Consider an unweighted, undirected, simple graph $G=(V,E)$. We have some subset $S \subseteq V$, and we want to determine the shortest distance from any vertex $v\in V$ to some vertex $s\in S$. To be clear, each $v\in V$ has a distance $d_1, d_2,\dots , d_{|S|}$ to each $s\in S$, and we want to find the smallest value of all $d_1, d_2,\dots , d_{|S|}$, and do this for every $v\in V$. So every $s\in S$, for example, would have a distance of $0$. The algorithm should run in $O(|V|+|E|)$ time. Note that this is independent of $|S|$. I was thinking of running simultaneous BFS's on each $s\in S$ and only assign a vertex a distance if it doesnt already have a distance estimate (i.e. no other node in $s$ reached to $v$ earlier). I'm not really sure how to make the runtime work out though, since we would have to cut off the BFS at some point to prevent a factor of $|S|$ in the runtime.Also, this is not a homework problem, it's a question from a midterm from a class a few years ago that I am using to study for my final coming up, and the solutions are not available. Thanks for the help!
Shortest distance from a set of points
algorithms;graphs;shortest path