# Computing the jth solution of a first-order query

Guillaume Bagan; Arnaud Durand; Etienne Grandjean; Frédéric Olive

RAIRO - Theoretical Informatics and Applications (2008)

- Volume: 42, Issue: 1, page 147-164
- ISSN: 0988-3754

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topBagan, Guillaume, et al. "Computing the jth solution of a first-order query ." RAIRO - Theoretical Informatics and Applications 42.1 (2008): 147-164. <http://eudml.org/doc/250274>.

@article{Bagan2008,

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 yj in some specific linear order y0 < y1 < ... < yn-1 where R(x) = \{y0,...,yn-1\}; compute the solution yj of rankj in the linear order <.
Algorithms of “minimal" data complexity are presented for the following problems: given any first-order formula $\varphi(\bar\{v\})$ and any structure S of bounded degree:
(1) compute a random element of $\varphi(S)=\\{\bar\{a\}: (S,\bar\{a\})\models\varphi(\bar\{v\})\\}$;
(2) compute the jth element of $\varphi(S)$ for some linear order of $\varphi(S)$;
(3) enumerate the elements of $\varphi(S)$ 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.
},

author = {Bagan, Guillaume, Durand, Arnaud, Grandjean, Etienne, Olive, Frédéric},

journal = {RAIRO - Theoretical Informatics and Applications},

keywords = {Complexity of enumeration; first-order queries; structures of bounded degree; linear time; constant time; constant delay},

language = {eng},

month = {1},

number = {1},

pages = {147-164},

publisher = {EDP Sciences},

title = {Computing the jth solution of a first-order query },

url = {http://eudml.org/doc/250274},

volume = {42},

year = {2008},

}

TY - JOUR

AU - Bagan, Guillaume

AU - Durand, Arnaud

AU - Grandjean, Etienne

AU - Olive, Frédéric

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

JO - RAIRO - Theoretical Informatics and Applications

DA - 2008/1//

PB - EDP Sciences

VL - 42

IS - 1

SP - 147

EP - 164

AB -
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 yj in some specific linear order y0 < y1 < ... < yn-1 where R(x) = {y0,...,yn-1}; compute the solution yj of rankj in the linear order <.
Algorithms of “minimal" data complexity are presented for the following problems: given any first-order formula $\varphi(\bar{v})$ and any structure S of bounded degree:
(1) compute a random element of $\varphi(S)=\{\bar{a}: (S,\bar{a})\models\varphi(\bar{v})\}$;
(2) compute the jth element of $\varphi(S)$ for some linear order of $\varphi(S)$;
(3) enumerate the elements of $\varphi(S)$ 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.

LA - eng

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

UR - http://eudml.org/doc/250274

ER -

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