We study the problem of local alignment, which is finding pairs of similar subsequences with gaps. The problem exists in biosequence databases. BLAST is a typical software for finding local alignment based on heuristic, but could miss results. Using the Smith-Waterman algorithm, we can find all local alignments in O(mn) time, where m and n are lengths of a query and a text, respectively. A recent exact approach BWT-SW improves the complexity of the Smith-Waterman algorithm under constraints, but still much slower than BLAST. This paper takes on the challenge of designing an accurate and efficient algorithm for evaluating local-alignment searches, especially for long queries. In this paper, we propose an efficient software called ALAE to speed up BWT-SW using a compressed suffix array. ALAE utilizes a family of filtering techniques to prune meaningless calculations and an algorithm for reusing score calculations. We also give a mathematical analysis and show that the upper bound of the total number of calculated entries using ALAE could vary from 4.50mn0.520 to 9.05mn0.896 for random DNA sequences and vary from 8.28mn0.364 to 7.49mn0.723 for random protein sequences. We demonstrate the significant performance improvement of ALAE on BWT-SW using a thorough experimental study on real biosequences. ALAE guarantees correctness and accelerates BLAST for most of parameters.