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New algorithms, such as Dualminer, apply both types of constraints at the same time (Bucila, Gehrke, Kifer, & White, 2002). It considers these two types of constraints in a double process, one mirroring the other for each type of constraint, hence its name. However, monotone and antimonotone constraints do not necessarily apply in duality. Especially when considering the mining process as a set of distinct phases, such as the building of structures to compress the data and the mining of these structures, the application of these constraints differ by type.

In VLDB, (pp. 323-333). Fayyad, U. M. (1991). On the induction of decision trees for multiple concept learning. Unpublished doctoral thesis, University of Michigan. Fayyad, U. , & Irani, K. (1993). Multi-interval discretization of continuous-valued attributes for classification learning. In Proceedings of the 12th International Joint Conference of Artificial intelligence, (pp. 1022-1027). Fisher, D. (1987). Knowledge acquisition via incremental conceptual clustering. Machine Learning, 2, 139-172.

Ql } replace every value in Qi by i for 0 ≤ i ≤ l. The while loop runs for as long as candidate boundary points exist, and it is possible to find a new cut point p such that the distance d ( A | BP* ) is less than the previous distance d ( A | BP* ). An experiment performed on a synthetic database shows that a substantial amount of time (about 78% of the total time) is spent on decreasing the distance by the last 1%. 01d . ,Pn} and the partition of obP jects determined by the cut points B* let p ∈ BP be a new cut point added to the set P.

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