We consider the problem of inferring the topology of a network with M sources and N receivers (an M-by- N network), by sending probes between the sources and receivers. Prior work has shown that this problem can be decomposed into two parts: first, infer smaller subnetwork components (1-by- N's or 2-by-2's) and then merge them to identify the M-by- N topology. We focus on the second part, which had previously received less attention in the literature. We assume that a 1-by- N topology is given and that all 2-by-2 components can be queried and learned using end-to-end probes. The problem is which 2-by-2's to query and how to merge them with the given 1-by- N, so as to exactly identify the 2-by- N topology, and optimize a number of performance metrics, including the number of queries (which directly translates into measurement bandwidth), time complexity, and memory usage. We provide a lower bound, [N 2], on the number of 2-by-2's required by any active learning algorithm and propose two greedy algorithms. The first algorithm follows the framework of multiple hypothesis testing, in particular Generalized Binary Search (GBS). The second algorithm is called the Receiver Elimination Algorithm (REA) and follows a bottom-up approach. It requires exactly N-1 steps, which is much less than all (2N) possible 2-by-2's. Simulation results demonstrate that both algorithms correctly identify the 2-by- N topology and are near-optimal, but REA is more efficient in practice.
Topology inference problems have also been studied in the context of phylogenetic trees @cite_45 @cite_4 . The work in @cite_1 built on @cite_40 and proposed robust algorithms for multiple source tree topology inference. The work in @cite_28 inferred the topology of sparse random graphs using end-to-end measurements between a small subset of nodes. However, the quartet structures and the way we measure them are different in our case due to the nature of active probing in network tomography (see problem formulation in ).
@cite_4 @cite_28 @cite_1 @cite_40 @cite_45
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We consider the task of topology discovery of sparse random graphs using end-to-end random measurements (e.g., delay) between a subset of nodes, referred to as the participants. The rest of the nodes are hidden, and do not provide any information for topology discovery. We consider topology discovery under two routing models: (a) the participants exchange messages along the shortest paths and obtain end-to-end measurements, and (b) additionally, the participants exchange messages along the second shortest path. For scenario (a), our proposed algorithm results in a sub-linear edit-distance guarantee using a sub-linear number of uniformly selected participants. For scenario (b), we obtain a much stronger result, and show that we can achieve consistent reconstruction when a sub-linear number of uniformly selected nodes participate. This implies that accurate discovery of sparse random graphs is tractable using an extremely small number of participants. We finally obtain a lower bound on the number of participants required by any algorithm to reconstruct the original random graph up to a given edit distance. We also demonstrate that while consistent discovery is tractable for sparse random graphs using a small number of participants, in general, there are graphs which cannot be discovered by any algorithm even with a significant number of participants, and with the availability of end-to-end information along all the paths between the participants. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2013 We consider the problem of inferring the topology of an M-by-N network by sending probes between M sources and N receivers. Prior work has shown that this problem can be decomposed into two parts: first, infer smaller subnetwork components (i.e., 1-by-N's or 2-by-2's) and then merge these components to identify the M-by-N topology. In this paper, we focus on the second part. In particular, we assume that a 1by-N topology is given and that all 2-by-2 components can be queried and learned using end-to-end probes. The problem is which 2-by-2's to query and how to merge them with the 1-byN, so as to exactly identify the 2-by-N topology, and optimize a number of performance metrics including measurement traffic, time complexity, and memory usage. We provide a lower bound, ⌈N 2⌉, on the number of 2-by-2's required by any active learning algorithm and we also propose a greedy algorithm that is nearoptimal and efficient in practice. It follows a bottom-up approach: at every step, it selects two receivers, queries the corresponding 2-by-2, and merges it with the given 1-by-N. The algorithm requires exactly N - 1 steps, which is much less than all (N:2) possible 2-by-2's, and it correctly identifies the 2-by-N topology. Several popular methods for phylogenetic inference (or hierarchical clustering) are based on a matrix of pairwise distances between taxa (or any kind of objects): The objective is to construct a tree with branch lengths so that the distances between the leaves in that tree are as close as possible to the input distances. If we hold the structure (topology) of the tree fixed, in some relevant cases (e.g., ordinary least squares) the optimal values for the branch lengths can be expressed using simple combinatorial formulae. Here we define a general form for these formulae and show that they all have two desirable properties: First, the common tree reconstruction approaches (least squares, minimum evolution), when used in combination with these formulae, are guaranteed to infer the correct tree when given enough data (consistency); second, the branch lengths of all the simple (nearest neighbor interchange) rearrangements of a tree can be calculated, optimally, in quadratic time in the size of the tree, thus allowing the efficient application of hill climbing heuristics. The study presented here is a continuation of that by Mihaescu and Pachter on branch length estimation [Mihaescu R, Pachter L (2008) Proc Natl Acad Sci USA 105:13206–13211]. The focus here is on the inference of the tree itself and on providing a basis for novel algorithms to reconstruct trees from distances. In this paper, we combine network coding and tomographic techniques for topology inference. Our goal is to infer the topology of a network by sending probes between a given set of multiple sources and multiple receivers and by having intermediate nodes perform network coding operations. We combine and extend two ideas that have been developed independently. On one hand, network coding introduces topology-dependent correlation, which can then be exploited at the receivers to infer the topology [1]. On the other hand, it has been shown that a traditional (i.e., without network coding) multiple source, multiple receiver tomography problem can be decomposed into multiple two source, two receiver subproblems [2]. Our first contribution is to show that, when intermediate nodes perform network coding, topological information contained in network coded packets allows to accurately distinguish among all different 2-by-2 subnetwork components, which was not possible with traditional tomographic techniques. Our second contribution is to use this knowledge to merge the subnetworks and accurately reconstruct the general topology. Our approach is applicable to any general Internet-like topology, and is robust to the presence of delay variability and packet loss.