Network topology plays an important role in many network operations. However, it is very difficult to obtain the topology of public networks due to the lack of internal cooperation. Network tomography provides a powerful solution that can infer the network routing topology from end-to-end measurements. Existing solutions all assume that routes from a single source form a tree. However, with the rapid deployment of Software Defined Networking (SDN) and Network Function Virtualization (NFV), the routing paths in modern networks are becoming more complex. To address this problem, we propose a novel inference problem, called the weight inference problem, which infers the finest-granularity information from end-to-end measurements on general routing paths in general topologies. Our measurements are based on emulated multicast probes with a controllable "width". We show that the problem has a unique solution when the multicast width is unconstrained; otherwise, we show that the problem can be treated as a sparse approximation problem, which allows us to apply variations of the pursuit algorithms. Simulations based on real network topologies show that our solution significantly outperforms a state-of-the-art network tomography algorithm, and increasing the width of multicast substantially improves the inference accuracy.