Suppose we have sets $X$ and $Y$, $|X|=m$, $|Y|=n$. $H$ is a universal family of hash functions from $X$ to $Y$. Let $S\subsetneq X$ be a proper subset of $X$. Does there exist some "partial" universal hash family that is universal on $S$ to $Y$, but not necessarily universal on $X$ to $Y$? The circuit complexity of that "partial" universal hashing should be preferably lower then $H$.
Update: Does there exist construction such that taking the union of two "partial" universal hash families from S1 and S2 to Y we can get a universal hash family from S1∪S2 to Y?