Abstract. In this paper we consider word equations with one variable (and arbitrary many occur-rences of it). A recent technique of recompression, which is applicable to general word equations, is shown to be suitable also in this case. While in general case the recompression is non-deterministic it determinises in case of one variable and the obtained running time is O(n + #X logn), where #X is the number of occurrences of the variable in the equation. This matches the previously-best algorithm due to Dąbrowski and Plandowski. Then, using a couple of heuristics as well as more detailed time analysis, the running time is lowered to O(n) in the RAM model. Unfortunately, no new properties of solutions are shown. 1.

We present an application of a local recompression technique, previously developed by the author in the context of compressed membership problems and compressed pattern matching, to word equations. The technique is based on local modification of variables (replacing X by aX or Xa) and replacement of pairs of letters appearing in the equation by a ‘fresh ’ letter, which can be seen as a bottom-up compression of the solution of the given word equation, to be more specific, building an SLP (Straight-Line Programme) for the solution of the word equation. Using this technique we give new self-contained proofs of many known results for word equations: the presented nondeterministic algorithm runs in O(n log n) space and in time polynomial in log N and n, where N is the size of the length-minimal solution of the word equation. It can be easily generalised to a generator of all solutions of the word equation. A further analysis of the algorithm yields a doubly exponential upper bound on the size of the length-minimal solution. The presented algorithm does not use exponential bound on the exponent of periodicity. Conversely, the analysis of the algorithm yields a new proof of the exponential bound on exponent of periodicity. For O(1) variables with arbitrary many appearances it works in linear space.

In this paper, a compressed membership problem for finite automata, both deterministic (DFAs) and non-deterministic (NFAs), with compressed transition labels is studied. The compression is represented by straight-line programs (SLPs), i.e. context-free grammars generating exactly one string. A novel technique of dealing with SLPs is introduced: the SLPs are recompressed, so that substrings of the input text are encoded in SLPs labelling the transitions of the NFA (DFA) in the same way, as in the SLP representing the input text. To this end, the SLPs are locally decompressed and then recompressed in a uniform way. Furthermore, in order to reflect the recompression in the NFA, we need to modify it only a little, in particular its size stays polynomial in the input size. Using this technique it is shown that the compressed membership for NFA with compressed labels is in NP, thus confirming the conjecture of Plandowski and Rytter [21] and extending the partial result of Lohrey and Mathissen [14]; as this problem is known to be NP-hard, we settle its exact computational complexity. Moreover, the same technique applied to the compressed membership for DFA with compressed labels yields that this problem is in P, and this problem is known to be P-hard.

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