A NEW CHARACTERISATION OF INFORMATION CAPACITY EQUIVALENCE IN NESTED RELATIONS

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1 A NEW CHARACTERISATION OF INFORMATION CAPACITY EQUIVALENCE IN NESTED RELATIONS Millist W Vincent Advanced Computing Research Centre, School of Computer and Information Science, University of South Australia, Adelaide, Australia millistvincent@unisaeduau Mark Levene Department of Computer Science, University College London, Gower Street, London, WC1E 6BT, UK MLevene@ukacuclcs Abstract Nested relations are an important subclass of object-relational systems that are now being widely used in industry Also, a subclass of nested relations, partitioned normal form (PNF) relations, is important because the semantics and manipulation of these relations are more transparent than general nested relations In this paper we address the question of determining when every PNF relation stored under one nested relation scheme can be transformed into another PNF relation stored under a different nested relation scheme without loss of information, referred to as the two schemes having equivalent information capacity This issue is important in many database application areas such as view processing, schema integration and schema evolution The main result of the paper provides a new characterisation of information capacity equivalence for single nested schemes It shows that two schemes are information capacity equivalent if and only if the two sets of multivalued dependencies induced by the two corresponding scheme trees are equivalent 1 INTRODUCTION The nested relational model [8] was introduced to provide a more natural way of modelling data in new application areas such as those involving textual, spatial or scientific data Since then, many aspects of the nested relational model have been investigated and several prototypes developed[1, 7, 13-15] More recently, nested relations are supported by object-relational systems that are predicted to become the industry standard An important subclass of nested relations is the class of partitioned normal form (PNF) nested relations [14] In a PNF relation the set of atomic attributes at every level of the nested relation must be a key It has been shown that PNF relations possess several desirable properties that nested relations in general do not possess, such as nesting and unnesting operations commuting, which results in the semantics of PNF nested relations and their query languages being more transparent than for general nested relations [12] The issue that is investigated in this paper is determining

2 when every PNF nested relation stored under one nested relation scheme can be transformed into another PNF nested relation stored under a different nested relation scheme without loss of semantic content, referred to as the two schemes being information capacity equivalent This is an important issue in all data models, not only the nested relational model, and has impact on a variety of areas such as view processing, scheme integration, schema evolution and schema translation in heterogeneous database systems In particular, the recent work in [9, 10] has highlighted the importance of information capacity in the development of correct techniques for schema integration and schema translation The approach to formalising the notion of information capacity equivalence used in this paper is based on the seminal work in [4, 6] The essential idea is that two database schemes are information capacity equivalent if there exists a bijection which maps from every instances of one scheme to an instance in the other If this property is satisfied, then every instance of one scheme can be mapped to an instance in the other scheme and then recovered since the mapping is 1-1 If no restriction is placed on the instance mapping apart from the requirement that it be a bijection, then this notion of information capacity equivalence is referred to as absolute equivalence Several more restrictive notions equivalence were also introduced in [4, 6] based on further restricting the instance mapping In particular, the other types equivalence introduced were (in increasing order of restrictiveness): internal equivalence (which does not allow the instance mapping 'invent' more than a fixed set of constants), generic equivalence (which requires that the instance mapping commutes with permutations of the data values which leave the instances unchanged) and query equivalence (which requires that the instance mapping be a query in the data model) It has been shown that in the case of the relational model, the different notions of data equivalence are not identical whereas in the format model introduced in [4, 6] absolute equivalence, internal equivalence and generic equivalence are identical The main result of this paper extends a previous result [18] and provides a new characterisation of absolute equivalence for PNF schemes It shows that two PNF schemes are absolutely equivalent if and only if the sets of multivalued dependencies (MVDs) induced by the corresponding scheme trees are equivalent This result confirms the importance of considering dependencies in restructuring, an observation that had been previously made but not fully investigated [5] A consequence of this result is also that absolute equivalence and query equivalence are identical for single nested schemes Finally, we use these results to develop a polynomial time algorithm for testing whether two nested schemes are absolutely equivalent We also point out that in characterising absolute equivalence we explicitly derive a 1-1 mapping between the PNF instances of the schemes expressed as a sequence of unnest and nest operations The benefit of deriving an instance mapping, and especially one that is expressible as a query, is that the instance mapping can be used in database translation tools to translate any query expressed against one scheme into another query expressed against an equivalent scheme [9, 10] The rest of the paper is organised as follows In Section 2 we introduce basic definitions and notations In Section 3 we derive the main result of the paper on characterising absolute equivalence on nested schemes and Section 4 contains concluding comments

3 2 DEFINITIONS AND NOTATION We now introduce the definitions and notation to be used in later sections More complete presentations can be found in standard references such as [12] 21 Trees A tree is a finite, acyclic, directed graph in which there is a unique node, called the root, with indegree 0 and every other node has indegree 1 A node n' is a child of a node n (or equivalently, n is the parent of n') if there is a directed edge from n to n' A pair of nodes are siblings if they are children of the same node A node is a leaf if it has no children A node n' is recursively defined to be a descendant of a node n (or equivalently, n is an ancestor of n') if either n' is a child of n, or n has a child n" such that n' is a descendant of n" The height of a tree T is the number of nodes on the longest path from the root to a leaf node A tree T' is a subtree of a tree T if the nodes of T' are a subset of those of T and for every pair of nodes n' and n, n' is a child of n in T if and only if n' is a child of n in T' A subtree T' is a principal subtree of T if the root of T' is a child of the root of T A tree T is balanced if every non-leaf node has at least two children 22 Scheme trees and nested relations Let U be a fixed countable set of atomic attribute names Associated with each attribute name A U is a countably infinite set of values denoted by DOM(A) and the set DOM is defined by DOM = DOM(A i ) for all A i U We assume that DOM(A j ) DOM(A j ) if i j A scheme tree is a tree containing at least one node and whose nodes are labelled with nonempty sets of attributes that form a disjoint partition of U If n denotes a node in a scheme tree T then: ATT(n) is the set of attributes associated with n; ΑΤΤ(Τ) is the union of ATT(n) for all n T; ANC(n) is the set of all ancestor nodes of n, including n; A(n) is the union of ATT(n') for all n' ANC(n); DESC(n) is the set of all descendant nodes of n, including n; D(n) is the union of ATT(n') for all n' DESC(n); ROOT(T) denotes the root node of T; HEIGHT(T) denotes the height of T Figure 21 illustrates an example scheme tree defined over the set of attributes {STUDENT, MAJOR, CLASS, EXAM, PROJECT}

4 STUDENT MAJOR CLASS EXAM PROJECT Fig 21 An example scheme tree If S and T are two scheme trees, then S and T are isomorphic if there exists a bijection τ from the nodes of S to those of T such that: (i) for any pair of nodes n' and n in S, n' is a child of n in T iff τ (n') is a child of τ (n) in S; (ii) for every node n S, ATT(n) = ATT(τ(n)) We note that it follows from this definition that if scheme trees S and T are isomorphic then ATT(S) = ATT(T) It also follows that the scheme tree resulting from interchanging in a scheme tree any pair of subtrees whose roots are siblings is isomorphic to the original scheme tree If a scheme tree T has leaf nodes {n 1,, n m } then the path set of T, P(T), is defined by P(T) = {A(n 1 ),, A(n m )} For instance, the path set of the scheme tree in Figure 21 is {{STUDENT, MAJOR}, {STUDENT, CLASS, EXAM}, {STUDENT, CLASS, PROJECT}} A nested relation scheme (NRS) for a scheme tree T, denoted by N(T), is the set defined recursively by: (i) If T consists of a single node u then N(T) = ATT(u); (ii) If A = ATT(ROOT(T)) and T 1,, T k, k 1, are the principal subtrees of T then N(T) = A {N(T 1 )} {N(T k )} For example, for the scheme tree T shown in Figure 21, N(T) = {STUDENT, {MAJOR}, {CLASS, {EXAM}, {PROJECT}}} We now recursively define the domain of a scheme tree T, denoted by DOM(N(T)), by: (i) If T consists of a single node n with ATT(n) = {A 1,, A n } then DOM(N(T)) = DOM(A 1 ) DOM(A n ); (ii) If A = ATT(ROOT(T)) and T 1,, T p are the principal subtrees of T, then DOM(N(T) = DOM(A) P(DOM(N(T 1 ))) P(DOM(N(T p ))) where P(S) denotes the set of all nonempty, finite subsets of a set S We note in the above definition that, in contrast to the VERSO model defined in [1], we do not allow empty sets

5 The set of atomic attributes in N(T), denoted by Z(N(T)), is defined by Z(N(T)) = N(T) U The set of higher order attributes in N(T), denoted by H(N(T)), is defined by H(N(T)) = N(T) - Z(N(T)) For instance, for the example shown in Figure 21, Z(N(T)) = {STUDENT} and H(N(T)) = {{MAJOR}, {CLASS, {EXAM}, {PROJECT}}} Also, if X is a subset of N(T) given by X = {A 1,, A k, {Y 1 },, {Y p }} where A 1,, A k are the atomic attributes of X and {Y 1 },, {Y p } are the higher order attributes, then the atomic attributes in X, denoted by ATOM (X), is recursively defined by ATOM(X) = {A 1,, A k } ATOM(Y 1 ) ATOM(Y p ) Finally we define a nested relation over a nested relation scheme N(T), denoted by r*(n(t)), or often simply by r* when N(T) is understood, to be a finite nonempty set of elements from DOM(N(T)) If t is a tuple in r* and Y is a nonempty subset of N(T), then t[y] denotes the restriction of t to Y and the restriction of r* to Y is then the nested relation defined by r*[y] = { t[y] t r*} An example of a nested relation over the scheme tree shown in Figure 21 is shown in Figure 22 The active domain of a nested relation r*, denoted by ADOM(r*), is the subset of DOM containing all the atomic values appearing in r* STUDENT {MAJOR} {CLASS {EXAM} {PROJECT}} Anna Maths CS100 mid-year Project A Computing final Project B Project C Bill Physics P100 final Prac Test 1 Prac Test 2 Chemistry CH 200 Test A Experiment 1 Test B Experiment 2 Test C Experiment 3 Fig 22 A nested relation 23 Nested Operators We now define the two restructuring operators for nested relations, nest and unnest, first introduced in [16] Let Y be a nonempty proper subset of N(T) Then the operation of nesting a relation r* on Y, denoted by νy(r*), is defined to be a nested relation over the scheme (N(T) - Y) {Y} and a tuple t νy(r*) iff: (1) there exists t' r* such that t[n(t) - Y] = t'[n(t) - Y] and (2) t[{y}] = {t"[y] t" r* and t"[n(t) - Y] = t[n(t) - Y]} Unnesting is defined as follows Let r*(n(t)) be a relation and {Y} an element of H(N(T)) Then the unnesting of r* on {Y}, denoted by µ{y}(r*) is a relation over the nested scheme (N(T) - {Y}) Y and a tuple t µ{y}(r*) iff there exists t' r* such that t'[n(t) - {Y}] = t[n(t) - {Y}] and t[y] t'[{y}]

6 More generally, one can define the total unnest of a relation, µ*(r*), as the flat relation defined recursively by: (1) if r* is a flat relation then µ*(r*) = r*; (2) otherwise µ*(r*) = µ*(µ{y}(r*)) where {Y} is a higher order attribute in the NRS for r* It can be shown [16] that the order of unnesting is immaterial and so µ* is uniquely defined 24 Constraints We now define dependencies in a nested relation in a similar fashion to the way they are defined in flat relations If r*(n(t)) is a nested relation and Y and Z are subsets of N(T) then r* satisfies the functional dependency (FD) Y Z iff for all tuples t, t' r*, if t[y] = t'[y] then t[z] = t'[z] The MVD Y Z is satisfied in r* iff for all t, t' r* such that t[y] = t'[y], there exists s r* such that s[y] = t[y], s[z] = t[z] and s[n(t) - Z] = t'[n(t) - Z] The set of MVDs induced by a scheme tree T is defined as follows [11] If (n, n') is an edge in T, then the MVD associated with this edge is the MVD A(n) D(n') and MVD(T) is the set of all such MVDs associated with all the edges in T For example, if T is the scheme tree in Figure 21 then MVD(T) = {STUDENT MAJOR, STUDENT CLASS EXAM PROJECT, STUDENT CLASS EXAM, STUDENT CLASS PROJECT} A join dependency (JD), denoted by ><[R 1,, R p ] where R i U for 1 i p, is satisfied in a flat relation r if r = π R1 (r) >< ><π Rp (r) where π denoted the projection operator The set of all flat relations which satisfy a set Σ of FDs and MVDs is denoted by SAT(Σ) Given a set Σ of FDs and MVDs and an FD Z W (or MVD Z W), Σ implies the FD Z W (or MVD Z W) if every relation in SAT(Σ) also satisfies Z W (or Z W) The closure of a set Σ of FDs and MVDs, denoted by Σ +, is the set of FDs and MVDs implied by Σ Two sets of dependencies, Σ and Ψ, are equivalent, written as Σ Ψ, if Σ + = Ψ + The subclass of nested relations we investigate in this paper, partitioned normal form (PNF) relations, was introduced in [14] If T is a scheme tree and PNF(T) denotes the set of all PNF nested relations over T, then a relation r* PNF(T) iff the following conditions hold: (i) r* is a nested relation defined over N(T); (ii) Z(N(T)) is a key for r*, ie r* satisfies the FD Z(N(T)) H(N(T)); (iii) For every Y H(N(T)) and t r*, t[y] is in PNF 3 ABSOLUTE EQUIVALENCE FOR PNF SCHEMES In this section we establish the main result of the paper on characterising absolute equivalence for scheme trees which extends the results given in [18] and gives a new characterisation of absolute equivalence For the sake of completeness and to assist the reader, we also reproduce the relevant results from [18] Details of proofs are omitted for lack of space and can be found in a full length version of this paper [19] Firstly, we recall the definition of absolute equivalence between schemes [4]

7 Definition 31 Let T and S be scheme trees where ATT(T) = ATT(S) = U and let X DOM The set DOM X (T) is defined by DOM X (T) = {r* r* PNF(T) and ADOM(r*) X} Then T is absolutely equivalent to S if there exists a positive integer N such that for all X DOM such that X DOM(A i ) N for every A i U, there exists a bijection from DOM X (T) to DOMX(S) A simple consequence of Definition 31 is the following alternative characterisation [4] of absolute equivalence in terms of domain sizes This characterisation is fundamental to the later results in this paper Lemma 31 Let T and S be scheme trees where ATT(T) = ATT(S) and let X DOM Then T and S are absolutely equivalent iff there exists a positive integer N such that for all X DOM such that X DOM(A i ) N for every A i U, DOM X (T) = DOM X (S) (where X denotes the cardinality of a set X) We now introduce a restructuring operator for PNF scheme trees, called COMPRESS Definition 32 Let T be a scheme tree and n a node in T with a single child node n' having children n 1,, n k Then the COMPRESS operator results in replacing the subtree with root n by a subtree with root containing ATT(n) ATT(n') and having children n 1,, n k The COMPRESS operator is illustrated in Figure 32 n A A 1 j COMPRES S n" A A 1 m n' A A j+1 m n 1 n k n 1 n k Fig 32 COMPRESS Operator We note that the COMPRESS operator can be equivalently defined, though less transparently, by a total unnest operator followed by an appropriate sequence on nest operators Next, we show that the nested scheme obtained by applying an arbitrary COMPRESS operator is absolutely equivalent the original scheme tree

8 Lemma 32 If T is a scheme tree then T and COMPRESS(T),where COMPRESS(T) denotes the scheme tree obtained by applying the COMPRESS operator to an arbitrary node in T, are absolutely equivalent It follows from this result that by repeatedly applying the COMPRESS operator to a scheme tree T until no more applications can be applied, a balanced tree is obtained which is absolutely equivalent to the original tree In the terminology of [6], this tree can be considered as a normal form tree for T Also, we have the following result that the (balanced) normal that the (balanced) normal form tree for T, denoted by B(T), is unique (up to an isomorphism) We write S T if there exists a sequence of scheme trees S 1,, S n such that S 1 = S,, S n = T and S i = COMPRESS(S i-1 ), n i > 1 Lemma 33 If S is a scheme tree and S 1 and S 2 are balanced scheme trees such that S S 1 and S S 2, then S 1 and S 2 are isomorphic We also recall the main result from [18] which characterises absolute equivalence in terms of the normal form trees Theorem 31 If T and S are scheme trees where ATT(S) = ATT(T), then S and T are absolutely equivalent iff B(S) and B(T) are isomorphic We now show that absolute equivalence can also be characterised in terms of the MVDs induced by the scheme tree Firstly, we follow the approach of [17] and define a generalised nest operator and then show that it can be used to construct a bijection between PNF instances of two scheme trees whose induced sets of MVDs are equivalent Definition 33 Let T be a scheme tree and r a flat relation defined over U Then nesting r according to T, denoted by ν * T (r), is defined by ν* T (r) = ν X k ν X0 (r) such that: (i) For every non-root node n T there exists one and only one X i such that N(T n ) = X i, where T n is the subtree of T with root n; (ii) If node n is a descendent of node n' in T, then the nest operation on the set corresponding to n is performed before the nest operation on the set corresponding to n ' For example, given the scheme tree T in Figure 21, then one nest sequence to construct ν * T (r) is ν MAJOR ν CLASS, {EXAM}, {PROJECT} ν EXAM ν PROJECT (r) We then can show that ν * T (r) has the following properties:

9 Lemma 34 If r is a flat relation in MVD(T), then: (i) ν * T (r) is uniquely defined; (ii) If r* PNF(T), then ν * T (µ*(s*)) = r*; (iii) µ*(ν * T (r)) = r We use this result to show that if MVD(S) and MVD(T) are equivalent, then there exists a bijection from PNF(S) to PNF(T) Lemma 35 If S and T are scheme trees where ATT(T) = ATT(S) and MVD(S) MVD(T), then the mapping from PNF(S) to PNF(T) defined by F(s*) = ν * T (µ*(s*)), s* PNF(S), is a bijection Combining these two preliminary results gives the main result of this paper which characterises absolute in terms of the induced MVDs Theorem 32 If T and S are scheme trees where ATT(S) = ATT(T), then S and T are absolutely equivalent iff MVD(S) MVD(T) The significance of this theorem is that it follows from it and Lemma 35 that if S and T are absolutely equivalent then there exits a bijection between instances of the schemes which can be expressed as a query in the standard nested relational query algebra In other words, if S and T are absolutely equivalent then they are also query equivalent and so absolute equivalence and query equivalence (and also generic and internal equivalence coincide for PNF schemes since it is well known that query equivalence generic equivalence internal equivalence absolute equivalence [5]) The practical importance of being able to express the instance mapping as a nested relational query was discussed in more detail in the Introduction Another consequence of Theorems 31 and 32 is that they provide two polynomial time algorithms for testing scheme tree equivalence, one by checking for the equivalence of the sets of MVDs and the other by converting each scheme tree to a balanced tree and then checking if the trees are isomorphic Using the MVD approach, it follows from the results in [3] that the implication problem of testing whether a set of MVDs Σ implies an MVD Z W can be solved in at worst in O(N log U ) time, where N is the total number of occurrences of attributes in Σ Testing whether two sets of MVDs are equivalent then reduces to running the membership test for each MVD in both sets and can be solved in at worst in O( Σ N log U ) time, assuming that Σ and N are upper bounds for both MVD sets The alternative approach can be implemented by first imposing an arbitrary total ordering on U and thereafter sorting first the attributes in each node of the scheme tree and then the children of each node according to the first attribute in a child This sort can be done in at worst in time t = (c i log c i + k i log k i ) where the sum is taken i over all the nodes in the tree and c i and k i represent the number of attributes in node i of the tree and the number of children of node i, respectively Using the properties of the log function we can derive t = log( i c i c i k i k i ) and so t O( U log U ) because

10 c i = U and k i U After sorting the scheme tree, compressing them can i i be done in at worst in O( U ) time since there are at most U nodes in the tree Finally, testing if the trees are isomorphic can be done in at worst O( U ) time since each attribute needs to be checked at most once and appears only once in a scheme tree Hence the total cost of testing for absolute equivalence using this method is at worst O( U log U ) time This compares favourably with the MVD approach when U < Σ N, which one expects normally to hold The next lemma summarises this result Lemma 36 Testing whether two PNF scheme trees are absolutely equivalent can be done in O( U log U ) time 4 CONCLUSIONS In this paper we have investigated the problem of when two nested relational schemes are information capacity equivalent and a proved a new characterisation of quivalence - that two nested schemes are equivalent if and only if the sets of MVDs induced by the corresponding scheme trees are isomorphic Also, a consequence of this result shows that absolute equivalence and query equivalence coincide for PNF scheme trees There are several other related topics that also warrant further investigation Data equivalence can be viewed as a special case of data dominance [8] which holds if there is a 1-1 (but not necessarily onto) mapping from the instances of one scheme to the instances of another scheme Finding then a complete set of restructuring operators for a scheme tree which preserve data dominance is an important question that arises in contexts such as schema integration where one often wants to ensure that the integrated scheme dominates the original schemes The approach in this paper has also assumed that attribute domains are disjoint and it would be useful to characterise data equivalence and dominance if this assumption is dropped Another aspect that needs further investigation is to extend the approach in this paper, which has only characterised data equivalence for a single nested scheme, to characterising absolute equivalence and dominance for nested database schemes References [1] S Abiteboul and N Bidoit, Non First Normal Form Relations: An Algebra Allowing Data Restructuring, Journal of Computer and System Sciences 33(3)(1986) [2] S Abiteboul and R Hull, Restructuring Hierarchical Database Objects, Theoretical Computer Science 62(1-2)(1988) 3-38 [3] Z Galil, An Almost Linear-Time Algorithm for Computing a Dependency Basis in a Relational Database, Journal of the ACM 29(1)(1982) [4] R Hull, Relative Information Capacity of Simple Relational Database Schemata, SIAM Journal of Computing 15(3)(1986) [5] R Hull, A survey of theoretic research on typed complex database objects, in: Databases, J Paredaens ed, (Academic Press, London, 1987)

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