Abstract
In this paper, a compressed membership problem for finite automata, both deterministic and non-deterministic, 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, such recompression induces only small changes in the automaton, in particular, the size of the automaton remains polynomial. 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 and extending the partial result of Lohrey and Mathissen; as it is already known, that this problem is 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; for this problem, only trivial upper-bound PSPACE was known.
Highlights
1.1 Compression and Straight-Line ProgrammesDue to ever-increasing amount of data, compression methods are widely applied in order to decrease the data’s size
The earlier work focused on the properties of strings described by straight-line programs (SLPs) and tried to use this knowledge in order to analyse the automaton and its input, hopefully this should result in an efficient algorithm
We focus on the SLPs, and not on the encoded strings
Summary
Due to ever-increasing amount of data, compression methods are widely applied in order to decrease the data’s size. There is a large demand for algorithms working directly on the compressed representation of the data, without explicit decompression. A different approach is explored: for some applications and for most of theory-oriented considerations it would be useful to model the practical compression standard by a more mathematically well-founded method. This idea lay at the foundations of the notion of Straight-Line Programms (SLP), whose instances can be seen as context-free grammars generating exactly one string. The recent stateof-the-art efficient algorithms for pattern matching in LZ compressed text essentially use the reformulation of LZ methods in terms of SLPs [11]. For more information about the SLPs and their applications, please look at the recent survey of Lohrey [29]
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