Computing the jth solution of a first-order query
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Volume 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
Keywords: 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;
@article{ITA_2008__42_1_147_0,
     author = {Bagan, Guillaume and Durand, Arnaud and Grandjean, Etienne and Olive, Fr\'ed\'eric},
     title = {Computing the jth solution of a first-order query},
     journal = {RAIRO - Theoretical Informatics and Applications - Informatique Th\'eorique et Applications},
     pages = {147--164},
     publisher = {EDP-Sciences},
     volume = {42},
     number = {1},
     year = {2008},
     doi = {10.1051/ita:2007046},
     zbl = {1149.68028},
     mrnumber = {2382549},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/ita:2007046/}
}
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 - Informatique Théorique et Applications
PY  - 2008
DA  - 2008///
SP  - 147
EP  - 164
VL  - 42
IS  - 1
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/ita:2007046/
UR  - https://zbmath.org/?q=an%3A1149.68028
UR  - https://www.ams.org/mathscinet-getitem?mr=2382549
UR  - https://doi.org/10.1051/ita:2007046
DO  - 10.1051/ita:2007046
LA  - en
ID  - ITA_2008__42_1_147_0
ER  - 
%0 Journal Article
%A Bagan, Guillaume
%A Durand, Arnaud
%A Grandjean, Etienne
%A Olive, Frédéric
%T Computing the jth solution of a first-order query
%J RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications
%D 2008
%P 147-164
%V 42
%N 1
%I EDP-Sciences
%U https://doi.org/10.1051/ita:2007046
%R 10.1051/ita:2007046
%G en
%F ITA_2008__42_1_147_0
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, Volume 42 (2008) no. 1, pp. 147-164. doi : 10.1051/ita:2007046. http://www.numdam.org/articles/10.1051/ita:2007046/

[1] G. Bagan, Mso queries on tree decomposable structures are computable with linear delay, in Proc. 15th Annual Conference of the EACSL (CSL'06). Lect. Notes Comput. Sci. 4207 (2006) 167-181. | MR

[2] G. Bagan, A. Durand and E. Grandjean, On acyclic conjunctive queries and constant delay enumeration, in Proc. 16th Annual Conference of the EACSL (CSL'07). Lect. Notes Comput. Sci. (2007) 208-222.

[3] B. Courcelle, Linear delay enumeration and monadic second-order logic. Discrete Appl. Math. (2007) (to appear).

[4] A. Durand and E. Grandjean, First-order queries on structures of bounded degree are computable with constant delay. ACM T. Comput. Logic (2007) 1-18. | MR

[5] A. Durand and F. Olive, First-order queries over one unary function, in Proc. 15th Annual Conference of the EACSL (CSL'06). Lect. Notes Comput. Sci. 4207 (2006) 334-348. | MR

[6] J. Flum, M. Frick and M. Grohe, Query evaluation via tree decompositions. J. ACM 49 (2002) 716-752. | MR

[7] M. Frick and M. Grohe, Deciding first-order properties of locally tree decomposable structures. J. ACM 48 (2001) 1184-1206. | MR

[8] G. Gottlob, N. Leone and F. Scarcello, The complexity of acyclic conjunctive queries. J. ACM 48 (2001) 431-498. | MR

[9] S. Grigorieff, Décidabilité et complexité des théories logiques. Collection Didactique INRIA 8 (1991) 7-97. | MR

[10] E. Grandjean and T. Schwentick, Machine-independent characterizations and complete problems for deterministic linear time. SIAM J. Comput. 32 (2002) 196-230. | MR | Zbl

[11] L. Libkin, Elements of finite model theory. EATCS Series, Springer (2004). | MR | Zbl

[12] S. Lindell, A normal form for first-order logic over doubly-linked data structures. Int. J. Found. Comput. Sci. (2006) (to appear). | MR

[13] R. Motwani and P. Raghavan, Randomized algorithms. Cambridge University Press (1995). | MR | Zbl

[14] C. Papadimitriou and M. Yannakakis, On the complexity of database queries. J. Comput. Syst. Sci. 58 (1999) 407-427. | MR | Zbl

[15] M.Y. Vardi, On the complexity of bounded-variable queries. Proc. Principles of Databases Systems (PODS'95), ACM Press (1995) 266-276.

[16] D. Seese, Linear time computable problems and first-order descriptions. Math. Structures Comput. Sci. 6 (1996) 505-526. | MR | Zbl

[17] M. Yannakakis, Algorithms for acyclic database schemes. Proc. Very Large Data Bases Conference (VLDB'81) (1981) 82-94.

Cited by Sources: