  A pseudo-polynomial time $(1 + \varepsilon)$-approximation algorithm is
presented for computing the integral and average Fr\'{e}chet distance between
two given polygonal curves $T_1$ and $T_2$. In particular, the running time is
upper-bounded by $\mathcal{O}( \zeta^{4}n^4/\varepsilon^{2})$ where $n$ is the
complexity of $T_1$ and $T_2$ and $\zeta$ is the maximal ratio of the lengths
of any pair of segments from $T_1$ and $T_2$. The Fr\'{e}chet distance captures
the minimal cost of a continuous deformation of $T_1$ into $T_2$ and vice versa
and defines the cost of a deformation as the maximal distance between two
points that are related. The integral Fr\'{e}chet distance defines the cost of
a deformation as the integral of the distances between points that are related.
The average Fr\'{e}chet distance is defined as the integral Fr\'{e}chet
distance divided by the lengths of $T_1$ and $T_2$.
  Furthermore, we give relations between weighted shortest paths inside a
single parameter cell $C$ and the monotone free space axis of $C$. As a result
we present a simple construction of weighted shortest paths inside a parameter
cell. Additionally, such a shortest path provides an optimal solution for the
partial Fr\'{e}chet similarity of segments for all leash lengths. These two
aspects are related to each other and are of independent interest.
