Super-Key Classes for Updating. Materialized Derived Classes in Object Bases

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1 Super-Key Classes for Updating Materialized Derived Classes in Object Bases Shin'ichi KONOMI 1, Tetsuya FURUKAWA 1 and Yahiko KAMBAYASHI 2 1 Comper Center, Kyushu University, Higashi, Fukuoka 812, Japan fkonomi,furukawag@cc.kyushu-u.ac.jp 2 Integrated Media Environment Experimental Laboratory, Kyoto University, Sakyo, Kyoto 606, Japan yahiko@kuis.kyoto-u.ac.jp Abstract. We describe data structures that allow ecient updates of materialized classes derived from relationship of classes in object bases. Materialization of derived classes reduces costs of retrievals and increases costs of updates. Costs of updates increase remarkably when several paths of objects derive the same object. If object bases satisfy the superkey condition proposed in this paper, consistencies of object bases are maintained by local navigations and the remarkable increase of the costs is avoided. Any object base can be transformed to satisfy the superkey condition by adding extra classes and their objects. In this manner, increasing redundancies allows ecient updates. 1 Introduction In databases, redundant data structures introduced to reduce costs of retrievals often increase costs of updates. Database management systems should allow ecient retrievals with minimal increase of update costs. Views in databases are proposed to provide exibility witho aecting underlying data. Views in object bases are also proposed introducing imaginary objects, virtual classes and virtual class hierarchies [1]. Costs of retrievals that are required to produce views depend on access paths of underlying data and complexity of view denitions. Materialization of views allows fast retrievals of views regardless of access paths of underlying data and complexity of view denitions. However, when updates occur in object bases, extra costs are required to keep materialized views consistent. Compared with replicated data in distribed databases, materialized views cause more expensive update overheads since they may require recompation of views. The following example shows characteristics of updating materialized views. Example 1. Let us consider a video database. The whole sequences of frames (images) are partitioned into scenes. In Fig.1(a), Mick and Paul appear at Airport in Scene1, Mick, Paul, and a dog at Airport in Scene2, and Mick and John at Hotel in Scene3. Figure 1 (b) shows objects and their connections in the video database.

2 Presence1 and Presence2 are derived objects where Presence1 is Mick's presence at Airport and Presence2 is Mick's presence at Hotel. Airport Airport Hotel Mick Mick Mick Paul Paul dog John Scene 1 Scene 2 Scene 3 (a) Fig. 1. Video database Paul Mick John Scene1 Airport Scene2 Hotel Scene3 Presence1 Presence2 (b) Suppose the derived objects are materialized and Scene2 is deleted in editing the video. Since Mick is at Airport in Scene2, Mick may lose his presence at Airport. However, after retrieving connections of objects, we nd that he is still at Airport in Scene1. Thus, Presence1 is not deleted. There are three dierent approaches for ecient maintenance of materialized views: Optimal Refresh Frequency of Materialized Views: Analytical modeling and optimal refresh policies of materialized view maintenance are presented in [12]. Screening of Updates of Materialized Views: Blakeley et al.[2] present conditions for detecting when an update of a base relation cannot aect a derived relation and for detecting when a derived relation can be updated using only the data of the derived relation itself and the given update operation. Incremental Updates of Materialized Views: The algorithm presented in [9] derives the minimal incremental relational expressions that need recompation. Roussopoulos[10] discusses incremental updates of a stored collection of pointers pointing to records of underlying relations needed to materialize a view. Here we focus on the third approach and discuss ecient methods of incremental updates of materialized classes derived from relational expressions including projections. In this paper, data structures that permit ecient incremental updates of materialized derived classes are described. When classes are derived from a path of classes in a class schema, the key of the path is dened to be ilized for updates. Class schemas satisfy the super-key condition presented in this paper if derived classes and the keys of the paths from which the classes are derived are contained in the same subgraphs of class schemas. If class schemas satisfy

3 the super-key condition, incremental updates are performed eciently by detecting multiple paths of underlying objects ilizing objects of classes in keys. A procedure to transform class schemas so that they satisfy the super-key condition is presented. Though the procedure add extra classes and introduce more redundancies into object bases, it allows ecient deletions of objects. Section 2 gives the denition of our model of object bases. Section 3 introduces materialized derived classes in object bases. Section 4 gives a procedure to maintain consistencies of materialized derived classes. Section 5 discusses data structures that allow ecient update propagations introducing the super-key condition and super-key classes. Section 6 describes design of object bases for ecient update propagations. Section 7 gives our conclusion. 2 A Model of Object Bases The model described here focuses on connections of objects and relationships of classes from which classes are derived. Objects are values associated with their identiers. Values of objects are assumed to be classied as follows: basic values: The special symbol nil or values from domain D, the union of all the domains in a database. set-values: A set fid 1 ; id 2 ; : : : ; id n g is a set-value where id i (1 i n) is an object identier (OID). tuple-values: Let id i (1 i m) be an OID and a i (1 i m) be an attribe. < a 1 : id 1, a 2 : id 2, : : :, a m : id m > is a tuple-value. Objects belong to classes and classes are structured in class hierarchies with ISA relationships, denoted by -. Classes are dened when object bases are designed. One of the above types of values is assumed to be specied in each class. If tuple-values are specied in a class, attribes are also specied in the class. Relationships of classes and connections of objects are classied as follows: Set relationships and set connections A set relationship from a class C to a class C 0 is denoted by C! C 0. Objects in class C are set-value objects o = (id; fid 1 ; id 2 ; : : : ; id n g) and all the objects identied by id 1 ; id 2 ; : : : ; id n are in class C 0. A set connection is a set of m links from object o to an object identied by id i (1 i m). Reference relationships and reference connections A reference relationship from a class C to a class C 0 on attribe a i is denoted by C ; ai C 0. C;C 0 is used instead of C ; ai C 0 if the attribe is trivial. The set of all the attribe of class C is denoted by Att(C). Objects in class C are tuple-value objects o = (id; < a 1 : id 1 ; a 2 : id 2 ; : : : ; a m : id m >) and object o 0 identied by one of id i (1 i m) is in class C 0. A reference connection is a link from o to o 0. ISA relationships ISA relationship from a class C to a class C 0 is denoted by C - C 0.

4 A class schema is a labeled directed graph S(V; E) where set of vertices V represents the set of all the classes in the schema and set of edges E represents the set of all the relationships of classes in the schema. Example 2. Figure 2 shows a part of the class schema for the video database in Fig.1. Each vertex in the gure represents a class. ISA relationships in the gure show that classes and Animal are subclasses of class Creature. Arrows with attribe names represent reference relationships. Objects in class Scene references objects in class Location as values of attribe Place, objects in class Animal or s as values of attribe Appearance. Arrows with doubled arrowheads represent set relationships. Objects in class s references objects in class as sets. s Scene Appearance Place Appearance Creature Animal Fig. 2. Class schema for video database Location att :Class :ISA relationship :Reference relationship :Set relationship An object graph is a labeled directed graph O(v; e) where v is a set of vertices and e is a set of edges. Each vertex of v corresponds to one of a basic-value, a set-value or a tuple-value object. Each edge of e corresponds to a link in reference connections or in set connections. Example 3. Figure 3 shows an object graph corresponding to the class schema in Fig.2. Objects in each class are encircled under the name of the class. Denition 1. Let S 0 (V 0 ; E 0 ) be a connected subgraph of class schema S(V; E) and O(v; e) and O 0 (v 0 ; e 0 ) be object graphs of S(V; E) and S 0 (V 0 ; E 0 ), respectively. S 0 (V 0 ; E 0 ) and O 0 (v 0 ; e 0 ) are an class subschema of S(V; E) and an object subgraph of O(v; e), respectively. Note that O 0 is a subgraph of object graph O if O 0 is an object subgraph of O. Objects in object graph O(v; e) and classes in class schema S(V; E) are related by function inst. inst(c) represents all the object in class C. Paths of classes P S = (C 1 ; C 2 ; : : : ; C n ) is a path in the undirected graph that corresponds to class schema S. P is used instead of P S if S is trivial. inst(p S ) is a set of paths p = (o 1 ; o 2 ; : : : ; o n ) where o i 2 inst(c i ) (1 i n). Paths can be represented with lists of vertices assuming that only one edge exists between two adjacent objects.

5 Paul John Mick o1 o2 o3 s o4 o 5 Scene o 6 o7 o 8 Location o 9 o 10 Airport Hotel Fig. 3. Object graph Denition 2. Let S be a class schema and P S = (C 1 ; C 2 ; : : : ; C s ) be a path of classes. A relation of P S, denoted by R(P S ), is relation R(C 1 ; C 2 ; : : : ; C s ). Instances of R(P S ) is a set of tuples (o 1 ; o 2 ; : : : ; o s ) where p = (o 1 ; o 2 ; : : : ; o s ) and p 2 inst(p S ). R(C 1 ; C 2 ; : : : ; C s ) is a relation in the relational model whose attribes and tuples correspond classes and set of objects. Functional dependencies in relation R(P ) are functional dependencies in relation R(C 1 ; C 2 ; : : : ; C s ) in the relational model. R(P)[C i = o] and R(P )[ C k1 ; C k2 ; : : : ; C kt ] denote selection and projection as in the relation model. Let P 0 be a subgraph of P and fc k1 ; C k2 ; : : : ; C kt g be the set of all the classes in P 0. R(P)[P 0 ] denotes projection R(P )[ C k1 ; C k2 ; : : : ; C kt ]. Note that R[P][P 0 ] = R[P 0 ]. Example 4. Let P = (; s; Scene; Location) be a path of classes and functional dependencies Scene! s and Scene! Location be asserted. Table 1 shows an instance of relational schema R(P ). Since functional dependencies Scene! s and Scene! Location hold in R(P ), a key of R(P ) is fscene, g. Table 1. Relation of path (, s, Scene, Location) s Scene Location o 1 o 4 o 6 o 9 o 1 o 4 o 7 o 9 o 2 o 5 o 8 o 10 o 3 o 4 o 6 o 9 o 3 o 4 o 7 o 9 o 3 o 5 o 8 o 10

6 Functional dependencies in R(P ) depend on relationships in class schemas. There is no functional dependency except for the trivial ones in R(C 1 ; C 2 ) where C 1 and C 2 are classes such that C 1! C 2 in class schema S(V; E). Functional dependency C 1!C 2 holds in R(C 1 ; C 2 ) where C 1 and C 2 are classes such that C 1 ;C 2 in class schema S(V; E). Denition 3. Let P be a path of classes and C be a set of all the classes in P. A key of path P, denoted by key(p ), is a minimal subset C 0 of C such that C 0!C holds in R(P ). 3 Materialized Derived Classes in Object Bases When derived classes are materialized to allow fast retrievals, redundancies increase in object bases and costs of updates increase. Example 5. In the class schema of Fig.4, class Presence is added to the class schema in Fig.2. Correspondences of locations and persons are easily retrieved from objects in class Presence. Suppose object o 7 of Scene is deleted. Since navigations from o 7 gives o 4, o 1 (Paul), and o 9 (Airport), the connection between o 1 (Paul) and o 9 (Airport) may be lost after the deletion of o 7. If no connection between o 1 (Paul) and o 9 (Airport) exists, object o 12 of class Presence should be deleted to keep the class schema consistent. However, after navigating from all the objects in class Scene, i.e., o 6, o 7, and o 8, to class and to class Location we nd o 1 (Paul) and o 9 (Airport) is still connected through o 6 after the deletion of o 7. In this case, object o 12 is not deleted. An object derived from an object graph O is assumed to have tuple value < a 1 : id 1, a 2 : id 2, : : :, a s : id m > where id i (1 i m) identies an object in O. A class derived from a class schema S is a class of objects derived from an object graph of S. Denition 4. Let o be an object, C be the class of o, P be a path of classes, and p 2 inst(p ). If every OID that appears as a value of object o is in p, o is an object derived from p, denoted by o < p. If, for every object o that belongs to C, there exists p 2 inst(p ) such that o < p, C is a class derived from P, denoted by C < P. Derived class C can be materialized in the following manner for quick retrievals of objects in C. Given class schema S, path of classes P S and class C + < P S, C + can be added to S by merging S(V; E) and class schema S + (V + ; E + ) that is constructed as follows: (1) Let C + be the class of tuple-value objects o = (id; < a 1 : id 1 ; a 2 : id 2 ; : : : ; a n : id m >). Add C + to V +.

7 s Scene Appearance Place Location Location Presence (a) class schema Paul John Mick o 1 o2 o 3 s o4 o 5 Scene o 6 o7 o 8 Location o 9 o 10 Airport Hotel Presence o o o o (b) object graph Fig. 4. Introduction of materialized derived class and objects (2) Let C i be such a class in S that includes the object identied by id i for every object o in C +. Add an edge C + ;C i to E + for each C i. S 3 (V 3 ; E 3 ) denotes an class schema obtained by merging S(V; E) and S + (V + ; E + ), i.e., V 3 = V [ V + and E 3 = E [ E +. Object graphs corresponding to S + (V + ; E + ) and S 3 (V 3 ; E 3 ) are denoted by O + (v + ; e + ) and O 3 (v 3 ; e 3 ), respectively. Class C + is called a materialized derived class. Let P S be a path of classes, C + < P S be a materialized derived class, S 3 (V 3 ; E 3 ) be a class schema obtained by adding C + to S where S 3 (V 3 ; E 3 ) = S(V; E) [ S + (V + ; E + ). A + = V \ V + is the set of all the classes that belong to both S and S +. Let fp 1 ; P 2 ; : : : ; P n g be the set of all the paths in S +. R(S + ) = R(P 1 ) 1 R(P 2 ) P(P n ). R(S + )[A + ] is a relation on attribes A + which is obtained from class schema S +. R(P S )[A + ] is a relation on attribes A + which is obtained from class schema S. Denition 5. S 3 (V 3 ; E 3 ) is consistent i R(S + )[A + ] = R(P S )[A + ]. Example 6. In Fig.4(a), Presence is a materialized class derived from path (; s; Scene; Location), denoted by Presence < (; s; Scene; Location).

8 Objects in Presence are derived from paths in the undirected graph that corresponds to the object graph in Fig.4(b) as follows: o 11 <(o 3 ; o 4 ; o 6 ; o 9 ) o 11 <(o 3 ; o 4 ; o 7 ; o 9 ) o 12 <(o 1 ; o 4 ; o 6 ; o 9 ) o 12 <(o 1 ; o 4 ; o 7 ; o 9 ) o 13 <(o 3 ; o 5 ; o 8 ; o 10 ) o 14 <(o 2 ; o 5 ; o 8 ; o 10 ) Let S 3 (V 3 ; E 3 ) denote the class schema. S 3 (V 3 ; E 3 ) = S(V; E)[S + (V + ; E + ) where V = f ; s; Scene; Location g, E = f s! ; Scene ; s; Scene ; Locationg, V + = f; Presence; Locationg, and E + = f Presence ; ; Presence ; Locationg. 4 Maintaining Consistencies after Updates When an object is inserted, deleted or modied in class schema S 3 containing materialized derived classes, propagation of updates may occur to keep S 3 consistent after the insertion, deletion or modication of objects. Let C + be a materialized derived class derived from a path of classes P. C + is updated based on preceding updates of P. Processes required to update C + can be performed in three steps, i.e., candidate retrieval, derivation test, and update. Path p inserted into inst(p ) requires derivation of object o < p (candidate retrieval). Also, if o is not in inst(c + ) previously (derivation test), o must be inserted into inst(c + ) (update). To check if o is in inst(c + ), traversal of entire inst(c + ) may be required. Path p deleted from inst(p ) requires derivation of object o < p (candidate retrieval). Also, if p is the only path such that o < p in inst(p ) (derivation test), o must be deleted from inst(c + ) (update). Finding such a path except for p that o < p normally requires much cost since entire inst(p ) are retrieved and object o < p are derived for every path p 2 inst(p ) in the worst cases. In the following part of the paper, we reduce the costs of processes required to maintain consistencies after a deletion of an object. Suppose C + < P is updated. Followings are candidate retrieval and derivation test that are represented as relational expressions. Let object o of class C u in P be inserted, deleted or modied. Candidate retrieval: Obtain R c = 0 R(P )[C u = o] 1 R(S + ) 1 [C + ]. Derivation test: Obtain R c 0 0 R(P )[C u 6= o] 1 R(S + ) 1 [C + ] Following part of this section gives a procedure that is sucient to keep S 3 consistent after a deletion of an object in P. First, P 0 is dened to trim P for C +. Denition 6. Let C + < P be a materialized derived class. If P 0 is the minimal subgraph of P such that satises C + < P 0, P 0 is a corresponding path of C +, denoted by P 0 C +.

9 Only the updates of subgraph P 0 of P aect objects in C +. Lemma 7. Let P 0 C + be a subgraph of P. R(S + )[A + ] = R(P )[A + ] i R(S + )[A + ] = R(P 0 )[A + ]. Proof. Since P 0 is a subgraph of P, R(P 0 ) = R(P )[P 0 ]. Thus, R(P 0 )[A + ] = R(P )[P 0 ][A + ] = R(P)[A + ]. If R(S + )[A + ] = R(P 0 )[A + ], R(S + )[A + ] = R(P )[A + ]. Also, if R(S + )[A + ] = R(P )[A + ], R(S + )[A + ] = R(P 0 )[A + ]. The following procedure updates materialized derived class C + after a deletion of an object in P 0 such that P 0 C +. If V + contains more than one materialized derived classes, apply the procedure to each of the materialized derived class. After applying the procedure, S 3 is consistent. Procedure 8. Let C + < P be a materialized derived class and P 0 C + be a subgraph of P. Given class schema S 3 and deleted object o of class C u in P 0. (1) Candidate Retrieval: Navigate in P 0 to nd a path p 2 inst(p 0 ) that contains o and then navigate from p to obtain a set of objects in C + that are connected to p. The obtained objects are the candidate objects, denoted by o c. (2) Derivation Test: For each o c 2 o c, navigate in P 0 to nd all the paths in inst(p 0 ) that do not contain o and o c < p. (3) Deletion: If no path is found for o c 2 o c in the preceding derivation test, o and o c are deleted. Otherwise, o is deleted. Since navigations in Step 1 is performed with only the paths in inst(p 0 ) that contains o, there is no need to traverse entire inst(p 0 ). However, Step 2 may require to traverse entire inst(p 0 ) in the worst cases. Lemma 9. Let S 3 (V 3 ; E 3 ) be a consistent class schema, C and C + be classes in S +, and P and P 0 be paths of classes where C + < P and P 0 C +. Suppose o 2 inst(c) is to be deleted. S 3 is consistent if Procedure 8 is applied to S 3. Proof. Let R 1 (X) and R 2 (X) be relational schemas and Y be a subset of attribes X. Since R 1 [Y ] (R 1 0 R 2 )[Y ] (R 1 [Y ] 0 R 2 [Y ]), (R 1 0 R 2 )[Y ] = R 1 [Y ] 0 R 2 [Y ] + (R 1 0 R 2 )[Y ]. Let 1R(P 0 ) = R(P 0 )[C = o] be tuples to be deleted in R(P 0 ). After deleting o in R(P 0 ), R(P 0 ) becomes R(P 0 ) 0 1R(P 0 ): Since (R 1 0 R 2 )[Y ] = R 1 [Y ] 0 R 2 [Y ] + (R 1 0 R 2 )[Y ], 0 R(P 0 ) 0 1R(P 0 ) 1 [A + ] = R(P 0 )[A + ] 0 1R(P 0 )[A + ] + (R(P 0 ) 0 1R(P 0 ))[A + ]. In Procedure 8, followings are performed: (1) 1R(P 0 )[A + ] is obtained in Step 1 of Procedure 8. (2) 0 R(P 0 ) 0 1R(P 0 ) 1 [A + ] is obtained in Step 2 of Procedure 8. (3) R(S + )[A + ] is changed to R new (S + )[A + ] where R new (S + )[A + ] = R(S + )[A + ] 0 0 1R(P 0 )[A + ] 0 0 R(P 0 ) 0 1R(P 0 ) 1 [A + ] 1

10 Since S 3 is consistent before the 0 deletion, R(S + 0 )[A + ] = R(P 0 )[A + 1 ]: Then, R new (S + )[A + ] = R(S + )[A + ] 0 1R(P 0 )[A + ] 0 R(P 0 ) 0 1R(P 0 ) [A ] = R(P 0 ) 0 1R(P 0 ) [A + ]. Let R(P 0 )01R(P 0 ) be R new (P 0 ) and R(P )01R(P ) be R new (P ). The above equation is equivalent to R new (S + )[A + ] = R new (P 0 )[A + ]. R new (S + )[A + ] = R new (P )[A + ] follows from Lemma 7. Thus, S 3 is consistent after the deletion if Procedure 8 is applied to S 3. 5 Ecient Deletions of Objects In some cases, derivation test can be omitted in Procedure 8. This section includes a condition satised by class schemas which allow to omit derivation test or to perform derivation test with only navigations on a single relationship. First, an example of a class schema which does not require derivation test is shown. Example 7. Materialized derived objects are deleted when the objects they point to are deleted. Suppose object o 9 is deleted in Fig.4. Objects o 11 and o 12 which point to o 9 are deleted witho derivation test. In the class schema presented in Fig.5, derivation test for derived class 'sscene can be always omitted in applying Procedure 8. Suppose object o 5 is deleted. Navigations from o 5 gives (o 2 ; o 5 ; o 8 ) and (o 3 ; o 5 ; o 8 ) and candidate objects o 17 and o 20 are obtained. Since paths in the instances of path (; s; Scene) and objects in class 'sscene have one-to-one relationship, there is no such a path p in the instances of path (; s; Scene) such that o 17 < p or o 20 < p if o 5 is deleted. Therefore, derivation test is not required to know o 17 and o 20 must be deleted. The following procedure is the same as Procedure 8 except that it does not include derivation test. Procedure 10. (1) Step 1 of Procedure 8 (2) o and every object o c 2 o c obtained in the candidate retrieval are deleted. The next two lemmas show the condition of class schemas for always omitting derivation test. Lemma 11. Let S 3 (V 3 ; E 3 ) = S(V; E) [ S + (V + ; E + ) be a consistent class schema, P and its subgraph P 0 be such paths of classes that C + < P and P 0 C +, and C be a class in P 0. Suppose object o 2 inst(c) is deleted and Procedure 10 is applied to S 3. If functional dependency A +! C holds, S 3 is consistent.

11 s Scene Location Appearance Place Scene 'sscene (a) class schema Paul John Mick o 1 o2 o3 s o 4 o 5 o15 Scene o 6 o 7 o 8 Location o 9 o 10 Airport Hotel o16 o17 o18 o o 'sscene (b) object graph Fig. 5. Class schema containing super-key class Proof. If there exists such a pair (o 1 ; o 2 ) of objects in inst(c) that satises R(P 0 )[C = o 1 ][A + ] = R(P 0 )[C = o 2 ][A + ], A +! C does not hold. Thus, if A +! C, no path is found in derivation test of Procedure 8. Since Procedure 10 is the same as Procedure 8 except that it does not contain a step corresponding to Step 2 of Procedure 8, applying Procedure 10 is identical to applying Procedure 8. From Lemma 9, it follows that S 3 is consistent. Lemma 12. Let S 3 (V 3 ; E 3 ) = S(V; E) [ S + (V + ; E + ) be a consistent class schema, and P and P 0 be such paths of classes that C + < P and P 0 C +. Suppose object o in P 0 is deleted and Procedure 10 is applied to S 3. If A + key(p 0 ), S 3 is consistent. Proof. Let P 0 be (C 1 ; C 2 ; : : : ; C n ). From the denition of key(p 0 ), key(p 0 )! fc 1 ; C 2 ; : : : ; C n g. Since fc 1 ; C 2 ; : : : ; C n g fc i g (1 i n), fc 1 ; C 2 ; : : : ; C n g! C i. Thus, A +!C i (1 i n). From Lemma 11, it follows that as for a deletion of o 2 inst(c i ) (1 i n), S 3 is consistent if we apply Procedure 10 to S 3. In Fig.5, A + = fscene; g and key(p 0 ) = fscene; g. From Lemma

12 12, it follows that the class schema in Fig.5 is consistent if we apply Procedure 10 to the class schema. Let S 3 = S + S + be a class schema, C + i (1 i n) be a class in S +, S 3 i = S + S + i be the class schema obtained by adding C + i to S. We denote by A + i the set of all the classes in both class subschemas S and S + i of S 3 i. Denition 13. Let P 0 C + i (1 i n) be a path of classes in S 3. S 3 (V 3 ; E 3 ) satises the super-key condition i n[ i=1 A + i key(p 0 ) for every P 0 in S 3. If there exists a class C + i such that Att(C + i ) key(p 0 ), C + i is called a super-key class. From Lemma 12, it follows that if S 3 = S [ S + satises the super-key condition and S + contains only a super-key class except for the classes contained in S, S 3 is kept consistent by applying Procedure 10. If S + contains more than one classes that are not contained in S, performing Procedure 10 for classes in S + is not sucient to keep S 3 consistent. Even in this case, Procedure 8 is not required when S + is properly designed. Example 8. In Fig.6, 0 sscene < (; s; Scene; Location) and Presence < (; s; Scene; Location). Since Presence and 'sscene are derived from the same path of classes, an object of Presence can be updated according to objects of 'sscene by navigating on the reference relationship between them. Note that the class schema contains super-key class 'sscene and satises the super-key condition. s Scene Location Appearance Place Scene 'sscene Presence Presence Fig. 6. Class schema allowing ecient deletions Location Let fc + 1 ; C+ 2 ; : : : ; C n + g be a set of classes in S+ (V + ; E + ), and C + 1 be a superkey class. If there exist reference relationships C + 1 ;C + i (2 i n), they can be ilized in candidate retrievals and derivation test. Reference connections of reference relationships C + 1 ;C i + must be double-linked and satisfy relational expression 0 R(P ) 1 R(S + ) 1 [C + 1 ; C+ i ] = R(C+ 1 ; C+ i ).

13 Procedure 14. Let fc + 1 ; C+ 2 ; : : : ; C+ n g be a set of classes in S + (V + ; E + ), C + 1 be a super-key class in fc + 1 ; C+ 2 ; : : : ; C n + g, and C+ 1 ;C i + (1 i n) be reference relationships in S +. (1) Candidate Retrieval: Apply the Step 1 of Procedure 8 to C + 1, and let oc 1 be the set of candidate objects of C + 1. Navigate the links of C+ 1 ;C i + (2 i n) to nd objects linked to an object in o c 1 and let the resulting objects be oc i (2 i n). (2) Derivation Test: Navigate the links of C + 1 ;C + i (2 i n) from oc i (2 i n). Note that the connections of C + 1 ;C i + must be double-linked and 0 R(P ) 1 R(S + ) 1 [C + 1 ; C+ i ] = R(C+ 1 ; C + i ). satisfy (3) Deletion: Delete o c 2 o c i if only one object of C+ 1 is connected to oc. Delete all the object in o c 1 from C+ 1. Delete o from C u. Lemma 15. Let S 3 (V 3 ; E 3 ) = S(V; E) [ S + (V + ; E + ) be a consistent class schema where V + = fc + 1 ; C+ 2 ; : : : ; C+ n g, C V + be a super-key class, C + 1 ;C + i (2 i n) be reference relationships, 0and reference connections of C + 1 ;C i + (2 i n) be double-linked and satisfy R(PS ) 1 1 R(S + ) [C + 1 ; C+ i ] = R(C + 1 ; C+ i ). If object o is deleted in S3 and Procedure 14 is applied to S 3, S 3 is consistent. Proof. Let Si 3 = S [ S i + (1 i n) be the class schema obtained by adding class C i + to S. We prove that every Si 3 (1 i n) is consistent. S 3 is consistent since applying Procedure 14 to 1 C+ i is equivalent to applying Procedure 10 to C +. Applying Step 3 in Procedure 14 to 1 S3 i (2 i n) is the same as applying Step 3 in Procedure 8 to Si 3 (2 i n). In Step 1 of Procedure 14, o c 1 corresponds to tuples obtained by evaluat- 0 ing R c 1 = R(P 0 )[C = 0 o] 1 1 R(S + ) [C + ]. To obtain 1 oc i (2 i n) that corresponds to R c i = R(P 0 )[C = o] 1 1 R(S + ) [C + i ], navigations on C+ 1 ;C + i (2 0 i n) from o c 1 is performed producing pairs of objects corresponding to R c 1 )1 0 R(C + 1 ; C+ i [C i + ]. Since R(C+ 1 ; C+ i ) = R(P 0 ) 1 1 R(S + ) [C + 1 ; C+ i 0 ], R c 1 1 )1 0 R(C + 1 ; C+ i [C + i ] = R c R(P 0 ) 1 ]1 0 R(S + ))[C + 1 ; C+ i [C + i ] = R c 1 1 R(P 0 ) R(S + ) [C + 1 ; C+ i ] [C i + ] = R(P 0 )[C = o] 1 1 R(S+ ) [C + 1 ; C i + ] 0 = R(P 0 )[C = o] 1 1 R(S + ) [C i + ] = 0 Rc i. In Step 2 of Procedure 14, R c i = R(P 0 )[C 6= o] 1 1 R(S + ) [C + i [C + i ] ] (2 i n) need to be obtained. R c 1 = R(P 0 )[C 6= o] 1 R(S 0 + )[C + ] = 1 R(C+ ) 0 1 Rc 1. In this step, processes corresponding to evaluating R c 1 ] are per- 1 )1 R(C + 1 ; C+ i [C i + formed. Since (R(P) 1 0 R(S + ))[C + 1 ; C+ i ] = R(C+ ; C+ i ), R c 1 1 )1 R(C + 1 ; C i + [C i + ] = 0 R c 1 1 (R(P 0 ) 1 ]1 R(S + ))[C + 1 ; C+ i [C i + ] =0 R c 1 1 R(P 0 ) 1 R(S + ))[C + 1 ; C+ i ][C+ i ] = R(P 0 )[C 6= o] R(S + ) [C + 1 ; C+ i ][C i + ] = R(P 0 )[C 6= o] 1 1 R(S + ) [C i + ] = Rc i. Thus, applying Procedure 14 to S 3 is equivalent to applying Procedure 10 to C + and Procedure 1 to each of 1 C+ i (2 i n). In Procedure 8, Steps 1 and 2 require navigations on entire P 0. However in Procedure 14, Steps 1 and 2 require navigations on only a subgraph of P 0. Navigations on class schemas can be classied as follows:

14 (1) Single-step navigation A navigation on a single relationship. (2) Multi-step navigation A navigation on multiple relationships. Theorem 16. Let S 3 (V 3 ; E 3 ) be a consistent class schema where S 3 = S [ S +, fc + 1 ; C+ 2 ; : : : ; C+ n g be the set of all the classes in S +, C + 1 be a super-key class in S +, C + 1 ;C i + (2 i n) be reference relationships and reference connections of C + 1 ;C + i (2 i n) be double-linked and satisfy 0 R(P ) 1 R(S + ) 1 [C + 1 ; C+ i ] = R(C + 1 ; C+ i ). Suppose object o is deleted in S 3. No multi-step navigation is required in the derivation test to keep S 3 consistent. Proof. From Lemma 15, it follows that S 3 is consistent if Procedure 14 is applied to S 3. Clearly, Procedure 14 requires no multi-step navigation in the derivation test. Although candidate retrieval requires multi-step navigations, only objects connected to o need to be retrieved. Since the class schema in Fig.6 contains a super-key class 'sscene and Presence is referenced by class 'sscene, no multi-step navigation is required to keep the class schema consistent if the reference connections between 'sscene and Presence are double-linked. 6 Design of Object Bases Allowing Ecient Updates This section gives a procedure to design object bases that allow ecient retrievals and updates. Generally, objects in object bases are retrieved quickly by introducing redundancies into object bases. On the other hand, to reduce overheads to update redundant objects, redundancies should be reduced. We introduce more redundancies into object bases to reduce overheads to update redundant objects. Let S 3 = S [ S + be a class schema that contains materialized derived classes. When S 3 does not satisfy the super-key condition, multi-step navigations are required for each materialized derived class in S +. The next procedure transforms class schemas to satisfy the super-key condition and thus allow single-step navigations in keeping S 3 consistent after a deletion of an object. Procedure 17. Given a class schema S 3 (V 3 ; E 3 ), (1) Let C + 1 ; C+ 2 ; : : : ; C+ n be classes in S+ that have tuple-values and Si 3 be a object schema obtained by adding C i + to S. Obtain path Pi 0 C i + for 1 i n. Let the set of all path Pi 0 be fp k 0 1 ; Pk 0 2 ; : : : ; Pk 0 m g. (2) Let Ck K 1 ; Ck K 2 ; : : : ; Ck K m be super-key classes of Pk 0 1 ; Pk 0 2 ; : : : ; Pk 0 m, i.e., Att(Ck K i ) key(pk 0 i ). Construct an class schema from S 3 and Ck K 1 ; Ck K 2 ; : : : ; Ck K m. (3) Add reference relationship Ck K i ;C + k if j Pk 0 i C + k. Note that reference con- j nection of C K k i ;C + k j must be double-linked and satisfy C + i ] = R(C+ 1 ; C+ i ). 0 R(P) 1 R(S + ) 1 [C + 1 ;

15 (4) Let the resulting class schema be S 0 (V 0 ; E 0 ). Theorem 18. Let S 0 be a consistent class schema transformed by Procedure 17. No multi-step navigation is required to keep S 0 consistent after a deletion of object o in P 0 i. Proof. Let Sk 3 i = S [ S + k i (1 i m), fc + 1 ; C+ 2 ; : : : ; C s + g be the set of all the classes in S + k i where C + j < Pk 0 i for (1 j s), and Ck K i be super-key class in S + k i. Procedure 17 add Ck K i if it does not exist in fc + 1 ; C+ 2 ; : : : ; C+ s g and adds reference relationships Ck K i ;C j (1 j s, C j 6= Ck K i ), whose connection are double-linked. From Theorem 16, it follows that Sk 3 i (1 i m) is consistent after the deletion if Procedure 17 is applied to S 3. Procedure 17 may increase the costs in keeping S 3 consistent after an insertion of an object. However, in general, the increase of the costs of an insertion can be ignored compared to the decrease of the costs of a deletion. Costs of retrievals in S 3 is not increased by Procedure 17. Theorem 19. Let S 0 be a class schema transformed by Procedure 17, O 3 (v 3 ; e 3 ) and O 0 (v 0 ; e 0 ) be object graphs of class schema S 3 (V 3 ; E 3 ) and S 0 (V 0 ; E 0 ), respectively. If O(v; e) is an object subgraph of O 3 that is accessed to perform retrieval query Q in S 3, O(v; e) is also an object subgraph of O 0 (v 0 ; e 0 ). Proof. Since Procedure 17 either adds classes or reference relationships to S 3 to obtain S 0, S 0 subsumes S 3. Thus, an object subgraph of S 3 is also an object subgraph of S 0. Theorem 19 implies that retrieval query Q can be performed in the same object subgraph of both S 3 and S 0. Example 9. The class schema in Fig.6 is transformed from the class schema in Fig.4 by Procedure 17. Connection of objects between classes 'sscene and Presence is shown in Fig.7. 'sscene is a super-key class of the class schema in Fig.6. 'sscene o 15 o 16 o 17 o 18 o 19 o 20 o o o o Presence Fig. 7. Reference connection for ecient candidate retrieval and derivation test

16 Suppose object o 7 is deleted. Navigations from o 7 gives fo 7 ; o 4 ; o 3 g and fo 7 ; o 4 ; o 1 g. o 16 and o 19 are candidate objects in 'sscene since they refer fo 7 ; o 1 g and fo 7 ; o 3 g, respectively. Also, o 12 and o 11 are candidate objects in Presence since they refer o 16 and o 19, candidate objects in the super-key class 'sscene. o 16 and o 19 is deleted witho derivation test. o 12 and o 11 are not deleted from Presence since navigations from o 12 and o 11 in Presence give o 15 and o 18, respectively. Suppose object o 21 referencing o 4 and o 9 is inserted in class Scene. Navigations from o 21 gives fo 21 ; o 4 ; o 3 ; o 9 g and fo 21 ; o 4 ; o 1 ; o 9 g. First, o 22 referencing o 21 and o 3 and o 23 referencing o 21 and o 1 are inserted in class 'sscene. Since there exist object o 11 referencing o 3 and o 9 and object o 12 referencing o 1 and o 9, no object is inserted in class Presence. Retrievals of corresponding persons and locations in Fig.6 is performed in the same manner as in Fig.4 using class Presence. 7 Conclusion If objects derived from several paths of objects are materialized in object bases, elimination of duplicates is required in performing incremental updates of materialized derived classes as is required in incremental updates of materialized views produced by relational expressions that include projections. The elimination of duplicates complicates incremental updates. Roussopoulos[10] decided not to discuss projections in incremental maintenance of ViewCache. Also, Qian and Wiederhold[9] pointed o the diculty of the project operators for incremental recompation of views. In this paper, we have described the super-key condition that allows ecient maintenance of materialized derived classes. Duplicates can be eciently eliminated in performing incremental updates if object bases satisfy the super-key condition. We began by dening our model of object bases introducing class schemas, object graphs and functional dependencies de- ned on relations of connected objects. Class schemas always keep the condition of consistencies if we perform Procedure 8 that incrementally updates materialized derived classes after every update. Derivation test in Procedure 8, which requires to obtain inst(p 0 ) for every candidate object in some cases, is omitted or performed as single-step retrievals if class schemas satisfy the super-key condition and contain super-key classes. Finally, we presented a procedure that transforms class schemas to satisfy the super-key condition. Although it might be possible to dene derived set-value objects and derived basic-value objects, only tuple-value objects are dened as materialized derived objects to simplify our discussion. The idea of the paper can be applied to derived set-value and derived basic-value objects with small modications. It calls for further discussion of ISA relationships. Implementations of ISA relationships may aect eciencies of updates of materialized derived classes in object bases. For example, objects of superclasses may have to be retrieved from their subclasses in candidate retrieval and derivation test. Implementation related issues include directions of reference relationships, available access paths, e.g., pointers,

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