Complexity of Makanin's algorithm

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The exponent of periodicity is an important factor in estimates of complexity of word-unification algorithms. We prove that the exponent of periodicity of a minimal solution of a word equation is of order 2 1.07d , where d is the length of the equation. We also give a lower bound 2 0.29d so our upper bound is almost optimal and exponentially better than the original bound (6d) 22d4 + 2 . Consequently, our result implies an exponential improvement of known upper bounds on complexity of word-unification algorithms.