Computing the jth solution of a first-order query

Bagan, Guillaume; Durand, Arnaud; Grandjean, Etienne; Olive, Frédéric

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 42 (2008) no. 1, pp. 147-164.

We design algorithms of “optimal” data complexity for several natural problems about first-order queries on structures of bounded degree. For that purpose, we first introduce a framework to deal with logical or combinatorial problems $R\subset I\times O$ whose instances $x\in I$ may admit of several solutions $R\left(x\right)=\{y\in O:(x,y)\in R\}$. One associates to such a problem several specific tasks: compute a random (for the uniform probability distribution) solution $y\in R\left(x\right)$; enumerate without repetition each solution $y_j$ in some specific linear order $y_0<y_1<...<y_{n-1}$ where $R\left(x\right)=\{y_0,\cdots ,y_{n-1}\}$; compute the solution $y_j$ of rank $j$ in the linear order $<$. Algorithms of “minimal” data complexity are presented for the following problems: given any first-order formula $\varphi \left(\overline{v}\right)$ and any structure $S$ of bounded degree: $\left(1\right)$ compute a random element of $\varphi \left(S\right)=\{\overline{a}:(S,\overline{a})\vDash \varphi \left(\overline{v}\right)\}$; $\left(2\right)$ compute the $j$th element of $\varphi \left(S\right)$ for some linear order of $\varphi \left(S\right)$; $\left(3\right)$ enumerate the elements of $\varphi \left(S\right)$ in lexicographical order. More precisely, we prove that, for any fixed formula $\varphi$, the above problem $\left(1\right)$ (resp. $\left(2\right)$, $\left(3\right)$) can be computed within average constant time (resp. within constant time, with constant delay) after a linear $\left(O\right(\left|S\right|\left)\right)$ precomputation. Our essential tool for deriving those complexity results is a normalization procedure of first-order formulas on bijective structures.

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