Lossless Decomposition (2)

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Lossless Decomposition (2)
Lecture 17
CS157A

• Prof. Sin-Min Lee
• Department of Computer Science
• San Jose State University

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Normal form and normalization A normal form is
a property of a relational database. When a
relation is non-normalized (that is, does not
satisfy a normal form), then it presents
redundancies and produces undesirable behavior
during update operations. This principle can be
used to carry out quality analysis and
constitutes a useful tool for database design.
Normalization is a procedure that allows the
non-normalized schemas to be transformed into new
schemas for which the satisfaction of a normal
form is guaranteed.
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Example of a relation with anomalies Employee
Salary Project Budget Function Brown 20
Mars 2 technician Green 35 Jupiter 15
designer Green 35 Venus 15
designer Hoskins 55 Venus 15
manager Hoskins 55 Jupiter 15
consultant Hoskins 55 Mars 2
consultant Moore 48 Mars 2 manager Moore
48 Venus 15 designer Kemp 48 Venus 15
designer Kemp 48 Jupiter 15 manager
The key is made up of the attributes Employee
and Project
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Normal Forms

• Each normal form is a set of conditions on a
  schema that guarantees certain properties
  (relating to redundancy and update anomalies)
• First normal form (1NF) is the same as the
  definition of relational model (relations sets
  of tuples each tuple sequence of atomic
  values)
• Second normal form (2NF) a research lab
  accident has no practical or theoretical value
  wont discuss
• The two commonly used normal forms are third
  normal form (3NF) and Boyce-Codd normal form
  (BCNF)

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Anomalies in the example relation The value of
the salary of each employee is repeated in all
the tuples relating to it therefore there is a
redundancy. If the salary of an employee
changes, we have to modify the value in all the
corresponding tuples. This problem is known as
the update anomaly. If an employee stops
working on all the projects but does not
leave the company, all the corresponding tuples
are deleted and so, even the basic information,
name and salary is lost. This problem is known as
the deletion anomaly. If we have information on
a new employee, we cannot insert it until the
employee is assigned to a project. This is known
as the insertion anomaly.
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Why these undesirable phenomena?

• Intuitive explanation we have used a single
  relation to represent
• items of information of different types.
• In particular, the following independent
  real-world concepts are
• represented in the relation
• employees with their salaries,
• projects with their budgets,
• participation of the employees in the projects
  with their functions.
• To systematically study the principles
  introduced informally, it is
• necessary to use a specific notion the
  functional dependency.

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Functional dependencies

• Given a relation r on a schema R(X) and two
  non-empty subsets Y
• and Z of the attributes X, we say that there is a
  functional dependency
• on r between Y and Z, if, for each pair of tuples
  t 1 and t 2 of r having
• the same values on the attributes Y, t 1 and t 2
  also have the same
• values of the Z attributes.
• A functional dependency between the attributes
  Y and Z is indicated
• by the notation Y -gt Z

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Functional dependencies in the example schema

• Employee -gt Salary
• the salary of each employee is unique and thus
  each time a certain
• employee appears in a tuple, the value of his or
  her salary always
• remains the same.
• Project -gt Budget
• the budget of each project is unique and thus
  each time a certain
• project appears in a tuple, the value of its
  budget always remains
• the same.

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Non-trivial functional dependencies

• We then say that a functional dependency Y-gt Z
  is non-trivial if no
• attribute in Z appears among the attributes of Y.
• Employee -gt Salary is a non-trivial functional
  dependency
• Employee Project -gt Project is a trivial
  functional dependency

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Anomalies and functional dependencies

• In our example, the two properties causing
  anomalies correspond
• exactly to attributes involved in functional
  dependencies
• the property the salary of each employee is
  unique and depends
• only on the employee corresponds to the
  functional dependency
• Employee -gt Salary
• the property the budget of each project is
  unique and depends
• only on the project corresponds to the
  functional dependency
• Project -gt Budget.
• Moreover, the following property can be
  formalized by means of a
• functional dependency
• the property in each project, each of the
  employees involved can
• carry out only one function corresponds to the
  functional
• dependency Employee Project -gt Function.

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Dependencies generating anomalies

• The first two dependencies generate undesirable
  redundancies and
• anomalies.
• The third dependency however never generates
  redundancies
• because, having Employee and Project as a key,
  the relation cannot
• contain two tuples with the same values of these
  attributes.
• The difference is that Employee Project is a
  key of the relation.

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BCNF

• Definition A relation schema R is in BCNF if for
  every FD X? Y associated with R either
• Y ? X (i.e., the FD is trivial) or
• X is a superkey of R
• Example Person1(SSN, Name, Address)
• The only FD is SSN ? Name, Address
• Since SSN is a key, Person1 is in BCNF

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(non) BCNF Examples

• Person (SSN, Name, Address, Hobby)
• The FD SSN ? Name, Address does not satisfy
  requirements of BCNF
• since the key is (SSN, Hobby)
• HasAccount (AcctNum, ClientId, OfficeId)
• The FD AcctNum? OfficeId does not satisfy BCNF
  requirements
• since keys are (ClientId, OfficeId) and (AcctNum,
  ClientId) not AcctNum.

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Redundancy

• Suppose R has a FD A ? B, and A is not a
  superkey. If an instance has 2 rows with same
  value in A, they must also have same value in B
  (gt redundancy, if the A-value repeats twice)
• If A is a superkey, there cannot be two rows with
  same value of A
• Hence, BCNF eliminates redundancy

SSN ? Name, Address SSN Name
Address Hobby 1111 Joe 123 Main
stamps 1111 Joe 123 Main coins
redundancy
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Third Normal Form

• A relational schema R is in 3NF if for every FD
  X? Y associated with R either
• Y ? X (i.e., the FD is trivial) or
• X is a superkey of R or
• Every A? Y is part of some key of R
• 3NF is weaker than BCNF (every schema that is in
  BCNF is also in 3NF)

BCNF conditions
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3NF Example

• HasAccount (AcctNum, ClientId, OfficeId)
• ClientId, OfficeId ? AcctNum
• OK since LHS contains a key
• AcctNum ? OfficeId
• OK since RHS is part of a key
• HasAccount is in 3NF but it might still contain
  redundant information due to AcctNum ? OfficeId
  (which is not allowed by BCNF)

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3NF (Non) Example

• Person (SSN, Name, Address, Hobby)
• (SSN, Hobby) is the only key.
• SSN? Name violates 3NF conditions since Name is
  not part of a key and SSN is not a superkey

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Decompositions

• Goal Eliminate redundancy by decomposing a
  relation into several relations in a higher
  normal form
• Decomposition must be lossless it must be
  possible to reconstruct the original relation
  from the relations in the decomposition
• We will see why

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Decomposition

• Schema R (R, F)
• R is set a of attributes
• F is a set of functional dependencies over R
• Each key is described by a FD
• The decomposition of schema R is a collection of
  schemas Ri (Ri, Fi) where
• R ?i Ri for all i (no new attributes)
• Fi is a set of functional dependences involving
  only attributes of Ri
• F entails Fi for all i (no new FDs)
• The decomposition of an instance, r, of R is a
  set of relations ri ?Ri(r) for all i

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Example Decomposition
Schema (R, F) where R SSN, Name,
Address, Hobby F SSN? Name,
Address can be decomposed into R1 SSN,
Name, Address F1 SSN ? Name,
Address and R2 SSN, Hobby F2

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Lossless Schema Decomposition

• A decomposition should not lose information
• A decomposition (R1,,Rn) of a schema, R, is
  lossless if every valid instance, r, of R can be
  reconstructed from its components
• where each ri ?Ri(r)

r2
r r1
rn

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Lossy Decomposition
The following is always the case (Think why?)
r ? r1
r2
rn
...
But the following is not always true
r ? r1
r2
rn
...
?
r1
r2
r
Example
SSN Name Address SSN Name
Name Address 1111 Joe 1 Pine
1111 Joe Joe 1 Pine 2222 Alice
2 Oak 2222 Alice Alice 2
Oak 3333 Alice 3 Pine 3333 Alice
Alice 3 Pine
The tuples (2222, Alice, 3 Pine) and (3333,
Alice, 2 Oak) are in the join, but not in the
original
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Lossy Decompositions What is Actually Lost?

• In the previous example, the tuples (2222, Alice,
  3 Pine) and (3333, Alice, 2 Oak) were gained, not
  lost!
• Why do we say that the decomposition was lossy?
• What was lost is information
• That 2222 lives at 2 Oak In the
  decomposition, 2222 can live at either 2 Oak or 3
  Pine
• That 3333 lives at 3 Pine In the
  decomposition, 3333 can live at either 2 Oak or 3
  Pine

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Example
Schema (R, F) where R SSN, Name,
Address, Hobby F SSN ? Name,
Address can be decomposed into R1 SSN,
Name, Address F1 SSN ? Name,
Address and R2 SSN, Hobby F2
Since R1 ? R2 SSN and SSN ? R1
the decomposition is lossless
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Intuition Behind the Test for Losslessness

• Suppose R1 ? R2 ? R2 . Then a row of r1 can
  combine with exactly one row of r2 in the
  natural join (since in r2 a particular set of
  values for the attributes in R1 ? R2 defines a
  unique row)

R1 ? R2 R1 ? R2 . a
a ... a b
. b c .
c r1 r2
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Proof of Lossless Condition

• r ? r1

r2 this is true for any decomposition
r2

• r ? r1

If R1 ? R2 ? R2 then card (r1
r2) card (r1)
(since each row of r1 joins with exactly one
row of r2)
But card (r) ? card (r1) (since r1 is a
projection of r) and therefore card (r) ? card
(r1 r2)
Hence r r1
r2
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Dependency Preservation

• Consider a decomposition of R (R, F) into R1
  (R1, F1) and R2 (R2, F2)
• An FD X ? Y of F is in Fi iff X ? Y ? Ri
• An FD, f ?F may be in neither F1, nor F2, nor
  even (F1 ? F2)
• Checking that f is true in r1 or r2 is
  (relatively) easy
• Checking f in r1 r2 is harder requires
  a join
• Ideally want to check FDs locally, in r1 and
  r2, and have a guarantee that every f ?F holds
  in r1 r2
• The decomposition is dependency preserving iff
  the sets F and F1 ? F2 are equivalent F (F1
  ? F2)
• Then checking all FDs in F, as r1 and r2 are
  updated, can be done by checking F1 in r1 and F2
  in r2

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Dependency Preservation

• If f is an FD in F, but f is not in F1 ? F2,
  there are two possibilities
• f ? (F1 ? F2)
• If the constraints in F1 and F2 are maintained,
  f will be maintained automatically.
• f ? (F1 ? F2)
• f can be checked only by first taking the join
  of r1 and r2. This is costly.

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Example
Schema (R, F) where R SSN, Name,
Address, Hobby F SSN ? Name,
Address can be decomposed into R1 SSN,
Name, Address F1 SSN ? Name,
Address and R2 SSN, Hobby F2
Since F F1 ? F2 the decomposition
is dependency preserving
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Example

• Schema (ABC F) , F A ? B, B? C, C? B
• Decomposition
• (AC, F1), F1 A?C
• Note A?C ? F, but in F
• (BC, F2), F2 B? C, C? B
• A ? B ? (F1 ? F2), but A ? B ? (F1 ? F2).
• So F (F1 ? F2) and thus the decompositions
  is still dependency preserving

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BCNF Decomposition Algorithm
Input R (R F) Decomp R while there is S
(S F) ? Decomp and S not in BCNF do
Find X ? Y ? F that violates BCNF // X isnt
a superkey in S Replace S in Decomp with
S1 (XY F1), S2 (S - (Y - X) F2) //
F1 all FDs of F involving only attributes of
XY // F2 all FDs of F involving only
attributes of S - (Y - X) end return Decomp
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Simple Example

• HasAccount

(ClientId, OfficeId, AcctNum)
ClientId,OfficeId ? AcctNum AcctNum ? OfficeId

• Decompose using AcctNum ? OfficeId

(OfficeId, AcctNum)
(ClientId , AcctNum) BCNF (only trivial FDs)
BCNF AcctNum is key FD AcctNum ? OfficeId
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A Larger Example
Given R (R F) where R ABCDEGHK and F
ABH? C, A? DE, BGH? K, K? ADH, BH? GE step 1
Find a FD that violates BCNF Not ABH
? C since (ABH) includes all attributes
(BH is a key) A ? DE
violates BCNF since A is not a superkey (A
ADE) step 2 Split R into R1 (ADE, F1A?
DE ) R2 (ABCGHK F1ABH?C, BGH?K, K?AH,
BH?G) Note 1 R1 is in BCNF Note 2
Decomposition is lossless since A is a key of
R1. Note 3 FDs K ? D and BH ? E are not in
F1 or F2. But both can be derived from F1?
F2 (E.g., K? A and
A? D implies K? D) Hence,
decomposition is dependency preserving.
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Example (cont)
Given R2 (ABCGHK ABH?C, BGH?K, K?AH,
BH?G) step 1 Find a FD that violates
BCNF. Not ABH ? C or BGH ? K, since BH is a
key of R2 K? AH violates BCNF since K is not
a superkey (K AH) step 2 Split R2 into
R21 (KAH, F21K ? AH) R22 (BCGK
F22) Note 1 Both R21 and R22 are in
BCNF. Note 2 The decomposition is
lossless (since K is a key of R21) Note
3 FDs ABH? C, BGH? K, BH? G are not in F21
or F22 , and they cant be derived
from F1 ? F21 ? F22 . Hence the
decomposition is not dependency-preserving
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Properties of BCNF Decomposition Algorithm

• Let X ? Y violate BCNF in R (R,F) and R1
  (R1,F1),
• R2 (R2,F2) is the resulting decomposition.
  Then
• There are fewer violations of BCNF in R1 and R2
  than there were in R
• X ? Y implies X is a key of R1
• Hence X ? Y ? F1 does not violate BCNF in R1 and,
  since X ? Y ?F2, does not violate BCNF in R2
  either
• Suppose f is X ? Y and f ? F doesnt violate
  BCNF in R. If f ? F1 or F2 it does not violate
  BCNF in R1 or R2 either since X is a superkey
  of R and hence also of R1 and R2 .

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Properties of BCNF Decomposition Algorithm

• A BCNF decomposition is not necessarily
  dependency preserving
• But always lossless
• since R1 ? R2 X, X ? Y, and R1 XY
• BCNFlosslessdependency preserving is sometimes
  unachievable (recall HasAccount)

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BoyceCodd Normal Form (BCNF)

• A relation r is in BoyceCodd normal form if
  for every (non-trivial)
• functional dependency X -gt Y defined on it, X
  contains a key K of r.
• That is, X is a superkey for r.
• Anomalies and redundancies, as discussed above,
  do not appear in
• databases with relations in BoyceCodd normal
  form, because the
• independent pieces of information are separate,
  one per relation.

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Decomposition into BoyceCodd normal form

• Given a relation that does not satisfy
  BoyceCodd normal form, we
• can often replace it with one or more normalized
  relations using a
• process called normalization.
• We can eliminate redundancies and anomalies for
  the example
• relation if we replace it with the three
  relations, obtained by
• projections on the sets of attributes
  corresponding to the three
• functional dependencies.
• The keys of the relations we obtain are the
  left hand side of a
• functional dependency the satisfaction of the
  BoyceCodd normal
• form is therefore guaranteed.

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