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Compressed membership in automata with compressed labels
 CSR, volume 6651 of LNCS
, 2011
"... Abstract. The algorithmic problem of whether a compressed string is accepted by a (nondeterministic) finite state automaton with compressed transition labels is investigated. For string compression, straightline programs (SLPs), i.e., contextfree grammars that generate exactly one string, are used ..."
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Abstract. The algorithmic problem of whether a compressed string is accepted by a (nondeterministic) finite state automaton with compressed transition labels is investigated. For string compression, straightline programs (SLPs), i.e., contextfree grammars that generate exactly one string, are used. Two algorithms for this problem are presented. The first one works in polynomial time, if all transition labels are nonperiodic strings (or more generally, the word length divided by the period is bounded polynomially in the input size). This answers a question of Plandowski and Rytter. The second (nondeterministic) algorithm is an NPalgorithm under the assumption that for each transition label the period is bounded polynomially in the input size. This generalizes the NP upper bound for the case of a unary alphabet, shown by Plandowski and Rytter. 1
Efficient algorithms for highly compressed data: The Word Problem in Higman’s group is in P
"... Power circuits are data structures which support efficient algorithms for highly compressed integers. Using this new data structure it has been shown recently by Myasnikov, Ushakov and Won that the Word Problem of the onerelator Baumslag group is is decidable in polynomial time. Before that the bes ..."
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Power circuits are data structures which support efficient algorithms for highly compressed integers. Using this new data structure it has been shown recently by Myasnikov, Ushakov and Won that the Word Problem of the onerelator Baumslag group is is decidable in polynomial time. Before that the best known upper bound was nonelementary. In the present paper we provide new results for power circuits and we give new applications in algorithmic group theory: 1. We define a modified reduction procedure on power circuits which runs in quadratic time thereby improving the known cubic time complexity. 2. We improve the complexity of the Word Problem for the Baumslag group to cubic time thereby providing the first practical algorithm for that problem. 3. The Word Problem of Higman’s group is decidable in polynomial time. It is due to the last result that we were forced to advance the theory of power circuits.
Compressed word problems in HNNextensions and amalgamated products
, 811
"... Abstract. It is shown that the compressed word problem for an HNNextension 〈H,t  t −1 at = ϕ(a)(a ∈ A) 〉 with A finite is polynomial time Turingreducible to the compressed word problem for the base group H. An analogous result for amalgamated free products is shown as well. 1 ..."
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Abstract. It is shown that the compressed word problem for an HNNextension 〈H,t  t −1 at = ϕ(a)(a ∈ A) 〉 with A finite is polynomial time Turingreducible to the compressed word problem for the base group H. An analogous result for amalgamated free products is shown as well. 1
Compressed Membership for NFA (DFA) with Compressed Labels is in NP (P) ∗
"... In this paper, a compressed membership problem for finite automata, both deterministic (DFAs) and nondeterministic (NFAs), with compressed transition labels is studied. The compression is represented by straightline programs (SLPs), i.e. contextfree grammars generating exactly one string. A novel ..."
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In this paper, a compressed membership problem for finite automata, both deterministic (DFAs) and nondeterministic (NFAs), with compressed transition labels is studied. The compression is represented by straightline programs (SLPs), i.e. contextfree 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 NPhard, 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 Phard.
Compressed Conjugacy and the Word Problem for Outer Automorphism Groups of Graph Groups
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Compressed Labels is in NP (P) ∗
, 2012
"... HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte p ..."
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HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et a ̀ la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. Compressed Membership for NFA (DFA) with