The k-distinct language: Parameterized automata constructions

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(Σ) ⊆ Σ^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(Σ). 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 = |Σ|.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.