The K-Distinct Language: Parameterized Automata Constructions
PARAMETERIZED AND EXACT COMPUTATION, IPEC 2014(2014)
摘要
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(Sigma) subset of Sigma(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(Sigma). We develop non-deterministic automata (NFAs) of sizes 4(k+o(k)).n(O(1)) and (2e)(k+o(k)).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).n(O(1)), where n = vertical bar Sigma vertical bar. 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.
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关键词
Automaton Construction, Non-deterministic Finite Automata (NFA), Approximate Counting, Parameterized Algorithms, Narrow Sieve
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