Shortest distance from a set of points
Introduce a new node $s_0$ with edges to all $s \in S$ (but no others). Run BFS starting in $s_0$; the quantity you are looking for is 

$\qquad\displaystyle \min_{s \in S} d(s, v) = d(s_0,v) - 1$

for every $v \in V$.

Nota bene: The idea extends to weighted graphs; you have to a) give the new edges weight $1$ and b) use an SSSPP algorithm, of course.