Lossless Decomposition

Elias AsegedSE 157B - DB 2

What is Decomposition?

• Decomposition – the process of breaking down in parts or elements.

• Decomposition in database means breaking tables down into multiple tables

• From Database perspective means going to a higher normal form

Decomposition

Important that decompositions are “good”,

Two Characteristics of Good Decompositions•1) Lossless•2) Preserve dependencies

What is lossless?

Lossless means functioning without a loss.In other words, retain everything.

Important for databases to have this feature.

Formal Definition

• Let R be a relation schema.• Let F be a set of functional dependencies on R.• Let and form a decomposition of R.• The decomposition is a lossless-join

decomposition of R if at least one of the following functional dependencies are in F+

1) R1 ∩ R2 R12) R1 ∩ R2 R2

In Simpler Terms…

• R1 ∩ R2 R1• R1 ∩ R2 R2

If R is split into R1 and R2, for the decomposition to be lossless then at least one of the two should hold true.

Projecting on R1 and R2, and joining back, results in the relation you started with

Why lossless?

Ensures that attributes involved in the natural join (R1 ∩ R2) are a candidate key for at least one of the two relations.

This ensures we can never get the situation where false tuples are generated, as for any value on the join attributes there will be a unique tuple in one of the relations.

A decomposition is lossless if we can recover: R(A,B,C)

R1(A,B) R2(A,C)

R’(A,B,C) should be the same as R(A,B,C)

Must ensure R’ = R

Decompose

Recover

Lossless Decomposition

Lossless Decomposition• Sometimes the same set of data is reproduced:

• (Word, 100) + (Word, WP) (Word, 100, WP)• (Oracle, 1000) + (Oracle, DB) (Oracle, 1000, DB)• (Access, 100) + (Access, DB) (Access, 100, DB)

Name Price Category

Word 100 WP

Oracle 1000 DB

Access 100 DB

Name Price

Word 100

Oracle 1000

Access 100

Name Category

Word WP

Oracle DB

Access DB

Lossy Decomposition• Sometimes it’s not:

• (Word, WP) + (100, WP) = (Word, 100, WP)• (Oracle, DB) + (1000, DB) = (Oracle, 1000, DB)• (Oracle, DB) + (100, DB) = (Oracle, 100, DB)• (Access, DB) + (1000, DB) = (Access, 1000, DB)• (Access, DB) + (100, DB) = (Access, 100, DB)

Name Price Category

Word 100 WP

Oracle 1000 DB

Access 100 DB

Category Name

WP Word

DB Oracle

DB Access

Category Price

WP 100

DB 1000

DB 100

What’swrong?

Ensuring lossless decomposition

R(A1, ..., An, B1, ..., Bm, C1, ..., Cp) R(A1, ..., An, B1, ..., Bm, C1, ..., Cp)

If A1, ..., An B1, ..., Bm or A1, ..., An C1, ..., Cp

Then the decomposition is lossless

R1(A1, ..., An, B1, ..., Bm)R1(A1, ..., An, B1, ..., Bm) R2(A1, ..., An, C1, ..., Cp)R2(A1, ..., An, C1, ..., Cp)

Note: don’t need both

Identifying a Loss Decomposition

• Make a table for sub schemas of R• Fill in table with distinguished variables

(corresponding to the sub schemas)– If one row is full of distinguished variables, it’s

lossless– If no one row is full, add distinguished variables• To add distinguished variables1) 2 or more rows with distinguished variables on LHS2) 1 or more rows with distinguished variables on RHS3) 1 or more rows with non-distinguished variables on RHS

Example 1 (From Class)

R(A B C D E)•FD1 = (A B)•FD2 = (BC E)•FD3 = (ED A)R1=(AB);

R2=(ACDE);

Answer

a a

a a a A a

A B C D ER1R2

*This decomposition is lossless

Example 2Is this decomposition lossless?

•R (A B C D E)FD1 – AB CFD2 – C EFD3 – BDFD4 – EA

R1=(BCD); R2=(ACE);

Answer

• If you do this procedure and you don’t have one row full of distinguished variables, then the decomposition is lossy.

a a a a aa a a

A B C D ER1R2

*This decomposition is lossless

• R(A B C D E)FD1: A BCFD2: BD CEFD3: E ADFD4: CE A

R1(ABC) = R2 (BCDE) =

Conclusion

Decomposing is the act of breaking tables down in order to achieve higher normal form.

Decompositions should always be lossless.•This confirms that information in the original relation can be accurately reconstructed based on the decomposed relations.

Remember that for a decomposition to be considered “GOOD” it must also preserve functional dependencies.

Questions?

References

• http://infolab.stanford.edu/~ullman/fcdb/spr99/lec14.pdf

• http://www.cs.sfu.ca/CC/354/zaiane/material/notes/Chapter7/node7.html

• http://www.cs.sjsu.edu/~lee/cs157b/29SpCS157BL14HuffmanCode&LosslessDecomposition.ppt