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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 × 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 yj in some specific linear order y0 < y1 < · · · < yn−1 where R(x) = {y0,..., yn−1}; compute the solution yj 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 j th 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 problems 1 and 2 (resp. problem 3) can be computed within constant time (resp. 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

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1 GREYC, Université de Caen, ENSICAEN, CNRS, Campus 2, F-14032 Caen cedex, France

Year: 1999

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oai:CiteSeerX.psu:10.1.1.186.7112

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