  In this work, we consider the problem of computing triangular bases of
integral closures of one-dimensional local rings.
  Let $(K, v)$ be a discrete valued field with valuation ring $\mathcal{O}$ and
let $\mathfrak{m}$ be the maximal ideal. We take $f \in \mathcal{O}[x]$, a
monic irreducible polynomial of degree $n$ and consider the extension $L =
K[x]/(f(x))$ as well as $\mathcal{O}_{L}$ the integral closure of $\mathcal{O}$
in $L$, which we suppose to be finitely generated as an $\mathcal{O}$-module.
  The algorithm $\operatorname{MaxMin}$, presented in this paper, computes
triangular bases of fractional ideals of $\mathcal{O}_{L}$. The theoretical
complexity is equivalent to current state of the art methods and in practice is
almost always faster. It is also considerably faster than the routines found in
standard computer algebra systems, excepting some cases involving very small
field extensions.
