## RAIRO - Theoretical Informatics and Applications

### Computing the jth solution of a first-order query

a1 GREYC, Université de Caen, ENSICAEN, CNRS, Campus 2, 14032 Caen Cedex, France; gbagan@info.unicaen.fr; grandjean@info.unicaen.fr

a2 Équipe de Logique Mathématique, Université Denis Diderot, CNRS UMR 7056, 2 place Jussieu, 75251 Paris Cedex 05, France; durand@logique.jussieu.fr

a3 LIF, Université Aix-Marseille 1, CNRS, 39 rue Joliot Curie, 13453 Marseille Cedex 13, France; frederic.olive@lif.univ-mrs.fr

Abstract

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 ⊂ I x O whose instances x ∈ I may admit of several solutions R(x) = {y ∈ O : (x,y) ∈ R}. One associates to such a problem several specific tasks: compute a random (for the uniform probability distribution) solution y ∈ R(x); enumerate without repetition each solution y j in some specific linear order y0 < y1 < ... < yn-1 where R(x) = {y0,...,yn-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 and any structure S of bounded degree: (1) compute a random element of ; (2) compute the jth element of for some linear order of ; (3) enumerate the elements of in lexicographical order. More precisely, we prove that, for any fixed formula φ, the above problem (1) (resp. (2), (3)) can be computed within average constant time (resp. within constant time, with constant delay) after a linear (O(|S|)) precomputation. Our essential tool for deriving those complexity results is a normalization procedure of first-order formulas on bijective structures.

(Online publication January 18 2008)

Key Words:

• Complexity of enumeration;
• first-order queries;
• structures of bounded degree;
• linear time;
• constant time;
• constant delay

Mathematics Subject Classification:

• 68Q15;
• 68Q19