The Cost of Monotonicity in Distributed Graph Searching
Résumé
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.
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