1. The cost of monotonicity in distributed graph searching
- Author
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Nicolas Nisse, David Soguet, David Ilcinkas, Département d'Informatique et d'Ingénierie (DII), Université du Québec en Outaouais (UQO), Laboratoire de Recherche en Informatique (LRI), Université Paris-Sud - Paris 11 (UP11)-CentraleSupélec-Centre National de la Recherche Scientifique (CNRS), Additional supports from the project 'Fragile' of the ACI Sécurité Informatique, and from the project 'Grand Large' of INRIA., Eduardo Tovar, Philippas Tsigas, Hacène Fouchal, 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), Algorithms, simulation, combinatorics and optimization for telecommunications (MASCOTTE), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-COMmunications, Réseaux, systèmes Embarqués et Distribués (Laboratoire I3S - COMRED), Laboratoire d'Informatique, Signaux, et Systèmes de Sophia Antipolis (I3S), Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Laboratoire d'Informatique, Signaux, et Systèmes de Sophia Antipolis (I3S), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA), 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, Université Nice Sophia Antipolis (1965 - 2019) (UNS), and COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (1965 - 2019) (UNS)
- Subjects
Monotonicity ,Mathematical optimization ,Theoretical computer science ,Computer Networks and Communications ,Computer science ,[INFO.INFO-DS]Computer Science [cs]/Data Structures and Algorithms [cs.DS] ,050801 communication & media studies ,0102 computer and information sciences ,02 engineering and technology ,Strength of a graph ,Biconnected graph ,01 natural sciences ,Theoretical Computer Science ,law.invention ,0508 media and communications ,Graph power ,law ,Line graph ,0202 electrical engineering, electronic engineering, information engineering ,Graph property ,Complement graph ,Graph searching ,Distance-hereditary graph ,Mobile agent ,Discrete mathematics ,020203 distributed computing ,05 social sciences ,Voltage graph ,Directed graph ,Butterfly graph ,Competitive ratio ,Graph bandwidth ,Computational Theory and Mathematics ,010201 computation theory & mathematics ,Hardware and Architecture ,[INFO.INFO-DC]Computer Science [cs]/Distributed, Parallel, and Cluster Computing [cs.DC] ,Null graph ,Graph factorization ,MathematicsofComputing_DISCRETEMATHEMATICS - Abstract
International audience; Blin {\it et al.} (2006) proposed a distributed protocol that enables the smallest number of searchers to clear any unknown asynchronous graph in a decentralized manner. {\it Unknown} means that the searchers are provided no {\it a priori} information about the graph. However, the strategy that is actually performed lacks of an important property, namely the monotonicity. That is, the clear part of the graph may decrease at some steps of the execution of the protocol. Actually, the protocol of Blin {\it et al.} is executed in exponential time. Nisse and Soguet (2007) proved that, in order to ensure the smallest number of searchers to clear any $n$-node graph in a monotone way, it is necessary and sufficient to provide $\Theta(n \log n)$ bits of information to the searchers by putting short labels on the nodes of the graph. This paper deals with the smallest number of searchers that are necessary and sufficient to monotoneously clear any graph in a decentralized manner, when the searchers have no a priori information about the graph. The distributed graph searching problem considers a team of searchers that is aiming at clearing any connected contaminated graph. The clearing of the graph is required to be {\it connected}, i.e., the clear part of the graph must remain permanently connected, and {\it monotone}, i.e., the clear part of the graph only grows. The {\it search number} $\mcs(G)$ of a graph $G$ is the smallest number of searchers necessary to clear $G$ in a monotone connected way in centralized settings. We prove that any distributed protocol aiming at clearing any unknown n-node graph in a monotone connected way, in decentralized settings, has {\it competitive ratio} $\Theta(\frac{n}{\log n})$. That is, we prove that, for any distributed protocol $\cal P$, there exists a constant $c$ such that for any sufficiently large $n$, there exists a $n$-node graph $G$ such that $\cal P$ requires at least $c\frac{n}{\log n}\, \mcs(G)$ searchers to clear $G$. Moreover, we propose a distributed protocol that allows $O(\frac{n}{\log n})\, \mcs(G)$ searchers to clear any unknown asynchronous $n$-node graph $G$ in a monotone connected way.
- Published
- 2009