In this paper, we develop two new greedy algorithms for the empirical inter-polation and the reduced basis method. The first algorithm is a Saturation Assumption based greedy algorithm. With a simple and reasonable Saturation Assumption on the error estimator, a high percentage of the workload of the standard greedy algorithm is saved. The second algorithm is an adaptively enriching greedy algorithm. In this algorithm, the samples in the train set are adaptively removed and enriched, and a safety check step is added at the end of the algorithm to ensure the quality of the basis set. It can be applied to problems with high dimensional parameter spaces. Various numerical examples are presented for both algorithms.