ABSTRACT
We consider the problem of collaborative tree exploration posed by Fraigniaud, Gasieniec, Kowalski, and Pelc [8] where a team of k agents is tasked to collectively go through all the edges of an unknown tree as fast as possible and return to the root. Denoting by n the total number of nodes and by D the tree depth, the O(n/log(k) + D) algorithm of [8] achieves the best competitive ratio known with respect to the optimal exploration algorithm that knows the tree in advance, which takes order max {2n/k, 2D} rounds. Brass, Cabrera-Mora, Gasparri, and Xiao [1] consider an alternative performance criterion, the additive overhead with respect to 2n/k, and obtain a 2n/k + O((D + k)k) runtime guarantee. In this announcement, we present 'Breadth-First Depth-Next' (BFDN), a novel and simple algorithm that performs collaborative tree exploration in time 2n/k + O(D2 log(k)), thus outperforming [1] for all values of (n, D) and being order-optimal for fixed k and trees with depth D = o(√n). The proof of our result crucially relies on the analysis of a simple two-player game with balls in urns that could be of independent interest. We extend the guarantees of BFDN to: scenarios with limited memory and communication, adversarial setups where robots can be blocked, and exploration of classes of non-tree graphs. Finally, we provide a recursive version of BFDN with a runtime of Oℓ(n/k1/ℓ + log(k)D1+1/ℓ) for parameter ℓ ≥ 1, thereby improving performance for trees with large depth. A complete version of the paper is available online [2].
- Peter Brass, Flavio Cabrera-Mora, Andrea Gasparri, and Jizhong Xiao. 2011. Multirobot Tree and Graph Exploration. IEEE Trans. Robotics 27, 4 (2011), 707--717. Google ScholarDigital Library
- Romain Cosson, Laurent Massoulié, and Laurent Viennot. 2023. Breadth-First Depth-Next: Optimal Collaborative Exploration of Trees with Low Diameter. arXiv preprint arXiv:2301.13307 (2023).Google Scholar
- Dariusz Dereniowski, Yann Disser, Adrian Kosowski, Dominik Pajak, and Przemyslaw Uznanski. 2013. Fast Collaborative Graph Exploration. In Automata, Languages, and Programming - 40th International Colloquium, ICALP 2013, Riga, Latvia, July 8--12, 2013, Proceedings, Part II (Lecture Notes in Computer Science, Vol. 7966), Fedor V. Fomin, Rusins Freivalds, Marta Z. Kwiatkowska, and David Peleg (Eds.). Springer, 520--532. Google ScholarDigital Library
- Yann Disser, Frank Mousset, Andreas Noever, Nemanja Skoric, and Angelika Steger. 2017. A General Lower Bound for Collaborative Tree Exploration. In Structural Information and Communication Complexity - 24th International Colloquium, SIROCCO 2017, Porquerolles, France, June 19--22, 2017, Revised Selected Papers (Lecture Notes in Computer Science, Vol. 10641), Shantanu Das and Sébastien Tixeuil (Eds.). Springer, 125--139. Google ScholarCross Ref
- Miroslaw Dynia, Miroslaw Korzeniowski, and Christian Schindelhauer. 2006. Power-Aware Collective Tree Exploration. In Architecture of Computing Systems - ARCS 2006, 19th International Conference, Frankfurt/Main, Germany, March 13--16, 2006, Proceedings (Lecture Notes in Computer Science, Vol. 3894), Werner Grass, Bernhard Sick, and Klaus Waldschmidt (Eds.). Springer, 341--351. Google ScholarDigital Library
- Miroslaw Dynia, Jaroslaw Kutylowski, Friedhelm Meyer auf der Heide, and Christian Schindelhauer. 2006. Smart Robot Teams Exploring Sparse Trees. In Mathematical Foundations of Computer Science 2006, 31st International Symposium, MFCS 2006, Stará Lesná, Slovakia, August 28-September 1, 2006, Proceedings (Lecture Notes in Computer Science, Vol. 4162), Rastislav Kralovic and Pawel Urzyczyn (Eds.). Springer, 327--338. Google ScholarDigital Library
- Miroslaw Dynia, Jakub Lopuszanski, and Christian Schindelhauer. 2007. Why Robots Need Maps. In Structural Information and Communication Complexity, 14th International Colloquium, SIROCCO 2007, Castiglioncello, Italy, June 5--8, 2007, Proceedings (Lecture Notes in Computer Science, Vol. 4474), Giuseppe Prencipe and Shmuel Zaks (Eds.). Springer, 41--50. Google ScholarCross Ref
- Pierre Fraigniaud, Leszek Gasieniec, Dariusz R. Kowalski, and Andrzej Pelc. 2006. Collective tree exploration. Networks 48, 3 (2006), 166--177. Google ScholarCross Ref
- Yuya Higashikawa, Naoki Katoh, Stefan Langerman, and Shin-ichi Tanigawa. 2014. Online graph exploration algorithms for cycles and trees by multiple searchers. J. Comb. Optim. 28, 2 (2014), 480--495. Google ScholarDigital Library
- Christian Ortolf and Christian Schindelhauer. 2012. Online multi-robot exploration of grid graphs with rectangular obstacles. In 24th ACM Symposium on Parallelism in Algorithms and Architectures, SPAA '12, Pittsburgh, PA, USA, June 25--27, 2012, Guy E. Blelloch and Maurice Herlihy (Eds.). ACM, 27--36. Google ScholarDigital Library
- Christian Ortolf and Christian Schindelhauer. 2014. A Recursive Approach to Multi-robot Exploration of Trees. In Structural Information and Communication Complexity - 21st International Colloquium, SIROCCO 2014, Takayama, Japan, July 23--25, 2014. Proceedings (Lecture Notes in Computer Science, Vol. 8576), Magnús M. Halldórsson (Ed.). Springer, 343--354. Google ScholarCross Ref
Index Terms
- Brief Announcement: Efficient Collaborative Tree Exploration with Breadth-First Depth-Next
Recommendations
Brief Announcement: An Exponential Separation Between Randomized and Deterministic Complexity in the LOCAL Model
PODC '16: Proceedings of the 2016 ACM Symposium on Principles of Distributed ComputingOver the past 30 years numerous algorithms have been designed for symmetry breaking problems in the LOCAL model, such as maximal matching, MIS, vertex coloring, and edge-coloring. For most problems the best randomized algorithm is at least exponentially ...
Brief Announcement: Using Read-k Inequalities to Analyze a Distributed MIS Algorithm
PODC '16: Proceedings of the 2016 ACM Symposium on Principles of Distributed ComputingUntil recently, the fastest distributed MIS algorithm, even for simple graphs, e.g., unoriented trees, has been the simple randomized algorithm discovered in the 80s. This algorithm (commonly called Luby's algorithm) computes an MIS in O(log n) rounds (...
Brief Announcement: Symmetry Breaking in the CONGEST Model: Time- and Message-Efficient Algorithms for Ruling Sets
PODC '17: Proceedings of the ACM Symposium on Principles of Distributed ComputingWe study local symmetry breaking problems in the Congest model, focusing on ruling set problems, which generalize the fundamental Maximal Independent Set (MIS) problem. Our work is motivated by the following central question: can we break the long-...
Comments