Seleznev O. V.

Thalheim Bernhard

English

University of Szeged

On the numbers of shortest keys in relational databases on nonuniform domains O . SELESNJEW, B. THALHEIM The use of the relational model of data structures by E. F. Codd is a promising mathematical tool for handling data. In this model the user's data are expressed as relations where the rows denote the records and the columns represent domains or attributes. For the handling of relations the identification of sets of domains, called keys, is suggested. The keys uniquely determine the values of the rest of the domains. Delobel and Casey, Fadous and Forsyth, M. Fernandez, C. L. Lucchesi and S. L. Osborn, J. Demetrovics and V. Thi have given different algorithms for finding the set of all minimal keys in a relational database given by a set of functional dependencies on the database. For characterizing the complexity of this algorithms we need some combinatorial bounds. In this paper we consider the maximal numbers of shortest keys in relational databases on weighted domains and extend the result of J. Demetrovics who solved the problem for relational databases on uniform domains. [1] 1. Basic notions We recall briefly some definitions of the theory, of relational databases. Given sets Du D2, ..., D„, called domains, not necessarily distinct, an «-ary relation R. defined over Du ...,D„ is a subset of the cartesian product D1XD2XD„. An attribute is a name assigned to a domain of a relation. Any value associated with an attribute is called attribute value. The attribute names must be distinct. The symbol U will be used to denote the set of all n attributes of R. A set of attributes X, XQ U, is called a key of R if, for every «-tuple of R, the values of the attributes in X uniquely determine the values of the attributes in U. Now, suppose we are given some weight function (or complexity measure) g: U—N' and a system SR of keys of R. For XQU let g(X)= % g(A) the Com-d e x plexity of X. An element K of SR is called g-shortest if there does not exist an element K' of SR with g(K')^g(K). By SR (g) we denote the set of all ̂ -shortest elements of SR and by sR(g) its cardinality. For g= 1 the set SR(g) is called the set of all shortest keys or the set of shortest keys in.an unweighted database. It is obvious that 268 O. Selesnjew and B. Thalheim any set SR(g) is a subset of a set of minimal keys [1]. For any set S of minimal keys there exists a subset S(g) of shortest keys. This is well-known for g= 1. Theorem 1. [1] The maximal size of a set of shortest keys in a database with n attri-i M l n ] butes is n n j I. For any 1 s k s U n j I, there is an n-ary relation R with k shortest keys. 2. Maximal number of shortest keys in nonuniform databases In practical cases, keys are of different meaning and complexity. Domains for attributes have very different complexity. This is well-known in practice but it is not taken into consideration in the theory of minimal keys. Therefore, shortest keys are introduced. Lower and upper bounds for sR(g) are proved in [4]. The most interesting set of functions g is the set G + of functions g with giA^^giAj) for i^j. The other cases can be splitted in the case g(A)= 1 for A£X and in this case for A£ U\X. We introduce the following functions: s(g) = max sR(g), J(G') = max s(g) for sets G' of weight functions. g €G' Using the functions g!,g2,g3 with g1(Ai)=2i, g2(Ai)=3IiM, g3(Ai)=i, for /, 1 ^ / S n , by the definitions and the recursion formulas for g3, we get n Corollary 2.1. For weight functions g it holds 1 s s(g) s | f n' 2. J ( f t ) = l , s(gj=2™, s ( g 3 , [ 4 ] . [1]. Our next aim is to prove Theorem 3. j ( G + ) = - - = £ = (1 —o(l>). n From number theory [2] we get that functions g with j ( g ) = j ( G + ) must be regulary. W.l.o.g. we consider a subclass G* of G + , the class of all equidistant func-tions g with the property g(Ai)—g(Ai^1)=c for some c and any /, Lemma 4. 1. Given two equidistant functions g, g' from G + . Then s(g)=s(g'). 2. Let g be a function from G + . There exists an equidistant function g' in G* such that s(g)=s(g'). Proof. 1. Is obvious. 2. W.l.o.g. we consider only functions g from G + with g(A,)<g(Ai+1) for /, 1 ^ /< n. We prove the assertion by induction. For n= 2 the assertion is obvious. Let n be a fixed number. Now we assume that for a fixed function g there is no equi-On the numbers of shortest keys in relational databases on nonuniform domains 269 distant function g'£G* such that s(g)Ss(g'). Let be SR a key system with s(g)= =sR(g). Now we define S,= {KiS^AAK}, S2={K\{An}/KiSR, An£K}. By precondition of induction, we get for an equidistant function g" such that s(g')Ss(g"). It follows that there is an equidistant function g+ in G* such that and s(g)^s(g+). That is a contradiction. W.l.o.g. we can consider for s(G+) the function g3 of Corollary 2. Now we define independent random variables gk with two-point distribution for k= 1, 2, ..., n: and consider the distribution of S„= 2 ¿¡¡. i=i Corollary 5. P ( . S N = [ " ( " 4 + 1 ) ] ] ^ j r J (g3) for probability P(Sn=m). For the expectation ESn and the variance DS„ of Sn we get m c-c v w v k " ( " + ! ) Mn = ESn= 2 E£k= 2-K = —7— k = l fc=1 ^ ^ Bn = DS. = 2mk = 24 ~ i 2 We shall say that the sequence {S„} satisfies a local limit theorem iff sup \BnP(Sn = m)-<p(x„m)\ - 0 («-=») m m—M S —M where xnm——-—zn=——cp is the standard normal distribution density. We denote a ( a , q , N ) = -± r 2 = r(mod.g), \tk\ S N) ( + ) H -9 /2 -=rS9 /2 for %k=Zk—Z'k symmetrized random variable, where ££ is a random variable inde-pendent of and having the same distribution as relatively prime integers a, q with a s - | - and 1 In [3] is proved the following: If the distribution function of the sum of unboun-dedly increasing number of random variables converges to the standard normal distribution function, i.e. N(0, 1), (1) K exp { - ^ min 2 «»(«, <7, ^ t )} - 0 ( « - oo), (2) Nn is selected such that Hm 2 f x 2 d F l X x ) = / > 0, (3) then the sum satisfies a local limit theorem. • 270 O. Selesnjew and B. Thalheim Let Nn=n. Then we get 1 = Km 4 - ¿ / > 3 = 1 ^ 0 , pak = k) = pak=-k) = j , p(ifc = o) = i . By summation of ( + ) over representatives of q we get for /eg {1, ..., «}. Observe that if 1^=0 then r=0 and this summand can be eliminated and that if lk=k then ak=rk+qlk for the unique representative of q. We get q>n) = 4i 2 r2P(alk = r (mod q)) = q -9/2<r39/2 From number theory it is known that if {*} form a full system of representatives of q n then {ax} form a full system of representatives. Now A„Smin 21 ak(a, " )= "•« k = 1 1 " s m i n —j 2rl- Assume that q=2m. (For odd q proof is analogous.) Let 0 < 4 4q k=I . If then n m 2 rl S Z № S cm3 S ca3«3 t = i t=I • for the full system of representatives r t = — (m— 1), ..., 0, 1, ..., m and therefore • • 1 ccc3 w3 l a £ min - r - r ca3«3 s - r -ST - ? = P n ' $ > - " ? 4#a 4a24 /1 If l < m < a n then the full system of representatives {/•}"(„,_!) is contained in {1 «} at least — times. Consequently we get l q J f -1 ¿ k 2 A, S min - L J rl S m i n - ^ - 7 7 — S 0,4 4#2 a,« . 4q2 m m (n \ 2 k Z S min 1 r ' . s min (n-q)k „ s min (n-2an)c = a,« \q ) 4q2 i 4q3 9 = c ( l - 2 a ) n = fin > 0 , £ > 0. On the numbers of shortest keys in relational databases on nonuniform domains 271 We get that (2) holds because A„=n exp j — y exp{~y"}"*"0(n — Summarizing corollary 5, lemma 4 and the properties of S„ we get »(ft) = 2"P(sn= [-^11]) = „~ 2" 2" 2" The proof of theorem 3 is complete. 1С rPClllt ЛХ/itb i \ r n 2" It is of interest to compare this result with s(g t) fo r^ 4 (^ )= 1 for AZU. Using a central limit theorem we get further 2" /M'+rt) с for some constant c. O. SELESNJEW MOSCOW STATE UNIVERSITY DEPT. OF MATHEMATICS A N D MECHANICS 117 234 MOSCOW USSR B. THALHEIM DRESDEN UNIVERSITY OF TECHNOLOGY DEPT. OF MATHEMATICS COMPUTER SCIENCE DIVISION 8027 DRESDEN GDR References [1] J. DEMETROVICS, On the equivalence of candidate keys with Sperner systems. Acta Cybernetica 4 (1979) , 2 4 7 — 2 5 2 . [2] K. KNOPP, I. SCHUR, Elementarer Beweis einiger asymptotischer Formeln der additiven Zahlen-theorie. Mathematische Zeitschrift 24, 1925, 559—574. [3] А. А. Миталаускас, В. JI. Статулявичус, Локальные предельные теоремы и аси-мптотические разломение для сумм независимых решетчатых случайных величин. Литовс-кий математический сборник 1966, Т. 6, N. 4, 569—583. [4] В. THALHEIM, Abhängigkeiten in Relationen. Dissertation В, Technische Universität Dresden, Dresden 1985. (Received March 4,1987) 

On the numbers of shortest keys in relational databases on nonuniform domains

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On the numbers of shortest keys in relational databases on nonuniform domains

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