The dense families of database relations were introduced by Järvinen [7]. The aim of this paper is to investigate some new properties of dense families of database relations, and their applications. That is, we characterize functional dependencies and minimal keys in terms of dense families. We give a necessary and sufficient condition for an abitrary family to be R— dense family. We prove that with a given relation R the equality set ER is an R—dense family whose size is at most m(m-1)/2, where m is the number of tuples in R. We also prove that the set of all minimal keys of relation R is the transversal hypergraph of the complement of the equality set ER. We give an effective algorithm finding all minimal keys of a given relation R. We also give an algorithm which from a given relation R finds a cover of functional dependencies that holds in R. The complexity of these algorithms is also esimated

Nguyen Hoang Son

Thi Vu Duc

English

University of Szeged

Acta Cybernetica 17 (2005) 173–182.Some Results Related to Dense Families ofDatabase RelationsVu Duc Thi∗ and Nguyen Hoang Son†AbstractThe dense families of database relations were introduced by Järvinen [7].The aim of this paper is to investigate some new properties of dense familiesof database relations, and their applications. That is, we characterize func-tional dependencies and minimal keys in terms of dense families. We give anecessary and sufficient condition for an abitrary family to be R− dense fam-ily. We prove that with a given relation R the equality set ER is an R−densefamily whose size is at most m(m−1)2, where m is the number of tuples in R.We also prove that the set of all minimal keys of relation R is the transversalhypergraph of the complement of the equality set ER. We give an effectivealgorithm finding all minimal keys of a given relation R. We aslo give an algo-rithm which from a given relation R finds a cover of functional dependenciesthat holds in R. The complexity of these algorithms is also esimated.1 Basic definitionsIn this section we present briefly the main concepts of the theory of relationaldatabases which will be needed in sequel. The concepts and facts given in thissection can be found in [1, 3, 4, 8, 9].Let U be a finite set of attributes (e.g. name, age etc). The elements of U willbe denoted by a, b, c, . . . , x, y, z, if an ordering on U is needed, by a1, . . . , an. Amap dom associates with each a ∈ U its domain dom(a). A relation R over U is asubset of Cartesian product∏a∈U dom(a).We can think of a relation R over U as being a set of tuples: R = {h1, . . . , hm},hi : U −→⋃a∈Udom(a), hi(a) ∈ dom(a), i = 1, 2, . . . , m.A functional dependency (FD for short) is a statement of form X → Y , whereX, Y ⊆ U . The FD X → Y holds in a relation R = {h1, . . . , hm} over U if(∀hi, hj ∈ R)((∀a ∈ X)(hi(a) = hj(a)) ⇒ (∀b ∈ Y )(hi(b) = hj(b))).∗Institute of Information Technology, Vietnamese Academy of Science and Technology, 18Hoang Quoc Viet, Hanoi, Vietnam.†Department of Mathematics, College of Sciences, Hue University, Vietnam.173brought to you by COREView metadata, citation and similar papers at core.ac.ukprovided by University of Szeged174 Vu Duc Thi and Nguyen Hoang SonWe also say that R satisfies the FD X → Y .Let FR be a family of all FDs that holds in R. Then F = FR satisfies(F1) X → X ∈ F,(F2) (X → Y ∈ F, Y → Z ∈ F ) ⇒ (X → Z ∈ F ),(F3) (X → Y ∈ F, X ⊆ V, W ⊆ Y ) ⇒ (V → W ∈ F ),(F4) (X → Y ∈ F, V → W ∈ F ) ⇒ (X ∪ V → Y ∪ W ∈ F ).A family of FDs satisfying (F1) - (F4) is called an f − family over U .Clearly, FR is an f -family over U . It is known [1] that if F is an arbitraryf -family, then there is a relation R over U such that FR = F .Given a family F of FDs over U , there exists a unique minimal f -family F+that contains F . It can be seen that F+ contains all FDs which can be derivedfrom F by the rules (F1) - (F4).A relation scheme s is a pair (U, F ), where U is a set of attributes and F is aset of FDs over U .Let U be a nonempty finite set and P(U) its power set. The mapping L :P(U) −→ P(U) is called a closure operation over U if it satisfies the followingconditions:(1) X ⊆ L(X),(2) X ⊆ Y implies L(X) ⊆ L(Y ),(3) L(L(X)) = L(X).Remark 1.1. It is clear that, if F is an f− family, and we define LF (X) asLF (X) = {a ∈ U : X → {a} ∈ F}then LF is a closure operation over U . Conversely, it is known [1, 3] that if L is aclosure operation, then there is exactly one f− family F over U so that L = LF ,whereF = {X → Y : X, Y ⊆ U, Y ⊆ L(X)}.Thus, there is a one-to-one correspondence between closure operations and f−families over U .Let R be a relation over U and K ⊆ U . Then K is a key of R if K → U ∈ FR.K is a minimal key of R if K is a key of R and any proper subset of K is not a keyof R.Denote KR the set of all minimal keys of R.Let I ⊆ P(U), U ∈ I, and A, B ∈ I ⇒ A∩B ∈ I. I is called a meet-semilatticeover U . Let M ⊆ P(U). Denote M+ = {∩M ′ : M ′ ⊆ M}. We say that M is agenerator of I if M+ = I. Note that U ∈ M+ but not in M , by convention it isthe intersection of the empty collection of sets.Denote N = {A ∈ I : A 	= ∩{A′ ∈ I : A ⊂ A′}}. It can be seen that N is theunique minimal generator of I.Some Results Related to Dense Families of Database Relations 1752 Hypergraphs and TransversalsLet U be a nonempty finite set and put P(U) for the family of all subsets of U .The family H = {Ei : Ei ∈ P(U), i = 1, 2, . . . , m} is called a hypergraph over U ifEi 	= ∅ holds for all i (in [2] it is required that the union of Eis is U , in this paperwe do not require this).The elements of U are called vertices, and the sets E1, . . . , Em the edges of thehypergraph H.A hypergraph H is called simple if it satisfies ∀Ei, Ej ∈ H : Ei ⊆ Ej ⇒ Ei = Ej .It can be seen that KR is a simple hypergraph.Let H be a hypergraph over U . Then min(H) denotes the set of minimal edgesof H with respect to set inclusion, i.e., min(H) = {Ei ∈ H :	 ∃Ej ∈ H : Ej ⊂ Ei},and max(H) denotes the set of maximal edges of H with respect to set inclusion,i.e., max(H) = {Ei ∈ H :	 ∃Ej ∈ H : Ej ⊃ Ei}.It is clear that, min(H) and max(H) are simple hypergraphs. Furthermore,min(H) and max(H) are uniquely determined by H.A set T ⊆ U is called a transversal of H (sometimes it is called hitting set) if itmeets all edges of H, i.e., ∀E ∈ H : T ∩E 	= ∅. Denote by Trs(H) the family of alltransversals of H. A transversal T of H is called minimal if no proper subset T ′ ofT is a transversal.The family of all minimal transversals of H called the transversal hypergraphof H, and denoted by Tr(H). Clearly, Tr(H) is a simple hypergraph.Proposition 2.1 ([2]). Let H and G two simple hypergraphs over U . Then(1) H = Tr(G) if and only if G = Tr(H),(2) Tr(H) = Tr(G) if and only if H = G,(3) Tr(Tr(H)) = H.By the definition of minimal transversal, the following proposition is obviousProposition 2.2. Let H be a hypergraph over U . ThenTr(H) = Tr(min(H)).The following algorithm finds the family of all minimal transversals of a givenhypergraph (by induction).Algorithm 2.3 ([5]).Input: let H = {E1, . . . , Em} be a hypergraph over U .Output: Tr(H).Method:Step 0. We set L1 := {{a} : a ∈ E1}. It is obvious that L1 = Tr({E1}).176 Vu Duc Thi and Nguyen Hoang SonStep q+1. (q < m) Assume thatLq = Sq ∪ {B1, . . . , Btq},where Bi ∩ Eq+1 = ∅, i = 1, . . . , tq and Sq = {A ∈ Lq : A ∩ Eq+1 	= ∅}.For each i (i = 1, . . . , tq) constructs the set {Bi ∪{b} : b ∈ Eq+1}. Denote themby Ai1, . . . , Airi(i = 1, . . . , tq). LetLq+1 = Sq ∪ {Aip : A ∈ Sq ⇒ A 	⊂ Aip, 1 ≤ i ≤ tq, 1 ≤ p ≤ ri}.Theorem 2.4 ([5]). For every q(1 ≤ q ≤ m)Lq = Tr({E1, . . . , Eq}), i.e., Lm =Tr(H).It can be seen that the determination of Tr(H) based on our algorithm doesnot depend on the order of E1, . . . , Em.Remark 2.5. Denote Lq = Sq ∪ {B1, . . . , Btq}, and lq(1 ≤ q ≤ m − 1) be thenumber of elements of Lq. It can be seen that the worst-case time complexity ofour algorithm isO(|U |2m−1∑q=0tquq),where l0 = t0 = 1 anduq ={lq − tq, if lq > tq;1, if lq = tq.Clearly, in each step of our algorithm Lq is a simple hypergraph. It is knownthat the size of arbitrary simple hypergraph over U cannot be greater than C[n/2]n ,where n = |U |. C[n/2]n is asymptotically equal to 2n+1/2/(π.n)1/2. From this, theworst-case time complexity of our algorithm cannot be more than exponential in thenumber of attributes. In cases for which lq ≤ lm(q = 1, ..., m − 1), it is easy to seethat the time complexity of our algorithm is not greater than O(|U |2|H||Tr(H)|2).Thus, in these cases this algorithm finds Tr(H) in polynomial time in |U |, |H| and|Tr(H)|. Obviously, if the number of elements of H is small, then this algorithm isvery effective. It only requires polynomial time in |R|.The following proposition is obviousProposition 2.6 ([5]). The time complexity of finding Tr(H) of a given hypergraphH is (in general) exponential in the number of elements of U .Proposition 2.6 is still true for a simple hypergraph.Some Results Related to Dense Families of Database Relations 1773 Dense FamiliesLet D ⊆ P(U) be a family of subsets of a U . We define a set FD over D as followsFD = {X → Y : (∀A ∈ D)X ⊆ A ⇒ Y ⊆ A}.Proposition 3.1 ([7]). If D is a family of subsets of a finite set U , then FD is anf− family over U .The notion of dense family of a database relation is defined in [7], as follows:Let R be a relation over U . We say that a family D ⊆ P(U) of attribute sets isR − dense (or dense in R) if FR = FD.The following proposition guarantees the existence of at least one dense family.In the sequel we denote LFR simply by LR.Proposition 3.2 ([7]). The family LR is R−dense.Proposition 3.3 ([7]). If D is R−dense, then D ⊆ LR.Note that by Proposition 3.2 and Proposition 3.3, LR is the greatest R−densefamily.For any A ⊆ U , we denote by A the complement of A with respect to the setU , that is, A = {a ∈ U : a 	∈ A}.Theorem 3.4 ([7]). Let R be a relation over U . If D ⊆ P(U) is R−dense, thenthe following conditions hold(1) K is a key of R if and only if it contains an element from each set in{A : A ∈ D, A 	= U}.(2) K is a minimal key of R if and only if it minimal with respect to the propertyof containing an element from each set in {A : A ∈ D, A 	= U}.Let U be a finite set and P(U) its power set. For every family D ⊆ P(U), thecomplement family of D is the family D = {A : A ∈ D} over U .Let R = {h1, . . . , hm} be a relation over U , and ER the equality set of R, i.e.,ER = {Eij : 1 ≤ i < j ≤ m}where Eij = {a ∈ U : hi(a) = hj(a)}.Proposition 3.5. The equality set ER is R−dense.Proof. Assume that X → Y ∈ FR. Let Eij ∈ ER such that X ⊆ Eij . This meansthat hi(X) = hj(X). From this, and according to the definition of FDs, we havehi(Y ) = hj(Y ). Thus, Y ⊆ Eij . By the definition of FER , that is,FER = {X → Y : (∀Eij ∈ ER)X ⊆ Eij ⇒ Y ⊆ Eij},we obtain X → Y ∈ FER .Conversely, let X → Y ∈ FER . Suppose that there are hi, hj ∈ R such thathi(X) = hj(X), 1 ≤ i < j ≤ m. Which means that X ⊆ Eij . By X → Y ∈ FER ,Y ⊆ Eij . Hence, we also obtain hi(Y ) = hj(Y ). Consequently, X → Y ∈ FR.The proposition is proved.178 Vu Duc Thi and Nguyen Hoang SonIt is easy to see that the dense family ER has at mostm(m−1)2 elements. ByProposition 3.3, we also have ER ⊆ LR.Theorem 3.6. Let R be a relation over U . ThenKR = Tr(min(ER)).Proof. By the definition of relation R, we have U 	∈ ER. From this, Proposition2.2, Proposition 3.5 and Theorem 3.4, the theorem is obvious.The proof is complete.Let R = {h1, . . . , hm} be a relation over U , and NR the nonequality set of R,i.e.,NR = {Nij : 1 ≤ i < j ≤ m}where Nij = {a ∈ U : hi(a) 	= hj(a)}.Note that, because R is a relation, ∅ 	∈ NR and U 	∈ ER. Moreover, NR = ER.From this, and Theorem 3.6, the following corollary is immediateCorollary 3.7. Let R be a relation over U . ThenKR = Tr(min(NR)).Corollary 3.7 was shown in [5].Proposition 3.8. If D is R− dense, thenmin(D − {∅}) = max(ER).Proof. According to Theorem 3.6, we have KR = Tr(ER). By Proposition 2.2, itis clear thatKR = Tr(max(ER)). (1)Because D is R− dense, and by Theorem 3.4, we have KR = Tr(D − {∅}). Fur-thermore, we haveTr(D − {∅}) = Tr(min(D − {∅})).HenceKR = Tr(min(D − {∅})). (2)From (1) and (2), we giveTr(min(D − {∅})) = Tr(max(ER)).By min(D− {∅}) and max(ER) are simple hypergraphs, thus according to Propo-sition 2.1 we havemin(D − {∅}) = max(ER).The proposition is proved.From Proposition 3.8, the following corollary is clearSome Results Related to Dense Families of Database Relations 179Corollary 3.9. If D is R− dense, thenmin(D − {∅}) = min(NR).Now we give a necessary and sufficient condition for an arbitrary family D isR− dense.Theorem 3.10. Let R be a relation, D ⊆ P(U) a family of subsets of a U . ThenD is R− dense iff for every X ⊆ ULR(X) =⎧⎨⎩⋂X⊆AA if ∃A ∈ D : X ⊆ A,U otherwise,where LR(X) = {a ∈ U : X → {a} ∈ FR}.Proof. First we prove that in an arbitrary family D ⊆ P(U) for all X ⊆ ULFD (X) =⎧⎨⎩⋂X⊆AA if ∃A ∈ D : X ⊆ A,U otherwise.Suppose that X is a set such that there is no A ∈ D with X ⊆ A. By thedefinition of FD, it is easy to see that X → U ∈ FD. Hence, LFD (X) = U .Since ∅ ⊆ ⋂A∈D A ⊆ A, according to the definition of FD and LFD we obtainLFD (∅) =⋂A∈DA.If X 	= ∅ and there is an A ∈ D such that X ⊆ A then we setG = {A : X ⊆ A, A ∈ D},B =⋂A∈GA.It is easy to see that X ⊆ B holds. If G = D or G 	= D, then we also obtainX → B ∈ FD.By the definition of LFD , we have B ⊆ LFD (X). Using X ⊆ B ⊆ LFD (X), weobtain B → LFD (X) ∈ FD.Now we suppose that b is an attribute such that b 	∈ B. Then, there is A ∈ Gso that b 	∈ A. Hence, by the definition of FD we have B → B ∪ {b} 	∈ FD.Consequently,LFD (X) =⋂A∈D(A).By Remark 1.1 it is easy to see that FR = FD holds iff LR = LFD does.The Theorem is proved.180 Vu Duc Thi and Nguyen Hoang SonFrom Theorem 3.10 and Proposition 3.5, the following proposition is obviousProposition 3.11. Let R = {h1, . . . , hm} be a relation over U = {a1, . . . , an}.Then(1) If D is R− dense, then D ∪ {U} also is R−dense, and thus ER ∪ {U} isR−dense.(2) If m = 1 or FR = {{a1} → U, . . . , {an} → U}, then families D1 = ∅,D2 = {∅} and D3 = {U} are R−denses.4 Finding the set of all minimal keys of a relationIn this section, we give the following algorithm finding all minimal keys of a givenrelation R. Remember that this problem is inherently exponential in the size of R[4].Algorithm 4.1.Input: a relation R = {h1, . . . , hm} over U .Output: KR.Method:Step 1. Construct the equality setER = {Eij : 1 ≤ i < j ≤ m}where Eij = {a ∈ U : hi(a) = hj(a)}.Step 2. Compute the complement of ER as followsER = {Eij : Eij ∈ ER}.Denote elements of ER by N1, . . . , NkStep 3. From ER compute the family min(ER) = {Ni ∈ ER :	 ∃Nj ∈ ER : Ni ⊆Nj}.Step 4. By Algorithm 2.3 we construct the set Tr(min(ER)).Based on Proposition 2.2, Algorithm 2.3 and Theorem 3.6, we have KR =Tr(min(ER)). It can be seen that the time complexity of this algorithm is thetime complexity of Algorithm 2.3. In many cases this algorithm is very effective(see Remark 2.5).It can be seen that, if the number of elements of the equality set ER is constant,i.e. |ER| ≤ k for some constant k, then the time complexity of finding KR of agiven relation R is polynomial time [9].The following example shows that for a given relation R, Algorithm 4.1 can beapplied to find all minimal keys of a given relation R.Some Results Related to Dense Families of Database Relations 181Example 4.2. Let us consider the relation R over U = {a, b, c, d} as followsR =a b c d0 0 0 00 0 0 12 0 0 03 3 0 04 0 4 45 5 5 0It can be seen that the equality set ER is the followingER = {∅, {b}, {c}, {d}, {b, c}, {c, d}, {a, b, c}, {b, c, d}}.HenceER = {{a}, {d}, {a, d}, {a, b}, {a, b, c}, {a, b, d}, {a, c, d}, U},min(ER) = {{a}, {d}}.From this, we obtainKR = {{a, d}}.5 Finding the cover of a relationFrom Proposition 3.5 and Theorem 3.10 we have an application, which is the fol-lowing algorithm finding a cover of FDs of a given relation R. Recall that thisproblem is inherently exponential in the size of R [6].Algorithm 5.1.Input: a relation R = {h1, . . . , hm} over U .Output: FR.Method:Step 1. Construct the equality setER = {Eij : 1 ≤ i < j ≤ m}where Eij = {a ∈ U : hi(a) = hj(a)}.Step 2. Compute the family E+R = {∩A : A ⊆ ER}. Denote the elements of E+R byX1, . . . , Xt.Step 3. Construct set of FDs as followsF = {K1 → X1 : K1 ∈ Key(X1)} ∪ · · · ∪ {Kt → Xt : Kt ∈ Key(Xt)}where Key(Xi) is a set of all minimal keys of ΠXi (R) (the projection of R onto theattributes set Xi).Obviously, F = FR. Note that LR = E+R . It is easy to see that the timecomplexity of this algorithm is exponential in the number of attributes.The following example shows that for a given relation R, Algorithm 5.1 can beapplied to find a cover of a given relation R.182 Vu Duc Thi and Nguyen Hoang SonExample 5.2. R is the following relation over U = {a, b, c, d}R =a b c0 0 00 1 01 1 0It can be seen that the equality set ER is the followingER = {{c}, {a, c}, {b, c}}.ThereforeE+R = {{c}, {a, c}, {b, c}, U}.From this, we haveF = {{a} → {c}, {b} → {c}, {a, b} → {c}}.It is obvious that F = FR.References[1] Armstrong W. W., Dependency structure of database relationship, InformationProcessing 74, North-Holland Pub. Co. , (1974) 580-583.[2] Berge C., Hypergraphs: combinatorics of finite sets, North - Holland, Amster-dam (1989).[3] Demetrocis J., On the equivalence of candidate keys with Sperner systems, ActaCybernetica 4, (1979), 247-252.[4] Demetrovics J., Thi V.D., Keys, antikeys and prime attributes, Annales Univ.Sci. Budapest Sect. Comp. 8, (1987), 35-52.[5] Demetrovics J., Thi V. D., Describing candidate keys by hypergraphs, Computersand Artificial Intelligence 18, 2 (1999), 191-207.[6] Gottlob G., Libkin L., Investigations on Armstrong relations, denpendency in-ference, and excluded functional dependencies, Acta Cybernetica Hungary 9, 4(1990), 385-402.[7] Järvinen J., Dense families and key functions of database relation instances,in: Freivalds R. (ed.), Fundamentals of Computation Theory, Proceedings ofthe 13th International Symposium, Lecture Notes in Computer Science 2138(Springer-Verlag, Heidelberg, 2001), 184-192.[8] Thi V. D., Minimal keys and antikeys, Acta Cybernetica 7 (1986), 361-371.[9] Thi V. D., Son N. H., Some problems related to keys and the Boyce-Codd normalform, Acta Cybernetica 16, 3 (2004), 473-483.Received December, 2004

Some results related to dense families of database relations

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Some results related to dense families of database relations

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