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Documents authored by Luttenberger, Michael

Document
DOI: 10.4230/LIPIcs.STACS.2018.48
Computing the Longest Common Prefix of a Context-free Language in Polynomial Time

Authors: Michael Luttenberger, Raphaela Palenta, and Helmut Seidl

Published in: LIPIcs, Volume 96, 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)

Abstract
We present two structural results concerning the longest common prefixes of non-empty languages. First, we show that the longest common prefix of the language generated by a context-free grammar of size N equals the longest common prefix of the same grammar where the heights of the derivation trees are bounded by 4N. Second, we show that each non-empty language L has a representative subset of at most three elements which behaves like L w.r.t. the longest common prefix as well as w.r.t. longest common prefixes of L after unions or concatenations with arbitrary other languages. From that, we conclude that the longest common prefix, and thus the longest common suffix, of a context-free language can be computed in polynomial time.
Cite as

Michael Luttenberger, Raphaela Palenta, and Helmut Seidl. Computing the Longest Common Prefix of a Context-free Language in Polynomial Time. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 48:1-48:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{luttenberger_et_al:LIPIcs.STACS.2018.48,
  author =	{Luttenberger, Michael and Palenta, Raphaela and Seidl, Helmut},
  title =	{{Computing the Longest Common Prefix of a Context-free Language in Polynomial Time}},
  booktitle =	{35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)},
  pages =	{48:1--48:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-062-0},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{96},
  editor =	{Niedermeier, Rolf and Vall\'{e}e, Brigitte},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2018.48},
  URN =		{urn:nbn:de:0030-drops-84828},
  doi =		{10.4230/LIPIcs.STACS.2018.48},
  annote =	{Keywords: longest common prefix, context-free languages, combinatorics on words}
}
Document
DOI: 10.4230/LIPIcs.STACS.2008.1351
Convergence Thresholds of Newton's Method for Monotone Polynomial Equations

Authors: Javier Esparza, Stefan Kiefer, and Michael Luttenberger

Published in: LIPIcs, Volume 1, 25th International Symposium on Theoretical Aspects of Computer Science (2008)

Abstract
Monotone systems of polynomial equations (MSPEs) are systems of fixed-point equations $X_1 = f_1(X_1, ldots, X_n),$ $ldots, X_n = f_n(X_1, ldots, X_n)$ where each $f_i$ is a polynomial with positive real coefficients. The question of computing the least non-negative solution of a given MSPE $vec X = vec f(vec X)$ arises naturally in the analysis of stochastic models such as stochastic context-free grammars, probabilistic pushdown automata, and back-button processes. Etessami and Yannakakis have recently adapted Newton's iterative method to MSPEs. In a previous paper we have proved the existence of a threshold $k_{vec f}$ for strongly connected MSPEs, such that after $k_{vec f}$ iterations of Newton's method each new iteration computes at least 1 new bit of the solution. However, the proof was purely existential. In this paper we give an upper bound for $k_{vec f}$ as a function of the minimal component of the least fixed-point $muvec f$ of $vec f(vec X)$. Using this result we show that $k_{vec f}$ is at most single exponential resp. linear for strongly connected MSPEs derived from probabilistic pushdown automata resp. from back-button processes. Further, we prove the existence of a threshold for arbitrary MSPEs after which each new iteration computes at least $1/w2^h$ new bits of the solution, where $w$ and $h$ are the width and height of the DAG of strongly connected components.
Cite as

Javier Esparza, Stefan Kiefer, and Michael Luttenberger. Convergence Thresholds of Newton's Method for Monotone Polynomial Equations. In 25th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 1, pp. 289-300, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2008)

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@InProceedings{esparza_et_al:LIPIcs.STACS.2008.1351,
  author =	{Esparza, Javier and Kiefer, Stefan and Luttenberger, Michael},
  title =	{{Convergence Thresholds of Newton's Method for Monotone Polynomial Equations}},
  booktitle =	{25th International Symposium on Theoretical Aspects of Computer Science},
  pages =	{289--300},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-06-4},
  ISSN =	{1868-8969},
  year =	{2008},
  volume =	{1},
  editor =	{Albers, Susanne and Weil, Pascal},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2008.1351},
  URN =		{urn:nbn:de:0030-drops-13516},
  doi =		{10.4230/LIPIcs.STACS.2008.1351},
  annote =	{Keywords: Newton's Method, Fixed-Point Equations, Formal Verification of Software, Probabilistic Pushdown Systems}
}
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