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The lower bound on the redundancy for lossless universal coding of regular memoryless sources with a bounded number of abrupt changes in the statistics is shown to be asymptotically achievable using a fixed per-letter computational complexity sequential compression scheme with fixed storage complexity. The scheme which outperforms any other known fixed-complexity scheme when regularity conditions hold is presented, and its redundancy is upper-bounded. Although the upper bounds are merely asymptotic, simulation results show that even for relatively short sequences, the redundancy obtained by asymptotically optimal schemes of higher complexity can still be achieved with fixed per-letter complexity. Furthermore, in practice, a fixed-complexity scheme based on the proposed scheme can in most cases achieve optimal redundancy even when the regularity conditions do not hold.