  Given a string $S$ of $n$ integers in $[0,\sigma)$, a range minimum query
RMQ$(i, j)$ asks for the index of the smallest integer in $S[i \dots j]$. It is
well known that the problem can be solved with a succinct data structure of
size $2n + o(n)$ and constant query-time. In this paper we show how to
preprocess $S$ into a compressed representation that allows fast range minimum
queries. This allows for sublinear size data structures with logarithmic query
time. The most natural approach is to use string compression and construct a
data structure for answering range minimum queries directly on the compressed
string. We investigate this approach in the context of grammar compression. We
then consider an alternative approach. Instead of compressing $S$ using string
compression, we compress the Cartesian tree of $S$ using tree compression. We
show that this approach can be exponentially better than the former, is never
worse by more than an $O(\sigma)$ factor (i.e. for constant alphabets it is
never asymptotically worse), and can in fact be worse by an $\Omega(\sigma)$
factor.
