Perpetual Graph Searching

Description

In graph searching, a team of mobile agents aims at clearing the edges of a contaminated graph. To clear an edge, an agent has to slide along it, however, an edge can be recontaminated if there is a path without agents from a contaminated edge to a clear edge. To goal of graph searching is to clear the graph, i.e., all edges are clear simultaneously, using the fewest number of agents. We study this problem in the minimal CORDA model of distributed computation. This model has very weak hypothesis: network nodes and agents are anonymous, have no memory of the past, and agents have no common sense of orientation. Moreover, all agents execute the same algorithm in the Look-Compute-Move manner and in an asynchronous environment. One interest of this model is that, if the clearing can be done by the agents starting from arbitrary positions (e.g., after faults or recontamination), the lack of memory implies that the clearing is done perpetually and then provides a first approach of fault-tolerant graph searching. Constraints due to the minimal CORDA model lead us to define a new variant of graph searching, called graph searching without collisions, where more than one agent cannot occupy the same node. We show that, in a centralized setting, this variant does not have the same behavior as classical graph searching. For instance, it not monotonous nor close by subgraph. We show that, in a graph with maximum degree $\Delta$, the smallest number of agents required to clear a graph without collisions is at most $\Delta$ times the number of searchers required when collisions are allowed. Moreover, this bound is tight up to a constant ratio. Then, we fully characterize graph searching without collisions in trees. In a distributed setting, i.e., in the minimal CORDA model, the question we ask is the following. Given a graph $G$, does there exist an algorithm that clears $G$, whatever be the initial positions of the agents on distinct vertices. In the case of a path network, we show that it is not possible is the number of agents is even in a path of odd order, or if there are at most two agents in a path with at least three vertices. We present an algorithm that clears all paths in all remaining cases. Finally, we propose an algorithm that clears any tree using a sufficient number of agents.

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