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The \(k\)-Distinct Language: Parameterized Automata Constructions

  • Ran Ben-Basat
  • Ariel Gabizon
  • Meirav ZehaviEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8894)

Abstract

In this paper, we pioneer a study of parameterized automata constructions for languages relevant to the design of parameterized algorithms. We focus on the \(k\) -Distinct language \(L_k(\varSigma )\subseteq \varSigma ^k\), defined as the set of words of length \(k\) whose symbols are all distinct. This language is implicitly related to several breakthrough techniques, developed during the last two decades, to design parameterized algorithms for fundamental problems such as \(k\) -Path and \(r\) -Dimensional \(k\) -Matching. Building upon the well-known color coding, divide-and-color and narrow sieves techniques, we obtain the following automata constructions for \(L_k(\varSigma )\). We develop non-deterministic automata (NFAs) of sizes \(4^{k+o(k)}\!\cdot \! n^{O(1)}\) and \((2e)^{k+o(k)}\!\cdot \! n^{O(1)}\), where the latter satisfies a ‘bounded ambiguity’ property relevant to approximate counting, as well as a non-deterministic xor automaton (NXA) of size \(2^k\!\cdot \! n^{O(1)}\), where \(n=|\varSigma |\). We show that our constructions lead to a unified approach for the design of both deterministic and randomized algorithms for parameterized problems, considering also their approximate counting variants. To demonstrate our approach, we consider the \(k\) -Path, \(r\) -Dimensional \(k\) -Matching and Module Motif problems.

Notes

Acknowledgement

We thank Hasan Abasi, Nader Bshouty, Michael Forbes and Amir Shpilka for helpful conversations.

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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Department of Computer ScienceTechnionHaifaIsrael

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