  Given a smooth closed embedded self-shrinker $S$ with index $I$ in $\mathbb
R^{n}$, we construct an $I$-dimensional family of complete translators
polynomially asymptotic to $S\times\mathbb R$ at infinity. We further prove
that $\mathbb R^{n+1}$ can be decomposed in many ways into a one-parameter
family of closed sets $\coprod_{a\in \mathbb R} T_a$, and each closed set $T_a$
contains a complete translator asymptotic to $S\times\mathbb R$ at infinity. If
the closed set $T_a$ fattens, namely it has nonempty interior, then there are
at least two translators asymptotic to each other at an exponential rate, which
can be viewed as a kind of nonuniqueness. We show that this fattening
phenomenon is non-generic but indeed happens.
