Abstract
Blin et al. (2006) proposed a distributed protocol that enables the smallest number of searchers to clear any unknown asynchronous graph in a decentralized manner. Unknown means that the searchers are provided no 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 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 Θ(n logn) 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 connected, i.e., the clear part of the graph must remain permanently connected, and monotone, i.e., the clear part of the graph only grows. The search number
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 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 
searchers to clear G. Moreover, we propose a distributed protocol that allows 
searchers to clear any unknown asynchronous n-node graph G in a monotone connected way.
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Ilcinkas, D., Nisse, N., Soguet, D. (2007). The Cost of Monotonicity in Distributed Graph Searching. In: Tovar, E., Tsigas, P., Fouchal, H. (eds) Principles of Distributed Systems. OPODIS 2007. Lecture Notes in Computer Science, vol 4878. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-77096-1_30
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