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

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

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

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

A1 A2 A3 A4 A5

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)

By FD1: A1 --> A3 A5

we have a new result table

A1 A2 A3 A4 A5

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)

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.

By FD3: A5 --> A1 A4

we have a new result table

A1 A2 A3 A4 A5

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)

By FD4: A3 A4 --> A2

we get a new result table

A1 A2 A3 A4 A5

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

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.