Application of Lempel-Ziv Encodings to the Solution of Words Equations

One of the most intricate algorithms related to words is Makanin's algorithm solving word equations. The algorithm is very complicated and the complexity of the problem of solving word equations is not well understood. Word equations can be used to deene various properties of strings, e.g. general versions of pattern-matching with variables. This paper is devoted to introduce a new approach and to study relations between Lempel-Ziv compressions and word equations. Instead of dealing with very long solutions we propose to deal with their Lempel-Ziv encodings. As our rst main result we prove that each minimal solution of a word equation is highly compressible (exponentially compressible for long solutions) in terms of Lempel-Ziv encoding. A simple algorithm for solving word equations is derived. If the length of minimal solution is bounded by a singly exponential function (which is believed to be always true) then LZ encoding of each minimal solution is of a polynomial size (though the solution can be exponentially long) and solvability can be checked in nondeterministic polynomial time. As our second main result we prove that the solvability can be tested in polynomial deterministic time if the lengths of all variables are given in binary. We show also that lexicographically rst solution for given lengths of variables is highly compressible in terms of Lempel-Ziv encodings.