A note on distance-preserving graph sparsification

We consider problems of the following type: given a graph G, how many edges are needed in worst case for sparse subgraph H that approximately preserves distances between set node pairs P? Examples include pairwise spanners, distance preservers, reachability etc. There has been trend area simple constructions based on hitting technique, followed by somewhat more complicated improve over bounds obtained from sets roughly log factor. In this note, we point out simpler don't actually need an extra factor first place. This simplifies and unifies few proofs area, it improves size +4 spanner O˜(np2/7) (Kavitha (2017) [13]) to O(np2/7).