In practice, yes, the computation time required to compute edit distance will depend on the actual data. Similarly, 'sorting' a list that is already sorted can

In practice, yes, the computation time required to compute edit distance will depend on the actual data. Similarly, 'sorting' a list that is already sorted can be done in linear time, and near-sorted lists can be sorted much faster than a randomly ordered list. This is certainly useful in practice, but it doesn't affect whether the worst- or average-case complexity is quadratic. I'd like to see a quadratic (or exponential etc) time problem that couldn't be solved faster in many cases.