MSO queries on tree decomposable structures are computable with linear delay

Linear-Delay_lin is the class of enumeration problems computable in two steps: the first step is a precomputation in linear time in the size of the input and the second step computes successively all the solutions with a delay between two consecutive solutions y₁ and y₂ that is linear in |y₂|. We prove that evaluating a fixed monadic second order (MSO) query $\varphi(\bar{X})$ (i.e. computing all the tuples that satisfy the MSO formula) in a binary tree is a Linear-Delay_lin problem. More precisely, we show that given a binary tree T and a tree automaton Γ representing an MSO query $\varphi(\bar{X})$, we can evaluate Γ on T with a preprocessing in time and space complexity O(|Γ|³|T|) and an enumeration phase with a delay O(|S|) and space O(max|S|) where |S| is the size of the next solution and max|S| is the size of the largest solution. We introduce a new kind of algorithm with nice complexity properties for some algebraic operations on enumeration problems. In addition, we extend the precomputation (with the same complexity) such that the i^th (with respect to a certain order) solution S is produced directly in time O(|S|log(|T|)). Finally, we generalize these results to bounded treewidth structures.