  Selection on $X_1+X_2+\cdots + X_m$ is an important problem with many
applications in areas such as max-convolution, max-product Bayesian inference,
calculating most probable isotopes, and computing non-parametric test
statistics, among others. Faster-than-na\"{i}ve approaches exist for $m=2$:
Frederickson (1993) published the optimal algorithm with runtime $O(k)$ and
Kaplan \emph{et al.} (2018) has since published a much simpler algorithm which
makes use of Chazelle's soft heaps (2003). No fast methods exist for $m>2$.
Johnson \& Mizoguchi (1978) introduced a method to compute the single $k^{th}$
value when $m>2$, but that method runs in $O(m\cdot n^{\lceil\frac{m}{2}\rceil}
\log(n))$ time and is inefficient when $m \gg 1$ and $k \ll
n^{\lceil\frac{m}{2}\rceil}$.
  In this paper, we introduce the first efficient methods, both in theory and
practice, for problems with $m>2$. We introduce the ``layer-ordered heap,'' a
simple special class of heap with which we produce a new, fast selection
algorithm on the Cartesian product. Using this new algorithm to perform
$k$-selection on the Cartesian product of $m$ arrays of length $n$ has runtime
$\in o(k\cdot m)$. We also provide implementations of the algorithms proposed
and evaluate their performance in practice.
