We investigate the problem of identifying individual link metrics in a communication network from end-to-end path measurements, under the assumption that link metrics are additive and constant. To uniquely identify the link metrics, the number of linearly independent measurement paths must equal the number of links. Our contribution is to characterize this condition in terms of the network topology and the number/placement of monitors, under the constraint that measurement paths must be cycle-free. Our main results are: (i) it is generally impossible to identify all the link metrics by using two monitors; (ii) nevertheless, metrics of all the interior links not incident to any monitor are identifiable by two monitors if the topology satisfies a set of necessary and sufficient connectivity conditions; (iii) these conditions naturally extend to a necessary and sufficient condition for identifying all the link metrics using three or more monitors. We show that these conditions not only allow efficient identifiability tests, but also enable an efficient algorithm to place the minimum number of monitors in order to identify all link metrics. Our evaluations on both random and real topologies show that the proposed algorithm achieves identifiability using a much smaller number of monitors than a baseline solution.

Abstract-We investigate the problem of identifying individual link metrics in a communication network from accumulated end-to-end metrics over selected measurement paths, under the assumption that link metrics are additive and constant during the measurement, and measurement paths cannot contain cycles. We know from linear algebra that all link metrics can be uniquely identified when the number of linearly independent measurement paths equals n, the number of links. It is, however, inefficient to collect measurements from all possible paths, whose number can grow exponentially in n, as the number of useful measurements (from linearly independent paths) is at most n. The aim of this paper is to develop efficient algorithms for constructing linearly independent measurement paths and calculating link metrics. We show that whenever there exists a set of n linearly independent measurement paths, there must exist a set of three pairwise independent spanning trees. We exploit this property to develop an algorithm that can construct n linearly independent, cycle-free paths between monitors without examining all candidate paths, whose complexity is quadratic in n. A further benefit of the proposed algorithm is that the generated paths satisfy a nested structure that allows linear-time computation of link metrics without explicitly inverting the measurement matrix. Our evaluations on both synthetic and real network topologies verify the superior efficiency of the proposed algorithms, which are orders of magnitude faster than benchmark solutions for large networks.

We investigate the problem of localizing node failures in a communication network from end-to-end path measurements, under the assumption that a path behaves normally if and only if it does not contain any failed nodes. To uniquely localize node failures, the measurement paths must show different symptoms under different failure events, i.e., for any two distinct sets of failed nodes, there must be a measurement path traversing one and only one of them. This condition is, however, impractical to test for large networks. Our first contribution is a characterization of this condition in terms of easily verifiable conditions on the network topology with given monitor placements under three families of probing mechanisms, which differ in whether measurement paths are (i) arbitrarily controllable, (ii) controllable but cycle-free, or (iii) uncontrollable (i.e., determined by the default routing protocol). Our second contribution is a characterization of the maximum identifiability of node failures, measured by the maximum number of simultaneous failures that can always be uniquely localized. Specifically, we bound the maximal identifiability from both the upper and the lower bounds which differ by at most one, and show that these bounds can be evaluated in polynomial time. Finally, we quantify the impact of the probing mechanism on the capability of node failure localization under different probing mechanisms on both random and real network topologies. We observe that despite a higher implementation cost, probing along controllable paths can significantly improve a network's capability to localize simultaneous node failures.

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