Network tomography is an appealing technology to infer link delay distributions since it only relies on end-to-end measurements. However, most approaches in network delay tomography are usually computationally intractable. In this letter, we propose a Fast link Delay distribution Inference algorithm (FDI). It estimates the node cumulative delay distributions by explicit computations based on a subtree-partitioning technique, and then derives the individual link delay distributions from the estimated cumulative delay distributions. Furthermore, a novel discrete delay model where each link has a different bin size is proposed to efficiently capture the essential characteristics of the link delay. Combining with the variable bin size model, FDI can identify the characteristics of the network-internal link delay quickly and accurately. Simulation results validate the effectiveness of our method.