Active Learning of Multiple Source Multiple Destination Topologies

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, $\\lceil {{ N}\\over { 2}} \\rceil $, 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 $ { N\\choose 2}$ 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.