CHAPTER-2 A GRAPH BASED APPROACH TO FIND CANDIDATE KEYS IN A RELATIONAL DATABASE SCHEME **

Transcription

1 HPTR- GRPH S PPROH TO FIN NIT KYS IN RLTIONL TS SHM **. INTROUTION Graphs have many applications in the field of knowledge and data engineering. There are many graph based approaches used in database management system, ranging from -R modeling to database schema design and normalization [8][0][5][7][8] [4][5]. Functional dependencies among the entities of a context problem area can be easily represented as a graph called the functional dependence graph (FG). part from database technology, FGs have many applications in Web data analysis [], Language processing [], ata cleaning [6], Network Theory [7] and ata Mining [9]. There are several types of generalized graphs such as F graph, Implication graph, and eduction graphs, etc. used to study functional dependencies. Hyper graph is another generalized graph that can be used to study functional dependencies []. There are many algorithms for finding all candidate keys of a relational scheme based on Karnaugh map [3], oolean matrix [4] and other set theoretic approaches [0]. Hossein Saiedian, et.al, described a method for finding the candidate keys in a relational scheme [9]. In their approach, the functional dependencies among the attributes in a database table can be represented as a graph. Such graphs are also called attribute graphs. They use the attribute graph for analyzing the dependencies to find the candidate keys. Their approach cannot find all the candidate keys of a relational scheme ** This chapter has been published as a research paper in the journal International Journal of omputer ngineering &Technology (IJT), Vol-4, issue- 6, pp.9-3, 04

2 if the F graph is strongly connected. Their approach needs a minimal set Fs to compute the candidate keys. In this chapter, a generalized graph based method to find all candidate keys in a relational database scheme is presented.. FUNTIONL PNNIS N THIR GRPHS Let be a functional dependency, where, are sets of attributes. epending on the cardinality of and the following graph patterns are obtained for the F... If and both are singular, i.e., = = then the graph pattern shown in fig..(a) is obtained... If = and > and = {,, n, then the graph pattern shown in fig..(b) is obtained...3 a) If > and = and ={,, n then the graph pattern shown in fig..(c) is obtained b) If > and > and ={,, n and ={,, n,then the graph pattern shown in fig..(d) is obtained. (a): Graph for n (b): Graph for.. n n (c): Graph for (d): Graph for n n, n n :onnector Node n Figure.: Functional ependencies and Their Graphs

3 Note that there are dependencies from a connector node to all components (nodes) of it. In fig., the following dependencies are also existed (,,.., n )..3 FUNTIONL PNNY GRPHS (FG) FOR RLTIONL TS SHM The functional dependency graph (FG) for a relational scheme is drawn based on the set of functional dependencies available in the relational scheme using the rules..,.., and..3. One node is created for each attribute in the relation scheme. connector-node is created each time the Rule..3 is applied, i.e. for each distinct nonsingular attribute set which is in the left-hand side of a F. Therefore, for a relational scheme R with N attributes with a set of Fs F and having M distinct non-singular attribute sets in the left-hand side of the Fs in F there will be N+M nodes in the FG. For example, for a relational scheme R= (,,,, ) with the set Fs F= {,,, a FG, as shown in fig.., with a total of 6 nodes is obtained. Figure.: F Graph for a relational scheme R= (,,,, ) with the set Fs F= {,,,.4 FINING TH NIT KYS USING FG To find the candidate keys from the FG, the following transformations are performed in the FG.

4 T: If there is a dependency c i c j, c i, c j { then remove c j from {. elete the connector node from the G if { is singular. ssign all edges of to c. The transformation process has two phases- ugmentation and Reduction. fter the transformation, the resultant graph contains only the candidate nodes. candidate node is a node which alone or along with some other candidate nodes, can determine all other nodes in the graph. Therefore the resultant graph is called the andidate Graph (G ). So, the problem now is to find the candidate graph from an FG. Therefore the transformation process can be represented as follows: G G G..4. ugmentation: Finding G. In the augmentation phase new connector nodes and edges are added to the FG by performing the following transformations. T: Let there exists a connector-type node which connect the node set ={c i,,<i n, and has an outgoing edge to a node. Let there exists a node in the FG from which there is a path to any node c i in. S-I: If there is a path from to c i, i, i n,then an edge from to is added where n is the cardinality of the set. S-II: If there is a path from to a set {c k, <i<n and c k <n where n is the cardinality of, then for each c k,a new connector node is created connecting node and the nodes in the set {- c k and an edge from to is added. Fig..3 shows the transformation process. of the FG for a relational scheme R= (,,,, ) with the set Fs F= {,,,. Node has a path to both components and of the set represented by the connector node, so

5 an edge from to the connector node is added due to case-i of the transformation T. Node has a path to only one component of the set represented by the connector node, so a new connector node is created to connect node and node and an edge from to the is added due to case-ii of the transformation T. (G) (G + ) : dge due to T (case-i) : dge due to T (case-ii) : Node created due to T (case-ii) Figure.3: Transformation T (ugmentation).4. Reduction: Finding G. In the reduction phase certain nodes and edges are deleted from the augmented FG by performing the following transformations. T3: Let be a connector node in G. If there is a node in G which has a direct edge to node, then delete all edges from to the first node in the paths to any c i. T4: elete a node N from the augmented FG G if its in-degree >0 and out-degree =0. In fig..4(b), first node is deleted because its in-degree is non zero and outdegree is zero. eletion of this node causes the deletion of the nodes and from the graph. T5: elete a cycle K from G if in-degree (K) >0 and out-degree (K) =0.

6 (a): xample of Transformation T3 3 (b): eletion of nodes,, Using Transformation rule T4 F F G G (i) (ii) (c) : ycle deletion using Transformation T5 Figure.4: Transformations in Reduction Process The following example in fig..5 explains the transformation process. Let R= (,,,, ) be a relational scheme with the set Fs F= {,,,. With this set of Fs F, the F graph G is obtained by using rule.,. and.3 discussed in section- of this paper. fter the transformation process the candidate graph G is obtained. From G it is clear that there is cycle having all the nodes in the graph. The candidate nodes are,, and. Therefore R= (,,,, ) with the set of Fs F= {,,, has four candidate keys-,,, and. (G) T (G + ) T3 T4 (G ) Figure.5: The Transformation Process to find the candidate keys of relational scheme R= (,,,, ) with the set Fs F= {,,,.

7 .5 FINITION N THORMS The following definition and theorems are presented with respect to the andidate graph G..5. efinition : node in a cycle in a candidate graph G of a relational scheme R is either a candidate key or a part of the candidate keys of R..5. Theorem : G=G iff one of the following conditions is satisfied. i. F= ii. G has a cycle having all the nodes in the cycle and there is no connector node. Proof: Let R= {,, n ) be a relational scheme with a set of Fs, F. Let G be the F graph for R. ase (i): If F=, i,j, i j, i j F, so the FG G has n isolated nodes and G remains unchanged during the ugmentation process and the Reduction process. Therefore, G=G (q..) G + =G (q..) Hence G=G (from q.. and q..) ase (ii): Now if F and there is a chain i,j, i j, i j i then no new connector nodes can be added during the ugmentation process, which gives. G=G (q..3) Similarly, no node of the graph G + is deleted during the reduction process, because there exists no node α i, i n such that in-degree(α i )>0 and out-degree(α i )=0. Hence, G + =G (q..4) Therefore, from q..3 & q..4, it is found that G=G //

8 .5.3 Theorem : In a candidate graph G, a candidate node cannot have edges if the node does not belong to a cycle. Proof: Let be the only candidate node in the candidate graph, G. So, is an isolated node. Therefore, In-degree () =Out-degree () = (q..5) Now, let there are two candidate nodes and in G. Here, two possibilities are available: a) Node and node both may be isolated or b) and are connected. In case (b) it is found that, if in-degree () = and out-degree () =0 then node cannot be survived in G due to transformation T4. Similarly, if in-degree () = and outdegree () =0 then node cannot survive in G due to transformation T4. Therefore, both nodes and can survive in G if they form a cycle. in-degree ()=out-degree()= in-degree()= out-degree()= (q..6) Therefore, nodes and can have edges if they are in a cycle. Otherwise they are isolated nodes. //.5.4 Theorem 3: If the andidate graph, G, of a relational scheme R has only one cycle and if all the nodes of G are in the cycle, then every attribute of R is a candidate key. Proof: Let V V V 3.V n- V n- V n V is the cycle in the andidate graph G with n vertices. In this chain, from any node V i there is a path to any node V j, j i and < j n by transitivity rule. Therefore, from any V i, < i n, all other nodes V j,< j n and j i are reachable, where n is the degree of R.

9 (a) (b) Figure.6: yclic andidate Graphs having only one cycle In fig..6(a), nodes,, both are candidate nodes. From node, node is reachable and from node, is reachable. In fig..6(b), nodes,,, and all are candidate nodes. From node, nodes,, are reachable i.e., node can functionally determine,, and. So, node is a candidate key. Similarly, from node, nodes,, are reachable i.e., node can functionally determine,,. So, node is also a candidate key. //.5.5 Theorem 4: If the andidate graph G has disconnected components, then the artesian product of the candidate nodes of each component is the set of all candidate keys. Proof: Let us consider a relation scheme R= {,, n ) with a set of Fs, F=. Since F=, there are n disjoint components in the G for R (by Theorem.4.4). The candidate key k for R is the artesian product all attributes of the relational scheme. Therefore, k= n (q..7) Now, let F, then a candidate graph G for R is obtained. Let be the set of m<n (since F, some nodes will be deleted or there will be at least one cycle in G ) disjoint components in G c such that = {{c, {c,., {c m. The component c i can be considered as a set. y Theorem-.4.4, if the candidate graph contains disjoint components then they must be either isolated nodes or cycles. If these disjoint components are considered as nodes, then the set of functional dependencies F available

10 among these components, is null, i.e. F=. Therefore, by q.-.7, it is obvious that, the set of candidate keys K= {{c {c. {c m. // c c c c 4 (a) c 3 (b) Figure.7: andidate graphs with isjoint omponents c In fig..7, an FG with disjoint components is shown. In fig..7(a) {{ { { { = { is the candidate key. ut in fig..7(b), {{ {,, = {{, {, { is the set of candidate keys..5.6 Theorem 5: relational scheme R has multiple candidate keys if and only if the andidate graph G for R contains at least one cycle. Proof: Let G c be the candidate graph for a relational scheme R, with n components,, n. Let K be the set of candidate keys of R. ach component i can be considered as a set of nodes. y Theorem.4.3 K={ n.. (q..7) Now by definition the following equation is obtained. k > if k is a cycle. (q..8) From equation 3, K = if i, i n, i =. (q..9) K > if k, k n k > (q..0) From q.-.9 and q.-.0, it is clear that K is non- singular if and only if there exists at least one cyclic component k, k n, in G c. // Figure-.8 shows all possible cyclic candidate graphs for a relational scheme R= (,,, ).

11 (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) (m ) (n) Figure.8: andidate Graphs for R= (,,, ) having ycles.6 NLYSIS OF TH PPROH.6. Out Line Of The pproach In the approach, a candidate graph G is obtained from an FG G by using the transformation rules described here. ut G cannot be obtained from G since all functional dependencies except the dependencies among the candidate nodes are deleted, i.e., G does not include all dependencies which imply the original set of Fs F. Though G cannot be obtained from G, the transformations applied in the FG drawn from the set of Fs F do not generate any extra dependencies that are not members of the closure of F, F. Moreover, in this graph based approach, it is not required to calculate F, the closure of F and F, the canonical cover of F. The approach can be implemented by the following algorithms. The outline of the approach is shown in fig..9. raw the FG (G) from the F set ugment G by adding onnector Nodes. G G Reduce G to get G G G Figure.9: Outline of the pproach

12 .6. Procedure Make-FG=(R(,, n ),,F,G) { G= ; For each R create a node for in G; If ( F ) then { For all ( ) F { If ( = N =) then dd an edge from node to in G; If ( = N ) then For all b add an edge from node to b in G { If ( > N ) then { reate a new connector node to connect all { ; Put an undirected edge from all a { to ; dd an edge from node to each b { ; elete the edge from node to node from F; Figure.0: lgorithm to construct the FG.6.3 Procedure ugment-g (graph G, graph G + ) { G + =G; For all node G +, { and for all nodes c i { { If there is a path from to c i then dd an edge from to ; Repeat { For all node G +, { and for all nodes c i { If there is a path from to c i then reate a connector node to group node and nodes {-{c i add an edge from to ; Until no more connector node can be created; Figure.: lgorithm for the ugmentation Process of F Graph G

13 .6.4 Procedure Reduce-G (graph G +, graph G ) { G =G + ; For each connector node G { If ( { =) then { elete node from G ; ssign all edges of to c {; For each connector node G { If the node c { has no edges except the connecting edge then elete node c from G ; lse elete the connecting edge between c and ; For all node which has a direct edge to in G { elete the edge from to the path to c i {; For all ycle K G { Remove all redundant edges from K;/Removal does not break the cycle/ While (( node G ) N (in-degree()>0 N out- degree()=0)) elete node from G ; For all ycle K G { If (in-degree(k)>0 N out- degree(k)=0) then elete cycle K from G ; Figure.: lgorithm for Reduction Process of the ugmented FG G to G.7. xplanatory xamples..7. xample: Problem: Find the candidate keys of the relation schema R= (,,, G, H, I) with the set of functional dependencies F= {,, G H, G I, H

14 Solution: From the problem statement it is found that, the FG G has 7 vertices,,, G, H, I and G as shown in fig..3. From fig..3 it is found that the candidate node for R= (,,, G, H, I) with the set of functional dependencies F= {,, G H, G I, H is the connector node. Therefore the candidate key of R is { = (G). I G G H (G) (G + ) I H G H I G I H (G ).7. xample: Problem: Find the candidate keys of the relation scheme R= (,,,,, F, G) with the set of functional dependencies F= {,,, F, F G, G Solution: Figure.3: Finding candidate nodes for R= (,,, G, H, I) with the set of functional dependencies F= {,, G H, G I, H From the problem statement, the FG G with 3 vertices,,, G,, F, G and,,,,f,g, as shown in fig..4 is obtained. There are following six connector nodes in the FG: ={, ={, 3={, 4={, 5={F, 6={G. For node : Since so is removed from.

15 For node 3: Since so is removed from 3. For node : Since and so an edge from to node is added. For node 4, the connector node 7= { and 8= { are added to the graph. For node 6, the node 9= {G is added to the graph G F G F (G) (G ) F G 6 6 Reduction (Node eletion) Reduction (ycle eletion) Graph G ) andidate Graph G Figure.4: Finding the candidate keys of the relation scheme R= (,,,,, F, G) with the set of Fs F= {,,, F, F G, G

16 y doing so, the augmented graph G is obtained. fter the reduction process the andidate graph G is obtained from which it is decided that the relation schema R=(,,,,,F,G) with the set of Fs F={,,,, F,F G,G has four candidate keys- G,,G and..8 ONLUSION In this chapter, a simple graph based method to find candidate keys of a relation scheme is described. This method takes an FG, drawn based on the set of Fs available in a relational scheme, as an input and using five very simple graph transformations the input FG is converted into a candidate graph (Transformed FG). This candidate graph gives all the candidate keys of the relation scheme. This approach does not require computation of F, F c and, the closure, canonical cover of F and closure of an attribute set. This is a purely graph based approach. This approach can also be used to find the candidate factors in the problem areas like medical diagnosis, social problems where there are many factors with cause-effect dependencies among them.