Schema Normalization

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Chapter 13 14

• Schema Normalization

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Normalization

• A design process to reduce redundancies and
  update anomalies in a relational schema
• Result a set of decomposed relations that meet
  certain normal form tests
• Four most commonly used normal forms are first
  (1NF), second (2NF) and third (3NF) normal forms,
  and BoyceCodd normal form (BCNF)
• A process that is based on keys and functional
  dependencies among the attributes of a relation

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Data Redundancy

• Grouping attributes into relation schemas has an
  effect on storage space.
• A bad grouping may produce data redundancy
• Problems associated with data redundancy are
  illustrated by comparing the following Staff and
  Branch relations with the StaffBranch relation.
• StaffBranch is the Natural Join of Staff and
  Branch
• This relation has redundant data (Baddress)

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Data Redundancy
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Update Anomalies

• Types of update anomalies include
• Insertion
• Insert tuple (SL22, John Wayne, Assistant,
  12000, B003, 163 Main, Rapid City ) in
  StaffBranch
• Different address for Branch with BranchNo B003
• Deletion
• Delete tuple for StaffNo SA9 in StaffBranch
• Loose information about Branch with BranchNo
  B007
• Modification.
• Update Baddress for StaffNo SG14
• Different address for Branch with BranchNo B003

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Lossless-join and Dependency Preservation
Properties

• Two important properties of decomposition
• - Lossless-join property enables us to find any
  instance of original relation from corresponding
  instances in the smaller relations.
• - Dependency preservation property enables us to
  enforce a constraint on original relation by
  enforcing some constraint on each of the smaller
  relations.

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Functional Dependency (FD)

• Main concept associated with normalization.
• Functional Dependency (FD)
• Describes relationship among relation attributes.
• 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.
• A maps to a single value of B
• Given a Branchs BranchNo, I know its address
• BranchNo ? Baddress

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Functional Dependency (cont.)

• 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.

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Example - Functional Dependency
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Functional Dependency (cont.)

• Main characteristics of FDs used in
  normalization
• have a 11 relationship between attribute(s) on
  left and right-hand side of a dependency
• hold for all time
• are nontrivial. (StaffNo?StaffNo is trivial)

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Functional Dependency (cont.)

• 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 a smaller set of functional
  dependencies (X) for a relation
• A bigger set of FDs, Y can be found, so that
  every FD in Y can be implied by X.

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Functional Dependency (cont.)

• Set of all functional dependencies implied by a
  given set of functional dependencies F called
  closure of F (written F).
• Set of inference rules, called Armstrongs
  axioms, specifies how new functional dependencies
  can be inferred from given ones.

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Functional Dependency Armstrongs axioms

• Let A, B, and C be subsets of the attributes of
  relation R. Armstrongs axioms (AA) are as
  follows
•  1. Reflexivity
• If B is a subset of A, then A B
• 2. Augmentation
• If A B, then A,C B,C
• ( Written also as AC ? BC )
• 3. Transitivity
• If A B and B C, then A C

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Reasoning About FDs

• Given some FDs, we can infer new FDs
• F StaffNo?BranchNo, BranchNo?Baddress
• Implies StaffNo?Baddress
• F is the set of all FDs that are implied by F
• Additional rules that follows from AA
• Union IF A?B and A?C, then A?BC
• Decomposition If A?BC, then A?B and A?C

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Example of Implied FDs using Inference rules

• Relation PROJECT(EmpID, ProjID, Budget,
  TimeSpent)
• Each project has a single budget ProjID?Budget
  FD1
• The relation keeps the time spent for an employee
  on each project EmpID, ProjID?TimeSpent FD2
• Implied FDs
• By augmentation on FD1
• EmpID,ProjID ? EmpID,Budget FD3
• By decomposing FD3
• EmpID,ProjID ? EmpID FD4
• EmpID,ProjID ? Budget FD5

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Determine whether a set of FDs F implies X?Y

• Computing F is expensive. Instead,
• Compute X (the closure of X) and check whether
  X includes all the attributes in Y.
• Algorithm for X
• X X
• repeat
• oldX X
• for each FD T?Z in F do
• if X ? T then X X ? Z
• until (X oldX)
• If X includes all attributes, then X is a key of
  the relation

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Example of Implied FDs using X

• Relation Contracts(EmpID, ProjID, Budget,
  TimeSpent) with set F FD1 ProjID?Budget FD2
  EmpID, ProjID?TimeSpent
• Find out if F implies FDx EmpID, ProjID?Budget
• Using the algorithm for X
• Initial X EmpID, ProjID
• X EmpID, ProjID Budget (using FD1)
• X EmpID, ProjID, Budget TimeSpent (using
  FD2)
• X includes Y therefore FDx is implied by F
• Is EmpID ? Budget implied by F?

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The Process of Normalization

• Formal technique for designing 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.

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Relationship Between Normal Forms
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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.

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First Normal Form (1NF)

• A relation in which intersection of each row and
  column contains one and only one value.
• An unnormalized relation must be converted to a
  1NF relation
• First identify a primary key, then
• place repeating data along with copy of the
  original key attribute(s) into a separate
  relation.

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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.
• EMP_PROJ(SSN, Pnum, Hours, Ename, Pname, Ploc)
• SSN,Pnum?Hours, SSN?Ename, Pnum?Pname, Ploc
• Relation is not in 2NF

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1NF to 2NF

• Identify FDs 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.
• EMP_PROJ(SSN, Pnum, Hours, Ename, Pname, Ploc)
• SSN,Pnum?Hours, SSN?Ename, Pnum?Pname, Ploc
• EMP_PROJ decomposed into
• EMP1(SSN, Pnum, Hours) , SSN,Pnum?Hours
• EMP2(SSN, Ename) , SSN?Ename
• EMP3(Pnum,Pname, Ploc) , Pnum?Pname, Ploc

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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.

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2NF to 3NF

• Identify FDs 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.
• EMP_DEPT(Enum, Ename, Sal, Dnum, Dname, Mgr)
• Enum?Ename, Sal, Dnum, Dnum?Dname, Mgr
• second FD is transitive through Dnum
• Decompose EMP_DEPT into
• EMP(Enum, Ename, Sal, Dnum) and
• DEPT(Dnum, Dname, Mgr)

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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.
• These definitions may find hidden redundancies
  (see example

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BoyceCodd 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.

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BoyceCodd 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.

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BoyceCodd 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 (ie. have at least one
  attribute in common).

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Example 14.2

• Relation ClientInterview(clientNo, interviewDate,
  interviewTime, staffNo, roomNo) with FDs
• FD1 clientNo, interviewDate ? interviewTime,
  staffNo, roomNo PK
• FD2 staffNo, interviewDate, interviewTime ?
  clientNo CK
• FD3 roomNo, interviewDate, interviewTime ?
  staffNo, clientNo CK
• FD4 staffNo, interviewDate ? roomNo
• Last FD makes the relation not in BCNF
• Possible anomaly to change roomNo for staffNo
  SG5 on 13-May-05 we must update two tuples.
• ClientInterview must be decomposed into 2
  relations (pp. 421)

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Example of decomposition (example 13.9)

• R( A, B, C, D, E, F, G, H, I)
• F AB?EF, A?C, B?DGHI, H?I, AE?BDFGHI, BE?ACF
• fd1 fd2 fd3 fd4
  fd5 fd6
• PK of R AB, candidate keys AE, BE
• R is not in 2NF because of fd2 and fd3, decompose
  it
• R1(A, C), R21(B, D, G, H, I), R22( A, B, E, F)
  (R1 is Client relation)
• fd2 fd3, fd4
  fd1, fd5, fd6 (R21 is PropertyOwner)
• R21 is not in 3NF because of fd4, decompose it
• R211(H, I), R212( B, D, G, H) (R211 is
  Owner)
• fd4 fd3
  (R212 is PropertyForRent)
• Decomposition is also in BCNF
• Relation is ClientRental(clientNo, propertyNo,
  cName, pAddress, rentStart, rentFinish, rent,
  ownerNo, oName)

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Review of Normalization (UNF to BCNF)
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Review of Normalization (UNF to BCNF)
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Review of Normalization (UNF to BCNF)
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Review of Normalization (UNF to BCNF)
Use fd2
Use fd3
Use fd4
fd3
fd4 fd1,fd6 fd2
(lost fd5)
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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.

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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.

B
M 1
A
1 M
C
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Fourth Normal Form (4NF)

• MVD between attributes A, B, and C in a relation
  using the following notation
• A ?? B
• A ?? C

1NF
Branchno Sname Oname
B003 Ann Carol
B003 David Carol
B003 Ann Tina
B003 David Tina
B005 Pete Rob
B005 John Rob
B005 Carmen Rob
B005 Pete Jed
B005 John Jed
B005 Carmen Jed
UNF
Branchno Sname Oname
B003 Ann David Carol Tina
B005 Pete John Carmen Rob Jed
?
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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.

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Fourth Normal Form (4NF)

• Defined as a relation that is in BCNF and
  contains no nontrivial MVDs.

BranchNo ??Sname BranchNo??Oname Decompose
BranchStaffOwner into two tables
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Lossless Join Decomposition

• Decomposition of R into R1, R2,Rm is
  lossless-join with respect to a set of FDs F in R
  if for every instance r of R, the following
  holds
• (?R1(r) X ?R2(r) XX ?Rm(r)) r
• The decomposition of R into X and Y is
  lossless-join with respect to F if and only if F
  contains
• (X ? Y) ?X, or
• (X ? Y) ? Y
• The decomposition of EMP_DEPT into EMP and DEPT
  is lossless-join.

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Fifth Normal Form (5NF)

• A relation decomposed into two relations must
  have lossless-join property, which ensures that
  no spurious tuples are generated when relations
  are reunited through a natural join.
• However, there are requirements to decompose a
  relation into more than two relations.
• Although rare, these cases are managed by join
  dependency and fifth normal form (5NF).

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Fifth Normal Form (5NF)

• A relation that has no join dependency.

Join dependency exists here
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5NF - Example
Eliminate join dependency by decomposing the
relation into three relations
Performing join on 2 relations produce spurious
tables Performing the join on all 3 relations
will produce the original relation