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The constrained multiple sequence alignment problem is to align a set of sequences subject to a given constrained sequence, which arises from some knowledge of the structure of the sequences. This paper presents new algorithms for this problem, which are more efficient in terms of time and space (memory) than the previous algorithms and with a worst-case guarantee on the quality of the alignment. Saving the space requirement by a quadratic factor is particularly significant as the previous O(n4)-space algorithm has limited application due to its huge memory requirement. Experiments on real data sets confirm that our new algorithms show improvements in both alignment quality and resource requirements.