Truth discovery is a key problem in data analytics which has received a great deal of attention in recent years. In this problem, we seek to obtain trustworthy information from data aggregated from multiple (possibly) unreliable sources. Most of the existing approaches for this problem are of heuristic nature and do not provide any quality guarantee. Very recently, the first quality-guaranteed algorithm has been discovered. However, the running time of the algorithm depends on the spread ratio of the input points and is fully polynomial only when the spread ratio is relatively small. This could severely restrict the applicability of the algorithm. To resolve this issue, we propose in this paper a new algorithm which yields a (1 + ε)-approximation in near quadratic time for any dataset with constant probability. Our algorithm relies on a data structure called range cover, which is interesting in its own right. The data structure provides a general approach for solving some high dimensional optimization problems by breaking them down into a small number of parametrized cases.