  We consider the problem of computing the Fr\'echet distance between two
curves for which the exact locations of the vertices are unknown. Each vertex
may be placed in a given uncertainty region for that vertex, and the objective
is to place vertices so as to minimise the Fr\'echet distance. This problem was
recently shown to be NP-hard in 2D, and it is unclear how to compute an optimal
vertex placement at all.
  We present the first general algorithmic framework for this problem. We prove
that it results in a polynomial-time algorithm for curves in 1D with intervals
as uncertainty regions. In contrast, we show that the problem is NP-hard in 1D
in the case that vertices are placed to maximise the Fr\'echet distance.
  We also study the weak Fr\'echet distance between uncertain curves. While
finding the optimal placement of vertices seems more difficult than the regular
Fr\'echet distance -- and indeed we can easily prove that the problem is
NP-hard in 2D -- the optimal placement of vertices in 1D can be computed in
polynomial time. Finally, we investigate the discrete weak Fr\'echet distance,
for which, somewhat surprisingly, the problem is NP-hard already in 1D.
