The k -Distinct Language: Parameterized Automata Constructions

Ran Ben-Basat, Ariel Gabizon, Meirav Zehavi

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.