 Given a parameter \epsilon \in (0, 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 2+\epsilon , or answering that there exists no curve in the input within distance 1.
<meaning-changed>  Given a parameter \epsilon \in (0, 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 2+\epsilon , or answering that there exists no curve in the input within distance 1.
 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 2+\epsilon , or answering that there exists no curve in the input within distance 1.