Thuan, Ho

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Ho, Thuan

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Some remarks on the algorithm of Lucchesi and Osborn

Thuan Ho

University of Szeged

Some remarks on the algorithm of Lucchesi and Osborn H o THUAN Let S=(Q, F) be a relation scheme, where Q—{A1,A2, ..., A„} is the uni-verse of attributes and F = {Li - Rt\Li, Rt g Q, i = 1, 2,..., m} is the set of functional dependencies. In [2] C. L. Lucchesi and S. L. Osborn provided a very interesting algorithm to determine the set of all keys for any relation scheme S= (Q, F>. Following our notation, the algorithm has time complexity 0 ( | F | № | | i 3 | ( № | + |i3|)), i.e. its running time is bounded by a polynomial of |i2|, |F | and \Jfs\, where |F | is the cardinality of F, and is the set of all keys for S. We reproduce here this algorithm with some modifications in accordance with our notation. Algorithm OL1. Set of all keys for S=<i2,F>; Comment. is the set of keys being accumulated in a sequence which can be scanned in the order in which the keys are entered; 3STs*- {Key (Q, F, £>)};1 for each K in do for each FD(Lt Ri) in F do T - LiUif^R,)-test -«- true; for each J in J f s do 1 Let Key (Q, F, X) be the algorithm Minimal Key in [2], which determines a key for S that is a subset of a specified superkey X. 192 H. Thuan if T includes J then test — false; if test then J f s - j f s U{Key (£2, F, T)} end end; return The following simple remarks, in some cases can be used to improve the per-formance of the algorithm of Lucchesi and Osborn. m m Remark 1. Let L, R, H be defined as: R= (J L= U U H= (J K}. ¡=1 i=l To find the first key for S = (£2, F) , instead of £2 it is better to use the superkey (i2\R)U(LDR) and Algorithm 1 in [1], and instead of the algorithm Key (£2, F, T) it is better to use Algorithm 2 in [1] for finding one key for S included in a given superkey T. Remark 2. In [1] it is shown that R\L Q Q\H, i.e. R\L consists only of non-prime attributes. Therefore, if Rt ^ R\L then Rif]K=0, \/K£jfs, and LtUiKXRJ^K. That means, when computing T=LtU U ( K \ R i ) , we can neglect all FDs (L^Ri) with R¡QR\L for every K e X s . Let us denote F = F\{LJ - Rj\Lj - RJ£ F and RJ Q R\L}. Remark 3. With a fixed A: in J f s , it is clear that if ^n¿? ¡=0 then L¡{J(K\R¡)^K. In that case, it is not necessary to continue to check whether T includes J for each J in So, it is better to compute T by the following order T = (tfVyUA. Remark 4. The algorithm of Lucchesi and Osborn is particularly effective when the number of keys for S= (Q, F) is small. But on what basis can we conclude that the number of keys for S is small? There is no general answer for all cases, and it is shown in [3] that the number of keys for a relation scheme (£2, F ) can be factorial in |F | or exponential in |Í2|, and that both of these upper bounds are attainable. However, it is shown in ([1], Corollary 1) that №1 ^ c r where h is the cardinality of Lf]R. Thus, if LOR has a few elements only, then it is a good criterion for saying that S has a small number of keys. In the case LC\R=0, Q\R is the unique key for S= (fí, F) as pointed out in ([1], Corol-lary 4). Example. We take up the example in [2], Appendix 1): Q= {a, b, c,d, e,f,g,h}, F — {a — b, c d, e - / , g - h}. Some remarks on the algorithm of Lucchesi and Osborn 193 It is clear that for this relation scheme LOR = 0, and it has exactly one key, namely Q\R—aceg. Taking Remarks 1—3 into account, the algorithm of Lucchesi and Osborn now can be presented as follows : Algorithm OL2. Set of all keys for S=(Q, F>; Xs - {Algo. 1 (Q, F, (Q\R) U (£ D R))} ;2 for each K in do for each FD(Lf - Rt) in F such that K\Rt ^¿K do T - (KXRdULr, test -«- true; for each J in do if T includes / then test false; if test then - j f s U {Algo. 2(Q, F, T)} end end; return Remark 5. The time complexity of Algorithm OL2 is 0(|Jfs||i2|(|Jfs||F|-HF||Ln/î|)). Abstract In [1] we have proposed two algorithms (Algorithm 1 and Algorithm 2) for finding one key of the relation scheme S=(Q, F) included in a given superkey. In this paper, we show that, using these algorithms and some simple remarks, the performance of the algorithm of Lucchesi and Osborn [2], in general, can be improved. References [1] Ho THUAN and LE VAN BAO, Some results about keys of relational schemes, Acta Cybernetica Tom. 7, Fasc. 1, Szeged, 1985, 99—113. [2] LUCCHESI, C. L. and OSBORN, S. L., Candidate keys for relations, J. of Computer and System S c i e n c e s , 1 7 , 1 9 7 8 , 2 7 0 — 2 7 9 . [3] OSBORN, S. L., Normal forms for relational databases, Ph. D. Dissertation, University of Water-loo, 1977. (Received Oct. 27, 1986) 2 Algo. 1 and Algo. 2 refer to Algorithm 1 and Algorithm 2 in [1] respectively. 

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Some remarks on the algorithm of Lucchesi and Osborn

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