H. Mannila and K.-J. Raiha. On the complexity of dependency inference. Discrete Applied Mathematics, 40:237 -- 243, 1992.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....be thousands or millions. The number of attributes p is typically somewhere from 5 to 50. However, for some data mining applications, p could easily be 1000. While the problem of finding the keys of a relation is simple to state, its algorithmic properties turn out to be surprising complex. See [33, 34] for a variety of results, and Section 8 for theoretically intriguing open problems. The algorithm for finding P I(d; p) in Section 3 can straightforwardly be applied to finding the keys of a relation. The patterns are sets of attributes. A pattern X R is frequent, if X is a superkey, and the ....

H. Mannila and K.-J. Raiha. On the complexity of dependency inference. Discrete Applied Mathematics, 40:237 -- 243, 1992.

....for computing Th(L; r; q) that accesses the data using only Isinteresting queries must use at least jBd(Th(L; r; q) j queries. 2 This result, simple as it seems, gives as a corollary a result about finding functional dependencies that in the more specific setting is not easy to find [ 19; 20 ] For simplicity, we present the result here for the case of finding keys of a relation. Given a relation r over schema R, a key of r is a subset X of R such that no two rows agree on X. Note that a superset of a key is always a key, and that X Y if and only Y X. Corollary 11 ( 20 ] ....

....[ 19; 20 ] For simplicity, we present the result here for the case of finding keys of a relation. Given a relation r over schema R, a key of r is a subset X of R such that no two rows agree on X. Note that a superset of a key is always a key, and that X Y if and only Y X. Corollary 11 ( 20 ] Given a relation r over schema R, finding the minimal keys that hold in r requires at least MAX(r) evaluations of the predicate Is X a key , where MAX(r) is the set of all maximal subsets (w.r.t. set inclusion) of R that do not contain a key. 2 The drawback of Theorem 10 is that the size ....

H. Mannila and K.-J. Raiha. On the complexity of dependency inference. Discrete Applied Mathematics, 40:237 -- 243, 1992.

....dependency is an expression X B, where X R and B 2 R. Such a dependency is true in the relation r, if for all pairs of rows t; u 2 r we have: if t and u have the same value for all attributes in X, they have the same value for B. For various algorithms for nding such dependencies, see [3, 24, 25, 26, 32]. Functional dependencies with a xed right hand side B can be found using the levelwise algorithm by considering the set of sentences fX j X Rg; and the selection predicate q: q(r; X) is true if and only if X B holds in r. The specialization relation is then the reverse of set inclusion: ....

....queries must use at least jBd(Th(L; r; q) j queries. We prove later that jBd(Th(L; r; q) j queries are necessary already for the veri cation of the result. This result gives as a corollary a result about nding functional dependencies that in the more speci c setting was not easy to nd [24, 25]. For simplicity, we present the result here for the case of nding keys of a relation. Given a relation r over schema R, a key of r is a subset X of R such that no two rows agree on X. Note that a superset of a key is always a key, and that X Y if and only Y X. Corollary 15 ( 25] Given a ....

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H. Mannila and K.-J. R#ih#. On the complexity of dependency inference. Discrete Applied Mathematics, 40:237 243, 1992.

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