Computing the k Nearest-Neighbors for all Vertices via Dijkstra

We are given a directed graph $G=(V,E)$ with $n$ vertices and $m$ edges, with positive weights on the edges, and a parameter $k>0$. We show how to compute, for every vertex $v\in V$, its $k$ nearest-neighbors. The algorithm runs in $O(k(n\log n+m))$ time, and follows by a somewhat careful modification of Dijkstra’s shortest path algorithm. This result is probably folklore, but we were unable to find a reference to it – thus, this note.