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.^[1]

Lossless-join Decomposition

Can also be called Nonadditive. If you decompose a relation $R$ into relations $R_{1},R_{2}$ you will guarantee a Lossless-Join if $R_{1}\times R_{2}=R$ .

If $R$ is split into $R_{1}$ and $R_{2}$ , for the decomposition to be lossless then at least one of the two should hold true.

Projecting on $R_{1}$ and $R_{2}$ , and joining back, results in the relation you started with.^[2]

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 or attribute sets in $F$ ):^[3]

$R_{1}\cap R_{2}\to R_{1}$
$R_{1}\cap R_{2}\to R_{2}$

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 $R_{1}\cap R_{2}=(A)$ , $A$ is a superkey in $R_{1}$ ( $A \rightarrow BC$ ) so $R_{1}\cap R_{2}\rightarrow R_{1}$ .

^[4]^[5]

References

↑ Pohler, K (2015). "Lossless-Join Decomposition: applications in quantitative computing metrics". International Journal of Applied Computer Science. 21 (4): 190–212.
↑ "Lossless Join Property". Stackoverflow.com. Retrieved 2016-02-07.
↑ "Lossless Join Decomposition" (PDF). University at Buffalo. Jan Chomicki. Retrieved 2012-02-08.
↑ "Lossless-Join Decomposition". Cs.sfu.ca. Retrieved 2016-02-07.
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[1] Pohler, K (2015). "Lossless-Join Decomposition: applications in quantitative computing metrics". International Journal of Applied Computer Science. 21 (4): 190–212.

[2] "Lossless Join Property". Stackoverflow.com. Retrieved 2016-02-07.

[3] "Lossless Join Decomposition" (PDF). University at Buffalo. Jan Chomicki. Retrieved 2012-02-08.

[4] "Lossless-Join Decomposition". Cs.sfu.ca. Retrieved 2016-02-07.

[5] ↑