Abstract
We use an algebra of preference strict-orders to give a formal derivation of the standard Block-Nested Loop (BNL) algorithm for computing the best or maximal objects w.r.t. such an order. This derivation is presented in terms of antichains, i.e., sets of mutually incomparable objects. We define an approximation relation between antichains that reflects the steps taken by the BNL algorithm. This induces a semilattice and the operator computing the maximal objects of a subset can be viewed as a closure operator in an associated pre-ordered set and hence yields a characterisation of antichains in terms of a Galois connection.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Börzsönyi, S., Kossmann, D., Stocker, K.: The skyline operator. In: Proceedings of the 17th International Conference on Data Engineering, pp. 421–430 (2001)
Dehne, F.: o(n 1/2) algorithms for the maximal elements and ECDF searching problem on a mesh-connected parallel computer. Inf. Process. Lett. 22(6), 303–306 (1986)
Desharnais, J., Möller, B., Struth, G.: Kleene algebra with domain. ACM Trans. Comput. Logic 7(4), 798–833 (2006)
Desharnais, J., Möller, B., Struth, G.: Algebraic notions of termination. Logical Methods in Computer Science 7(1) (2011)
Erne, M., Koslowski, J., Melton, A., Strecker, G.: A primer on Galois connections. In: Proc. 1991 Summer Conference on General Topology and Applications in Honor of Mary Ellen Rudin and Her Work. Annals of the New York Academy of Sciences, vol. 704, pp. 103–125. New York Academy of Sciences (1993)
Kozen, D.: A completeness theorem for Kleene algebras and the algebra of regular events. Information and Computation 110(2), 366–390 (1994)
Möller, B., Roocks, P.: An algebra of database preferences. Journal of Logical and Algebraic Methods in Programming 84, 456–481 (2015)
Möller, B., Roocks, P., Endres, M.: An Algebraic Calculus of Database Preferences. In: Gibbons, J., Nogueira, P. (eds.) MPC 2012. LNCS, vol. 7342, pp. 241–262. Springer, Heidelberg (2012)
Partsch, H.: Specification and Transformation of Programs - A Formal Approach to Software Development. Texts and Monographs in Computer Science. Springer (1990)
Stojmenovic, I., Miyakawa, M.: An optimal parallel algorithm for solving the maximal elements problem in the plane. Parallel Computing 7(2), 249–251 (1988)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this paper
Cite this paper
Möller, B. (2015). Towards Antichain Algebra. In: Kahl, W., Winter, M., Oliveira, J. (eds) Relational and Algebraic Methods in Computer Science. RAMICS 2015. Lecture Notes in Computer Science(), vol 9348. Springer, Cham. https://doi.org/10.1007/978-3-319-24704-5_21
Download citation
DOI: https://doi.org/10.1007/978-3-319-24704-5_21
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-24703-8
Online ISBN: 978-3-319-24704-5
eBook Packages: Computer ScienceComputer Science (R0)