Before revision:  There exists an efficient reduction from the (1+\varepsilon) -approximate-nearest-neighbor problem to ANNC, where in the former problem the answer to a query is a curve C\inC with \delta(Q,C)\le(1+\varepsilon)\cdot\delta(Q,C^*), where C^* is the curve of C closest to Q. Given a set C of n curves, each consisting of m points in d dimensions, we construct a data structure for ANNC that uses n\cdot O(1{\varepsilon})^{md} storage space and has O(md \log(\frac{nm{\varepsilon})
After revision:  There exists an efficient reduction from the (1+\varepsilon) -approximate nearest-neighbor problem to ANNC, where in the former problem the answer to a query is a curve C\inC with \delta(Q,C)\le(1+\varepsilon)\cdot\delta(Q,C^*), where C^* is the curve of C closest to Q. Given a set C of n curves, each consisting of m points in d dimensions, we construct a data structure for ANNC that uses n\cdot O(1{\varepsilon})^{md} storage space and has O(md \log(\frac{nm{\varepsilon})