Algorithms for Extracting Timeliness Graphs

We consider asynchronous message-passing systems in which some links are timely and processes may crash. Each run defines a timeliness graph among correct processes: (p, q) is an edge of the timeliness graph if the link from p to q is timely (that is, there is a bound on communication delays from p to q). The main goal of this paper is to approximate this timeliness graph by graphs having some properties (such as being trees, rings, . . . ). Given a family S of graphs, for runs such that the timeliness graph contains at least one graph in S then using an extraction algorithm, each correct process has to converge to the same graph in S that is, in a precise sense, an approximation of the timeliness graph of the run. For example, if the timeliness graph contains a ring, then using an extraction algorithm, all correct processes eventually converge to the same ring and in this ring all nodes will be correct processes and all links will be timely. We first present a general extraction algorithm and then a more specific extraction algorithm that is communication efficient (i.e., eventually all the messages of the extraction algorithm use only links of the extracted graph).