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Quality of Service (QoS) of large-scale distributed systems depends on the properties of the network connecting the nodes/hosts of the system. Topological information about the underlying network is beneficial for improving the performance, devising reliability schemes, ensuring low overhead, and enhancing the scalability of such systems. Topology information is often obtained with the support of the network infrastructure. Unfortunately, this support is often limited and sometimes not reliable. Various techniques have been proposed to infer useful information about the structure of the IP topology using strictly end-to-end measurements. In this paper, we rely on path sharing information between the nodes of a distributed system collected using end-to-end measurements and explore how much of the logical topology can be inferred using only this information. We propose an algorithm to construct such an inferred graph and evaluate this algorithm by simulations. In the synthetic topologies we considered, error in the estimated distances between the end nodes is on average a negligible fraction of the diameter for the tree topologies and less than 20% of the diameter for denser graphs.