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FUNCTIONAL DEPENDENCIES Definition • Let R be the relation, and let x and y be the arbitrary subset of the set of attributes of R. Then we say that Y is functionally dependent on x – in symbol. XY (Read x functionally determines y) – If and only if each x value in R has associated with it precisely one y value in R In other words Whenever two tuples of R agree on their x value, they also agree on their Y value. Deepak Gour, Faculty – DBMS, School of Engineering, SPSU Example (SCP Relation) S# City P# QTY S1 London P1 100 S1 London P2 100 S2 Paris P1 200 S2 Paris P2 200 S3 Delhi P2 300 S4 Kolkata P2 400 S4 Kolkata P2 400 S4 Kolkata P5 400 Deepak Gour, Faculty – DBMS, School of Engineering, SPSU Example (SCP Relation) (contd..) One FD : - ( { S#} {City}) • Because every tuple of that relation with a given S# value also has the same city value. • The left and right hand side of an FD are sometimes called determinant and the dependents respectively. Deepak Gour, Faculty – DBMS, School of Engineering, SPSU Exercise Check whether following relation satisfy FD as not • < S#, P# > <QTY> • <S#, P#> <City> • < S#, P#> <City, QTY> • <S#, P#> <S#> • <S#, P#> <S#, P#, QTY, City> • <OTY> <S#> Deepak Gour, Faculty – DBMS, School of Engineering, SPSU Extended definition over basic one • Let R be the relation variable, and let x and y be arbitrary subset of the set of attributes of R. Then we says that Y is functionally dependent on x – in symbol. XY (Read x functionally determines y) • If and only if, in every possible legal value of R, each x value has associated with it precisely one y value Or in other words • In every possible legal value of R, whenever two tuple agree on their X values, they also agree on their Y value. Deepak Gour, Faculty – DBMS, School of Engineering, SPSU TRIVIAL & NON-TRIVIAL DEPENDENCIES • One-way to reduce the size of the set of FD we need to deal with is to eliminate the trivial dependencies. • An FD is trivial if and only if the right hand side is a subset of the left hand side. e.g. <S#, P#> <S#>. (Trivial) • Nontrivial dependencies are the one, which are not trivial. Deepak Gour, Faculty – DBMS, School of Engineering, SPSU CLOSURE of a set of dependencies • The set of all FDs that are implied by a given set S of FDs is called the closure of S, denoted by S+ • So we need an algorithm which compute S+ from S. Deepak Gour, Faculty – DBMS, School of Engineering, SPSU Algorithm CLOSURE (Z, S): = Z; DO “ forever” For each FD X -> Y in S Do; if X < CLOSURE [Z, S] /* <= “subset of” */ then CLOSURE [Z,S] : = CLOSURE [Z, Z] U Y; end; If CLOSURE [Z, S] did not change on this iteration. Then leave loop; /* Computation complete */ End; Deepak Gour, Faculty – DBMS, School of Engineering, SPSU Example Suppose use are given R with attributes A, B, C, D, E, F, and FDs • A BC • E CF • BE • CD EF Then compute the closure (A, B)+ of the set of attributes under this set of FD’s Deepak Gour, Faculty – DBMS, School of Engineering, SPSU Solution 1. We initialize the result CLOSURE [Z, S] to <A, B> 2. We now go round the inner loop four times, once for each for the given FDs. An the first iteration (For FD A BC), we find that the left hand side is indeed a subset of CLOSURE (Z, S) as computed so for, so we add attributes (B and C) to the result. CLOSURE [Z, S] is now the set <A, B, C>. Deepak Gour, Faculty – DBMS, School of Engineering, SPSU Solution 3. On the second iteration (for FD E CF>. we find that the left hand side is not a subset of the result as computed so for, which than remain unchanged. 4. On the third iteration (For FD B E), we add E to the closure, which now has the value <A, B, C, E> 5. On the fourth iteration, (for FD CD EF), remains unchanged. Deepak Gour, Faculty – DBMS, School of Engineering, SPSU Solution 6. Inner loop times, on the first iteration no change, second, it expands to <A,B, C, E, F> third & fourth, no change. 7. Again inner loop four times, no change, and so the whole process terminates. Deepak Gour, Faculty – DBMS, School of Engineering, SPSU Armstrong rules 1. Reflexivity: if B is a subset of A, then A B. 2. Augmentation: if A B then AC BC 3. Transitivity: it A B and B C then A C. 4. Self – determination: A A. 5. Decomposition: If A BC, then AB, AC. 6. Union: it A B and A C, then A BC 7. Composition: if A B, C D then AC BD. 8. If A B and C D, then All (C – B) BD Deepak Gour, Faculty – DBMS, School of Engineering, SPSU Armstrong rules (contd..) • Now we define a set of FD to be irreducible as minimal; if and only if it satisfies the following two properties. (1) The right hand side of every FD in S involve just one attribute (i.e., it is a singleton set) (2) The left hand side of every FD in S is irreducible in turn meaning that no attribute can be discarded from the determinant without changing the CLUSURE S+. Deepak Gour, Faculty – DBMS, School of Engineering, SPSU Example • A BC, • BC • AB • AB C • AC D Compute an irreducible set of FD that is equivalent to this given set. Deepak Gour, Faculty – DBMS, School of Engineering, SPSU Solution (1) The step is to rewrite the FD such that each has a singleton right hand side. • AB • AC • BC • AB • AB C • AC D We observe that the FD A B occurs twice. So one occurrence will be eliminated. Deepak Gour, Faculty – DBMS, School of Engineering, SPSU Solution 2. Next, attributed C can be eliminated from the left hand side of the FD AC D • Because we have A C, • By augmentation A AC • And we are given AC D, • So A D by transitivity; Thus C on the left hand side is redundant. Deepak Gour, Faculty – DBMS, School of Engineering, SPSU Solution 3. Next, we observe that the FD AB C can be eliminated, because again we have AC By augmentation AB CB By decomposition AB C 4. Finally, the FD A C is implied by the FD A B and B C, so it can be eliminated. Now we have A B BC AD This set is irreducible. Deepak Gour, Faculty – DBMS, School of Engineering, SPSU Example • A BC • BE • CD EF Show that FD AD F for R. Deepak Gour, Faculty – DBMS, School of Engineering, SPSU Solution 1. A BC (given) 2. A C (1, decomposition) 3. AD CD (2, augmentation) 4. CD EF (given) 5. AD EF (3 & 4, transitivity) 6. AD F (5, decomposition Deepak Gour, Faculty – DBMS, School of Engineering, SPSU Normalization Learning Objectives • Definition of normalization and its purpose in database design • Types of normal forms 1NF, 2NF, 3NF, BCNF, and 4NF • Transformation from lower normal forms to higher normal forms • Design concurrent use of normalization and E-R modeling are to produce a good database design • Usefulness of denormalization to generate information efficiently Deepak Gour, Faculty – DBMS, School of Engineering, SPSU Normalization • Main objective in developing a logical data model for relational database systems is to create an accurate representation of the data, its relationships, and constraints. • To achieve this objective, must identify a suitable set of relations. Deepak Gour, Faculty – DBMS, School of Engineering, SPSU Normalization • Four most commonly used normal forms are first (1NF), second (2NF) and third (3NF) normal forms, and Boyce–Codd normal form (BCNF). • Based on functional dependencies among the attributes of a relation. • A relation can be normalized to a specific form to prevent possible occurrence of update anomalies. Deepak Gour, Faculty – DBMS, School of Engineering, SPSU Normalization • Normalization is the process for assigning attributes to entities – Reduces data redundancies – Helps eliminate data anomalies – Produces controlled redundancies to link tables • Normalization stages – 1NF - First normal form – 2NF - Second normal form – 3NF - Third normal form – 4NF - Fourth normal form Deepak Gour, Faculty – DBMS, School of Engineering, SPSU Data Redundancy • Major aim of relational database design is to group attributes into relations to minimize data redundancy and reduce file storage space required by base relations. • Problems associated with data redundancy are illustrated by comparing the following Staff and Branch relations with the StaffBranch relation. Deepak Gour, Faculty – DBMS, School of Engineering, SPSU Data Redundancy Deepak Gour, Faculty – DBMS, School of Engineering, SPSU Data Redundancy • StaffBranch relation has redundant data: details of a branch are repeated for every member of staff. • In contrast, branch information appears only once for each branch in Branch relation and only branchNo is repeated in Staff relation, to represent where each member of staff works. Deepak Gour, Faculty – DBMS, School of Engineering, SPSU Update Anomalies • Relations that contain redundant information may potentially suffer from update anomalies. • Types of update anomalies include: – Insertion, – Deletion, – Modification. Deepak Gour, Faculty – DBMS, School of Engineering, SPSU Functional Dependency • Main concept associated with normalization. • Functional Dependency – Describes relationship between attributes in a relation. – If A and B are attributes of relation R, B is functionally dependent on A (denoted A B), if each value of A in R is associated with exactly one value of B in R. Deepak Gour, Faculty – DBMS, School of Engineering, SPSU Functional Dependency • Property of the meaning (or semantics) of the attributes in a relation. • Diagrammatic representation: Determinant of a functional dependency refers to attribute or group of attributes on left-hand side of the arrow. Deepak Gour, Faculty – DBMS, School of Engineering, SPSU Example - Functional Dependency Deepak Gour, Faculty – DBMS, School of Engineering, SPSU Functional Dependency • Main characteristics of functional dependencies used in normalization: – have a 1:1 relationship between attribute(s) on left and right-hand side of a dependency; – hold for all time; – are nontrivial. Deepak Gour, Faculty – DBMS, School of Engineering, SPSU Functional Dependency • Complete set of functional dependencies for a given relation can be very large. • Important to find an approach that can reduce set to a manageable size. • Need to identify set of functional dependencies (X) for a relation that is smaller than complete set of functional dependencies (Y) for that relation and has property that every functional dependency in Y is implied by functional dependencies in X. Deepak Gour, Faculty – DBMS, School of Engineering, SPSU The Process of Normalization • Formal technique for analyzing a relation based on its primary key and functional dependencies between its attributes. • Often executed as a series of steps. Each step corresponds to a specific normal form, which has known properties. • As normalization proceeds, relations become progressively more restricted (stronger) in format and also less vulnerable to update anomalies. Deepak Gour, Faculty – DBMS, School of Engineering, SPSU Relationship Between Normal Forms Deepak Gour, Faculty – DBMS, School of Engineering, SPSU Unnormalized Form (UNF) • A table that contains one or more repeating groups. • To create an unnormalized table: – transform data from information source (e.g. form) into table format with columns and rows. Deepak Gour, Faculty – DBMS, School of Engineering, SPSU First Normal Form (1NF) • A relation in which intersection of each row and column contains one and only one value. Deepak Gour, Faculty – DBMS, School of Engineering, SPSU UNF to 1NF • Nominate an attribute or group of attributes to act as the key for the unnormalized table. • Identify repeating group(s) in unnormalized table which repeats for the key attribute(s). Deepak Gour, Faculty – DBMS, School of Engineering, SPSU UNF to 1NF • All key attributes defined • No repeating groups in table • All attributes dependent on primary key Deepak Gour, Faculty – DBMS, School of Engineering, SPSU Second Normal Form (2NF) • Based on concept of full functional dependency: – A and B are attributes of a relation, – B is fully dependent on A if B is functionally dependent on A but not on any proper subset of A. • 2NF - A relation that is in 1NF and every non- primary-key attribute is fully functionally dependent on the primary key (no partial dependency) Deepak Gour, Faculty – DBMS, School of Engineering, SPSU 1NF to 2NF • Identify primary key for the 1NF relation. • Identify functional dependencies in the relation. • If partial dependencies exist on the primary key remove them by placing them in a new relation along with copy of their determinant. Deepak Gour, Faculty – DBMS, School of Engineering, SPSU 2NF Conversion Results Figure 4.5 Deepak Gour, Faculty – DBMS, School of Engineering, SPSU Third Normal Form (3NF) • Based on concept of transitive dependency: – A, B and C are attributes of a relation such that if A B and B C, – then C is transitively dependent on A through B. (Provided that A is not functionally dependent on B or C). • 3NF - A relation that is in 1NF and 2NF and in which no non-primary-key attribute is transitively dependent on the primary key. Deepak Gour, Faculty – DBMS, School of Engineering, SPSU 2NF to 3NF • Identify the primary key in the 2NF relation. • Identify functional dependencies in the relation. • If transitive dependencies exist on the primary key remove them by placing them in a new relation along with copy of their determinant. Deepak Gour, Faculty – DBMS, School of Engineering, SPSU 3NF Conversion Results • Prevent referential integrity violation by adding a JOB_CODE PROJECT (PROJ_NUM, PROJ_NAME) ASSIGN (PROJ_NUM, EMP_NUM, HOURS) EMPLOYEE (EMP_NUM, EMP_NAME, JOB_CLASS) JOB (JOB_CODE, JOB_DESCRIPTION, CHG_HOUR) Deepak Gour, Faculty – DBMS, School of Engineering, SPSU General Definitions of 2NF and 3NF • Second normal form (2NF) – A relation that is in 1NF and every non- primary-key attribute is fully functionally dependent on any candidate key. • Third normal form (3NF) – A relation that is in 1NF and 2NF and in which no non-primary-key attribute is transitively dependent on any candidate key. Deepak Gour, Faculty – DBMS, School of Engineering, SPSU Boyce–Codd Normal Form (BCNF) • Based on functional dependencies that take into account all candidate keys in a relation, however BCNF also has additional constraints compared with general definition of 3NF. • BCNF - A relation is in BCNF if and only if every determinant is a candidate key. Deepak Gour, Faculty – DBMS, School of Engineering, SPSU Boyce–Codd normal form (BCNF) • Difference between 3NF and BCNF is that for a functional dependency A B, 3NF allows this dependency in a relation if B is a primary- key attribute and A is not a candidate key. • Whereas, BCNF insists that for this dependency to remain in a relation, A must be a candidate key. • Every relation in BCNF is also in 3NF. However, relation in 3NF may not be in BCNF. Deepak Gour, Faculty – DBMS, School of Engineering, SPSU Boyce–Codd normal form (BCNF) • Violation of BCNF is quite rare. • Potential to violate BCNF may occur in a relation that: – contains two (or more) composite candidate keys; – the candidate keys overlap (i.e. have at least one attribute in common). Deepak Gour, Faculty – DBMS, School of Engineering, SPSU 3NF Table Not in BCNF Figure 4.7 Deepak Gour, Faculty – DBMS, School of Engineering, SPSU Decomposition of Table Structure to Meet BCNF Deepak Gour, Faculty – DBMS, School of Engineering, SPSU BCNF Conversion Results Deepak Gour, Faculty – DBMS, School of Engineering, SPSU Review of Normalization (UNF to BCNF) Deepak Gour, Faculty – DBMS, School of Engineering, SPSU Review of Normalization (UNF to BCNF) Deepak Gour, Faculty – DBMS, School of Engineering, SPSU Review of Normalization (UNF to BCNF) Deepak Gour, Faculty – DBMS, School of Engineering, SPSU Review of Normalization (UNF to BCNF) Deepak Gour, Faculty – DBMS, School of Engineering, SPSU Fourth Normal Form (4NF) • Although BCNF removes anomalies due to functional dependencies, another type of dependency called a multi-valued dependency (MVD) can also cause data redundancy. • Possible existence of MVDs in a relation is due to 1NF and can result in data redundancy. Deepak Gour, Faculty – DBMS, School of Engineering, SPSU Fourth Normal Form (4NF) - MVD • Dependency between attributes (for example, A, B, and C) in a relation, such that for each value of A there is a set of values for B and a set of values for C. However, set of values for B and C are independent of each other. Deepak Gour, Faculty – DBMS, School of Engineering, SPSU Fourth Normal Form (4NF) • MVD between attributes A, B, and C in a relation using the following notation: A B A C Deepak Gour, Faculty – DBMS, School of Engineering, SPSU Fourth Normal Form (4NF) • MVD can be further defined as being trivial or nontrivial. – MVD A B in relation R is defined as being trivial if (a) B is a subset of A or (b) A B = R. – MVD is defined as being nontrivial if neither (a) nor (b) are satisfied. – Trivial MVD does not specify a constraint on a relation, while a nontrivial MVD does specify a constraint. Deepak Gour, Faculty – DBMS, School of Engineering, SPSU Fourth Normal Form (4NF) • Defined as a relation that is in BCNF and contains no nontrivial MVDs. Deepak Gour, Faculty – DBMS, School of Engineering, SPSU 4NF - Example Deepak Gour, Faculty – DBMS, School of Engineering, SPSU 3NF Table Not in BCNF Figure 4.7 Deepak Gour, Faculty – DBMS, School of Engineering, SPSU Decomposition of Table Structure to Meet BCNF Deepak Gour, Faculty – DBMS, School of Engineering, SPSU Decomposition into BCNF Figure 4.9 Deepak Gour, Faculty – DBMS, School of Engineering, SPSU 4NF Conversion Results Set of Tables in 4NF Multivalued Dependencies (an employee can work for many services and on many projects Deepak Gour, Faculty – DBMS, School of Engineering, SPSU Denormalization • Normalization is one of many database design goals • Normalized table requirements – Additional processing – Loss of system speed • Normalization purity is difficult to sustain due to conflict in: – Design efficiency – Information requirements – Processing Deepak Gour, Faculty – DBMS, School of Engineering, SPSU Unnormalized Table Defects • Data updates less efficient • Indexing more cumbersome • No simple strategies for creating views Deepak Gour, Faculty – DBMS, School of Engineering, SPSU Summary • We will use normalization in database design to create a set of relations in 3FN normal form: – Each entity has a unique primary key, and each attribute depends upon the primary key – No partial dependency – No transitive dependency Deepak Gour, Faculty – DBMS, School of Engineering, SPSU