Computing the jth solution of a first-order query
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 RI×O whose instances xI may admit of several solutions R(x)={yO:(x,y)R}. One associates to such a problem several specific tasks: compute a random (for the uniform probability distribution) solution yR(x); enumerate without repetition each solution y j in some specific linear order y 0 <y 1 <...<y n-1 where R(x)={y 0 ,,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 ϕ(v ¯) and any structure S of bounded degree: (1) compute a random element of ϕ(S)={a ¯:(S,a ¯)ϕ(v ¯)}; (2) compute the jth element of ϕ(S) for some linear order of ϕ(S); (3) enumerate the elements of ϕ(S) 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.

DOI : 10.1051/ita:2007046
Classification : 68Q15, 68Q19
Mots clés : complexity of enumeration, first-order queries, structures of bounded degree, linear time, constant time, constant delay
Bagan, Guillaume  ; Durand, Arnaud  ; Grandjean, Etienne  ; Olive, Frédéric 1

1 LIF, Université Aix-Marseille 1, CNRS, 39 rue Joliot Curie, 13453 Marseille Cedex 13, France;
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Bagan, Guillaume; Durand, Arnaud; Grandjean, Etienne; Olive, Frédéric. Computing the jth solution of a first-order query. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 42 (2008) no. 1, pp. 147-164. doi : 10.1051/ita:2007046. http://www.numdam.org/articles/10.1051/ita:2007046/

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