Before revision: In the (1+\varepsilon,r) -approximate-near-neighbor problem for curves (ANNC) under some distance measure \delta, the goal is to construct a data structure for a given set C of curves that supports approximate near-neighbor queries: Given a query curve Q, if there exists a curve C\inC such that \delta(Q,C)\le r, then return a curve C'\inC with \delta(Q,C')\le(1+\varepsilon)r.
After revision: In the (1+\varepsilon,r) -approximate near-neighbor problem for curves (ANNC) under some distance measure \delta, the goal is to construct a data structure for a given set C of curves that supports approximate near-neighbor queries: Given a query curve Q, if there exists a curve C\inC such that \delta(Q,C)\le r, then return a curve C'\inC with \delta(Q,C')\le(1+\varepsilon)r.