Deterministic Gathering with Crash Faults

Abstract

A team consisting of an unknown number of mobile agents, starting from different nodes of an unknown network, have to meet at the same node and terminate. This problem is known as {\em gathering}. We study deterministic gathering algorithms under the assumption that agents are subject to {\em crash faults} which can occur at any time. Two fault scenarios are considered. A {\em motion fault} immobilizes the agent at a node or inside an edge but leaves intact its memory at the time when the fault occurred. A more severe {\em total fault} immobilizes the agent as well, but also erases its entire memory. Of course, we cannot require faulty agents to gather. Thus the gathering problem for fault prone agents calls for all fault-free agents to gather at a single node, and terminate. When agents move completely asynchronously, gathering with crash faults of any type is impossible. Hence we consider a restricted version of asynchrony, where each agent is assigned by the adversary a fixed speed, possibly different for each agent. Agents have clocks ticking at the same rate. Each agent can wait for a time of its choice at any node, or decide to traverse an edge but then it moves at constant speed assigned to it. Moreover, agents have different labels. Each agent knows its label and speed but not those of other agents. We construct a gathering algorithm working for any team of at least two agents in the scenario of motion faults, and a gathering algorithm working in the presence of total faults, provided that at least two agents are fault free all the time. If only one agent is fault free, the task of gathering with total faults is sometimes impossible. Both our algorithms work in time polynomial in the size of the graph, in the logarithm of the largest label, in the inverse of the smallest speed, and in the ratio between the largest and the smallest speed.