The cost of monotonicity in distributed graph searching

Blin et al. (Theor Comput Sci 399(1–2):12–37, 2008) proposed a distributed protocol enabling the smallest possible number of searchers to clear any unknown graph in a decentralized manner. However, the strategy that is actually performed lacks of an important property, namely the monotonicity. This paper deals with the smallest number of searchers that are necessary and sufficient to monotonously clear any unknown graph in a decentralized manner. 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. We prove that a distributed protocol clearing any unknown n-node graph in a monotone connected way, in a decentralized setting, can achieve but cannot beat competitive ratio of Q(\fracnlogn){\Theta(\frac{n}{\log n})} , compared with the centralized minimum number of searchers. Moreover, our lower bound holds even in a synchronous setting, while our constructive upper bound holds even in an asynchronous setting.