Lossless decomposition
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CS157A Lecture 19. LOSSLESS DECOMPOSITION. Prof. Sin-Min Lee Department of Computer Science San Jose State University. Definition of Decomposition. A decomposition of a relation R is a set of relations { R1, R2,…, Rn } such that each Ri is a subset of R and the union of all of the Ri is R.

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LOSSLESS DECOMPOSITION

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Lossless decomposition

CS157A Lecture 19

LOSSLESS DECOMPOSITION

Prof. Sin-Min Lee

Department of Computer Science

San Jose State University


Definition of decomposition

Definition of Decomposition

A decomposition of a relation R is a set of relations { R1, R2,…, Rn } such that each Ri is a subset of R and the union of all of the Ri is R


Example of decomposition

Example of Decomposition

From R( A B C ) we can have two subsets as:

R1( A C ) and R2( B C )

if we union R1 and R2 we will get R

R = R1 U R2


Definition of lossless decompotion

Definition of Lossless Decompotion

A decomposition {R1, R2,…, Rn} of a relation R is called a lossless decomposition for R if the natural join of R1, R2,…, Rn produces exactly the relation R.


Example

Example

R( A1, A2, A3, A4, A5 )

R1( A1, A2, A3, A5 ); R2( A1, A3, A4 );

R3( A4, A5 ) are subsets of R.

We have FD1: A1 --> A3 A5

FD2: A2 A3 --> A2

FD3: A5 --> A1 A4

FD4: A3 A4 --> A2


Lossless decomposition

A1 A2 A3A4A5

a(1) a(2) a(3) b(1,4) a(5)

a(1) b(2,2) a(3) a(4) b(2,5)

b(3,1) b(3,2) b(3,3) a(4) a(5)


Lossless decomposition

By FD1: A1 --> A3 A5

we have a new result table

A1 A2A3A4A5

a(1) a(2) a(3) b(1,4) a(5)

a(1) b(2,2) a(3) a(4) a(5)

b(3,1) b(3,2) b(3,3) a(4) a(5)


Lossless decomposition

By FD2: A2 A3 --> A4

we don’t have a new result table because we don’t have any equally elements. Therefore, the result doesn’t change.


Lossless decomposition

By FD3: A5 --> A1 A4

we have a new result table

A1A2A3A4A5

a(1) a(2) a(3) a(4) a(5)

a(1) b(2,2) a(3) a(4) a(5)

b(3,1) b(3,2) b(3,3) a(4) a(5)


Lossless decomposition

By FD4: A3 A4 --> A2

we get a new result table

A1A2A3A4A5

a(1) a(2) a(3) a(4) a(5)

a(1) a(2) a(3) a(4) a(5)

b(3,1) b(3,2) b(3,3) a(4) a(5)

tuple1 and tuple2 are lossless because they have all a(I)


Summary

Summary

A decomposition { R1, R2,…, Rn } of a relation R is called a lossless decomposition for R if the natural join of R1, R2,…, Rn produces exactly the relation R

NOTE: not every decomposition is lossless. It is possible to produce a decomposition that is lossy, one that losses information.


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