11.
Fleischer, R., Trippen, G.: Exploring an unknown graph efficiently. In: Brodal, G.S., Leonardi, S. (eds.) ESA 2005. LNCS, vol. 3669, pp. 11–22. Springer, Heidelberg (2005)
12. 12.
Fraigniaud, P., Gąsieniec, L., Kowalski, D.R., Pelc, A.: Collective tree exploration. Netw. 48, 166–177 (2006)
13. 13.
Förster, K.-T., Wattenhofer, R.: Directed graph exploration. In: Baldoni, R., Flocchini, P., Binoy, R. (eds.) OPODIS 2012. LNCS, vol. 7702, pp. 151–165. Springer, Heidelberg (2012)
14. 14.
Gabriely, Y., Rimon, E.: Competitive on-line coverage of grid environments by a mobile robot. Comput. Geom. Theory Appl. 24(3), 197–224 (2003)
15. 15.
Higashikawa, Y., Katoh, N., Langerman, S., Tanigawa, S.-I.: Online graph exploration algorithms for cycles and trees by multiple searchers. Journal of Combinatorial Optimization, 1–16 (2012)Google Scholar
16. 16.
Kolenderska, A., Kosowski, A., Małafiejski, M., Żyliński, P.: An improved strategy for exploring a grid polygon. In: Kutten, S., Žerovnik, J. (eds.) SIROCCO 2009. LNCS, vol. 5869, pp. 222–236. Springer, Heidelberg (2010)
17. 17.
Ortolf, C., Schindelhauer, C.: Online multi-robot exploration of grid graphs with rectangular obstacles. In: Proceedings of the Twenty-fourth Annual ACM Symposium on Parallelism in Algorithms and Architectures, SPAA 2012, pp. 27–36. ACM, New York (2012)Google Scholar
18. 18.
Rao, N.S.V., Kareti, S., Shi, W., Iyengar, S.S.: Robot navigation in unknown terrains: Introductory survey of non-heuristic algorithms. Technical Report ORNL/TM-12410:1–58, Oak Ridge National Laboratory (July 1993)Google Scholar