Molecular biologists frequently align DNA sequences of entire genomes to detect important matched and mismatched regions. Even though ef.cient dynamic programming algorithms exist for this problem, the required computing time is still very high due to the size of these sequences (usually a few million base pairs in length). Because the number of sequenced organisms is increasing rapidly, fast and accurate solutions are of highest importance to research in this area. In this paper we present an algorithm to compute the optimal and near-optimal alignments of two sequences in linear space and quadratic time. We demonstrate how this algorithm can be parallelized efficiently on a PC cluster and on a computational grid in order to reduce its runtime significantly. The grid implementation uses a hierarchical approach combining inter-cluster and intra-cluster parallelism.