Before revision:  For an attainable approximation factor c>1 , the data structure can be used to preprocess n curves in R (aka time series), each of complexity m, to answer queries with a curve of complexity k by either returning a curve that lies within Fr\'echet distance c , or answering that there exists no curve in the input within distance 1.
After revision:  For an attainable approximation factor c>1 and a query radius r, an approximate-near-neighbor data structure can be used to preprocess n curves in R (aka time series), each of complexity m, to answer queries with a curve of complexity k by either returning a curve that lies within Fr\'echet distance c , or answering that there exists no curve in the input within distance 1.