Lossless Decomposition

Lossless decomposition refers to the process of decomposing a relation (or table) into two or more smaller relations (tables) in such a way that no information is lost when the tables are joined back together. The lossless property ensures that the original data can be fully reconstructed by joining the decomposed relations, preserving all the data and avoiding any redundancy.

Lossless decomposition is crucial in database normalization because it allows us to break down a complex relation into simpler, more manageable relations without losing data integrity or introducing inconsistencies.

Why is Lossless Decomposition Important?

1. Data Integrity: Lossless decomposition ensures that no data is lost during the decomposition process. This is essential for maintaining the integrity of the database.

2. Avoiding Redundancy: Decomposing relations helps eliminate redundancy and simplifies the database design.

3. Efficient Query Processing: By decomposing a large table into smaller, related tables, we can make queries more efficient and easier to process.

Lossless Join Property

A decomposition is considered lossless if the original relation can be fully reconstructed from the decomposed relations by performing a natural join on them. This means that no information is lost during the process of decomposition and recombination.

The lossless join property is achieved if the decomposition meets the following condition:

• If we decompose a relation R into two relations, R1 and R2, the join of R1 and R2 must yield the original relation R.

  More formally, if R1 and R2 are two decomposed relations, the decomposition is lossless if:

  R1 ⨝ R2 = R

Where:

• R1 and R2 are the decomposed relations.

• ⨝ represents the natural join between the two tables.

• R is the original relation.

When is Decomposition Lossless?

For a decomposition to be lossless, the following conditions must be satisfied:

1. The intersection of the decomposed relations must contain a candidate key of the original relation.

2. Alternatively, if the intersection contains a superkey of one of the decomposed relations, the decomposition will be lossless.

In other words, the shared attributes between the decomposed relations should be able to uniquely identify the rows in the original relation.

Example of Lossless Decomposition

Let’s consider the following relation Employee:

Functional Dependencies:

• EmpID → EmpName, DeptID

• DeptID → DeptName

We decompose the relation into two smaller relations:

1. Employee Table (stores the relationship between employees and departments):

   EmpID	DeptID
   101	D001
   102	D002
   103	D001

2. Department Table (stores department details):

   DeptID	DeptName
   D001	HR
   D002	IT

Joining the Decomposed Tables:

By performing a natural join between Employee and Department on DeptID, we can reconstruct the original relation:

Since the original relation is perfectly reconstructed by the join, this decomposition is lossless.

Lossless Decomposition Formula

The formula for ensuring that decomposition is lossless can be summarized as follows:

Let the original relation R have attributes A, B, C, D and the decomposition be into R1 (A, B) and R2 (B, C, D). The decomposition is lossless if:

• The intersection of R1 and R2 (i.e., {B}) contains a candidate key for the original relation R. In this case, B should be able to uniquely identify each row in R.

Lossless Decomposition Condition:

• If R1 ∩ R2 contains a candidate key or a superkey from R1 or R2, the decomposition is lossless.

In the above example, the DeptID is the key that can be used to join Employee and Department to get back the original data, ensuring that the decomposition is lossless.

Example of Non-Lossless Decomposition

Let’s consider another relation Book with the following attributes:

Now, suppose we decompose it into two relations:

1. BookAuthor Table:

   BookID	Author
   B001	Author1
   B002	Author2
   B003	Author1

2. BookPublisher Table:

   BookID	Publisher	Price
   B001	Pub1	20
   B002	Pub2	25
   B003	Pub1	20

Joining the Decomposed Tables:

When we perform a natural join between the two decomposed tables on BookID, we get:

This works fine for BookID but consider this:

• If we were to join the two tables based on a different attribute (say Author or Publisher), it might not give us the original data back correctly. Therefore, in some cases, this decomposition might not be lossless.

Conclusion

Lossless decomposition is crucial for ensuring that no information is lost when decomposing a relation into smaller relations. By ensuring that decomposed tables can be joined back together correctly, lossless decomposition guarantees the integrity of the data.

Key takeaways:

1. Lossless decomposition ensures that when decomposed tables are joined, the original data is fully reconstructed.

2. For a decomposition to be lossless, the intersection of the decomposed tables must contain a candidate key or a superkey of the original table.

3. Dependency-preserving decomposition ensures that functional dependencies are preserved in the decomposed tables.