LOSSLESS DECOMPOSITION

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LOSSLESS DECOMPOSITION
CS157A Lecture 16

• Prof. Sin-Min Lee
• Department of Computer Science
• San Jose State University

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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

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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

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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.

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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 --gt A3 A5
• FD2 A2 A3 --gt A2
• FD3 A5 --gt A1 A4
• FD4 A3 A4 --gt A2

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• A1 A2 A3 A4 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)

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• By FD1 A1 --gt A3 A5
• we have a new result table
• A1 A2 A3 A4
  A5
• 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)

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• By FD2 A2 A3 --gt A4
• we dont have a new result table because we dont
  have any equally elements. Therefore, the result
  doesnt change.

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• By FD3 A5 --gt A1 A4
• we have a new result table
• A1 A2 A3 A4
  A5
• 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)

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• By FD4 A3 A4 --gt A2
• we get a new result table
• A1 A2 A3 A4
  A5
• 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)

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