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### 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 A3A4 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)

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)

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

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)

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

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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