# Lossless-Join Decomposition

In computer science the concept of a Lossless-Join Decomposition is central in removing redundancy safely from databases while preserving the original data.

## Lossless-join Decomposition

Can also be called Nonadditive. If you decompose a relation $R$ into relations $R_1$ and $R_2$ you will guarantee a Lossless-Join if $R_1$$R_2$ = $R$.

Let $R$ be a relation schema.

Let $F$ be a set of functional dependencies on $R$.

Let $R_1$ and $R_2$ 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 $F$+(where $F$+ stands for the closure for every attribute in $F$):[1]

• $R_1$ ∩ $R_2$$R_1 - R_2$
• $R_1$ ∩ $R_2$$R_2 - R_1$
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## Example

• Let $R = (A, B, C, D)$ be the relation schema, with $A$, $B$, $C$ and $D$ attributes.
• Let $F = \{ A \rightarrow BC \}$ be the set of functional dependencies.
• Decomposition into $R_1 = (A, B, C)$ and $R_2 = (A, D)$ is lossless under $F$ because $R_1 \cap R_2 = (A)$ and $A \rightarrow BC$ so $R_1 \cap R_2 \rightarrow R_1 - R_2$.
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## Normal Forms

Decomposition into BCNF or 3NF must always be lossless.

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## References

1. ^ "Lossless Join Decomposition". University at Buffalo (Jan Chomicki). Retrieved 2012-02-08.
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Last modified on 2 May 2013, at 16:56