Number of reachable vertices in DAG for every vertex
The best exact algorithm will run in time O( min{mn, n^2.38} ) by using fast binary matrix multiplication. However, there is a random algorithm, which runs in time O(m+n) and estimates the number of reachable nodes from each node with a small relative error, please refer to paper "Size-Estimation Framework with Applications to Transitive Closure and Reachability" by Edith Cohen.