1. Remembering without memory: Tree exploration by asynchronous oblivious robots
- Author
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Nicola Santoro, David Ilcinkas, Andrzej Pelc, Paola Flocchini, Distributed Computing Research Group [Ottawa], University of Ottawa [Ottawa], Laboratoire Bordelais de Recherche en Informatique (LaBRI), Université de Bordeaux (UB)-Centre National de la Recherche Scientifique (CNRS)-École Nationale Supérieure d'Électronique, Informatique et Radiocommunications de Bordeaux (ENSEIRB), Algorithmics for computationally intensive applications over wide scale distributed platforms (CEPAGE), Université Sciences et Technologies - Bordeaux 1-Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-École Nationale Supérieure d'Électronique, Informatique et Radiocommunications de Bordeaux (ENSEIRB)-Centre National de la Recherche Scientifique (CNRS), Département d'Informatique et d'Ingénierie (DII), Université du Québec en Outaouais (UQO), School of Computer Science [Ottawa], Carleton University, See paper for details., ANR-07-BLAN-0322,ALADDIN,Algorithm Design and Analysis for Implicitly and Incompletely Defined Interaction Networks(2007), Université de Bordeaux (UB)-École Nationale Supérieure d'Électronique, Informatique et Radiocommunications de Bordeaux (ENSEIRB)-Centre National de la Recherche Scientifique (CNRS), Université Sciences et Technologies - Bordeaux 1 (UB)-Inria Bordeaux - Sud-Ouest, Partially supported by NSERC Discovery grant. Andrzej Pelc is also partially supported by the Research Chair in Distributed Computing at the Université du Québec en Outaouais. This work was done during the stay of David Ilcinkas at the Research Chair in Distributed Computing of the Université du Québec en Outaouais and at the University of Ottawa, as a postdoctoral fellow., and Alexander A. Shvartsman, Pascal Felber
- Subjects
Theoretical computer science ,General Computer Science ,Computer science ,[INFO.INFO-DS]Computer Science [cs]/Data Structures and Algorithms [cs.DS] ,0102 computer and information sciences ,02 engineering and technology ,exploration ,01 natural sciences ,Constructive ,Theoretical Computer Science ,Combinatorics ,Computer Science::Robotics ,oblivious ,mobile agent ,0202 electrical engineering, electronic engineering, information engineering ,Mathematics ,asynchronous ,Discrete mathematics ,Degree (graph theory) ,Swarm behaviour ,robot ,Mobile robot ,Binary logarithm ,tree ,Asymptotically optimal algorithm ,010201 computation theory & mathematics ,Asynchronous communication ,Robot ,Graph (abstract data type) ,020201 artificial intelligence & image processing ,[INFO.INFO-DC]Computer Science [cs]/Distributed, Parallel, and Cluster Computing [cs.DC] ,Computer Science(all) - Abstract
International audience; In the effort to understand the algorithmic limitations of computing by a swarm of robots, the research has focused on the minimal capabilities that allow a problem to be solved. The weakest of the commonly used models is {\sc Asynch} where the autonomous mobile robots, endowed with visibility sensors (but otherwise unable to communicate), operate in Look-Compute-Move cycles performed asynchronously for each robot. The robots are often assumed (or required to be) oblivious: they keep no memory of observations and computations made in previous cycles. We consider the setting when the robots are dispersed in an anonymous and unlabeled graph, and they must perform the very basic task of {\em exploration}: within finite time every node must be visited by at least one robot and the robots must enter a quiescent state. The complexity measure of a solution is the number of robots used to perform the task. We study the case when the graph is an arbitrary tree and establish some unexpected results. We first prove that, in general, exploration cannot be done efficiently. More precisely we prove that there are $n$-node trees where $\Omega(n)$ robots are necessary; this holds even if the maximum degree is $4$. On the other hand, we show that if the maximum degree is $3$, it is possible to explore with only $O(\frac{\log n} {\log\log n})$ robots. The proof of the result is constructive. We also prove that the size of the team used in our solution is asymptotically {\em optimal}: there are trees of degree $3$, whose exploration requires $\Omega(\frac{\log n}{\log\log n})$ robots. Our final result shows that the difficulty in tree exploration comes in fact from the symmetries of the tree. Indeed, we show that, in order to explore trees that do not have any non-trivial automorphisms, 4 robots are always sufficient and often necessary.
- Published
- 2010