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\documentclass{acmabs}

\begin{document}

\Journal{Journal of the ACM}

\refkey{KoscielskiP96}

\title{Complexity of {Makanin's} Algorithm}

\author{Antoni Ko{\'s}cielski \and Leszek Pacholski}

\Pages{670--684}

\Month{July}

\Year{1996}

\Volume{43}

\Number{4}

\maketitle

\begin{abstract}

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 $(6d)^{2^{2d^4}} +
2$. Consequently, our result implies an exponential improvement of
known upper bounds on complexity of word-unification algorithms.

\end{abstract}

\begin{categories}

F.2.2 [\textbf{Analysis of Algorithms and Problem Complexity}]:
Nonnumerical Algorithms and Problems---\emph{computations on discrete
structures}; I.1.2 [\textbf{Algebraic Manipulation}]:
Algorithms---\emph{algebraic algorithms}

\end{categories}

\begin{terms}

Algorithms, complexity, equations

\end{terms}

\begin{keywords}

Diophantine equation, periodicity, semantic unification, semigroups, word
  equations

\end{keywords}

\end{document}
