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.
The research leading to these results has received funding from the European Community’s Seventh Framework Programme (FP7/2007-2013) under grant agreement number 257575.
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Notes
- 1.
We consider vertex weights, rather than color similarity scores (see [41]), for the sake of clarity. Our algorithms can be easily modified to handle color similarity scores.
- 2.
We slightly improve the previous randomized algorithm that approximately counts only 3-dimensional \(k\)-matchings, and deterministic algorithm for Module Motif.
- 3.
We note that determining the minimal NFA size, even for a finite regular language, is computationally hard [20].
- 4.
If while reading a word we reach a state where we cannot progress by reading the next symbol, this run is rejected and the state we are at is not added to \(M(w)\).
- 5.
We assume NXAs do not contain directed cycles of only \(\epsilon \)-transitions, as their XOR-language is not properly defined in this case.
- 6.
Usually the Hankel matrix is defined as an infinite matrix, and we are defining a submatrix of it; for a finite language \(L\subseteq [n]^k\), that the rank of this submatrix is clearly the same as the rank of the entire matrix, and we are only concerned with its rank.
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Acknowledgement
We thank Hasan Abasi, Nader Bshouty, Michael Forbes and Amir Shpilka for helpful conversations.
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Ben-Basat, R., Gabizon, A., Zehavi, M. (2014). The \(k\)-Distinct Language: Parameterized Automata Constructions. In: Cygan, M., Heggernes, P. (eds) Parameterized and Exact Computation. IPEC 2014. Lecture Notes in Computer Science(), vol 8894. Springer, Cham. https://doi.org/10.1007/978-3-319-13524-3_8
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