How Difficult is it to Walk the Dog?

We study the complexity of computing the Fréchet distance (also called dog-leash distance) between two polygonal curves with a total number of n vertices. For two polygonal curves in the plane we prove an Ω(n log n) lower bound for the decision problem in the algebraic computation tree model allowing arithmetic operations and tests. Up to now only a O(n) upper bound for the decision problem was known. The Ω(n log n) lower bound extends to variants of the Fréchet distance such as the weak as well as the discrete Fréchet distance. For the one-dimensional case we give a linear-time algorithm to solve the decision problem for the weak Fréchet distance between one-dimensional polygonal curves.