  We study two fundamental problems dealing with curves in the plane, namely,
the nearest-neighbor problem and the center problem. Let $\mathcal{C}$ be a set
of $n$ polygonal curves, each of size $m$. In the nearest-neighbor problem, the
goal is to construct a compact data structure over $\mathcal{C}$, such that,
given a query curve $Q$, one can efficiently find the curve in $\mathcal{C}$
closest to $Q$. In the center problem, the goal is to find a curve $Q$, such
that the maximum distance between $Q$ and the curves in $\mathcal{C}$ is
minimized. We use the well-known discrete Frechet distance function, both
under~$L_\infty$ and under $L_2$, to measure the distance between two curves.
  For the nearest-neighbor problem, despite discouraging previous results, we
identify two important cases for which it is possible to obtain practical
bounds, even when $m$ and $n$ are large. In these cases, either $Q$ is a line
segment or $\mathcal{C}$ consists of line segments, and the bounds on the size
of the data structure and query time are nearly linear in the size of the input
and query curve, respectively. The returned answer is either exact under
$L_\infty$, or approximated to within a factor of $1+\varepsilon$ under~$L_2$.
We also consider the variants in which the location of the input curves is only
fixed up to translation, and obtain similar bounds, under $L_\infty$.
  As for the center problem, we study the case where the center is a line
segment, i.e., we seek the line segment that represents the given set as well
as possible. We present near-linear time exact algorithms under $L_\infty$,
even when the location of the input curves is only fixed up to translation.
Under $L_2$, we present a roughly $O(n^2m^3)$-time exact algorithm.
