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Theoretically, the fastest algorithm by Crochemore et al. for computing the alignment of two given strings of size n over a constant alphabet takes O(n²/log n) time. The algorithm uses Lempel–Ziv parsing to divide the dynamic programming matrix into blocks and utilizes the repetitive structure. It is the only previously known subquadratic-time algorithm that can handle scoring matrices of arbitrary weights. However, this algorithm takes O(n²/log n) space, and reducing the space while preserving the time complexity has been an open problem for more than 20 years. We present a solution to this issue by achieving an O(n) space algorithm that maintains O(n²/log n) time. The classical refinement by Hirschberg reduces the space complexity of the textbook O(n²) algorithm to O(n) while preserving the quadratic time. However, applying this technique to the algorithm of Crochemore et al. has been considered challenging because their method requires O(n² / log n) space even when computing only the alignment score. Our modification enables the application of Hirschberg’s refinement, allowing traceback computation in O(n) space while preserving the O(n² / log n) overall time complexity. Our algorithm can be applied to both global and local string alignment problems.