  Let $\mathbf{p}$ be a configuration of $n$ points in $\mathbb{R}^d$ for some
$n$ and some $d \ge 2$. Each pair of points defines an edge, which has a
Euclidean length in the configuration. A path is an ordered sequence of the
points, and a loop is a path that has the same endpoints. A path or loop, as a
sequence of edges, also has a Euclidean length, which is simply the sum of its
Euclidean edge lengths. We are interested in reconstructing $\mathbf{p}$ given
a set of edge, path and loop lengths. In particular, we consider the unlabeled
setting where the lengths are given simply as a set of real numbers, and are
not labeled with the combinatorial data describing which paths or loops gave
rise to these lengths.
  In this paper, we study the question of when $\mathbf{p}$ will be uniquely
determined (up to an unknowable Euclidean transform) from some given set of
path or loop lengths through an exhaustive trilateration process. Such a
process has been already been used for the simpler problem of unlabeled edge
lengths.
