Distributed Database Management Systems: Lecture 18
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Distributed Database
Management Systems
Lecture 18
In the previous lecture
• Continue with VF
– Information Requirement
– Attribute affinities
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In this Lecture
• Continue with VF
– Global Affinity Measure
– Bond Energy Algorithm
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n
n
∑ ∑ aff(Ai, Aj) [aff(Ai, Aj1) + aff(Ai, Aj+1)
+ aff(A
) + aff(A
i1, A
i+1, Aj) ]
A
A
A j
A
AM =
i = 1 j = 1
1
2
3
4
A1
45
0
45
0
A2
0
80
5
75
A3
45
5
53
3
A4
0
75
3
78
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The AA is symmetric so
A1
A2
A3
A4
A1
45
0
45
0
A2
0
80
5
75
A3
45
5
53
3
A4
0
75
3
78
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Rather than taking..
n
n
∑ ∑ aff(Ai, Aj) [aff(Ai, Aj1) + aff(Ai, Aj+1)
+ aff(Ai1, Aj) + aff(Ai+1, Aj) ]
AM =
i = 1 j = 1
We take only..
n
n
n
∑ aff(A , A ) aff(A , A
∑ ∑
aff(Ai, Aj) aff(Ai, Aj1) +
AM =
i = 1
j = 1 i = 1
n
Let
bond(Ax, Ay) =
∑
i
j
aff(Az, Ax)aff(Az, Ay)
z = 1
n
∑
bond(Aj, Aj1) +bond(Aj, Aj+1)
AM =
j = 1
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i
)
j+1
n
∑
AM =
j = 1
bond(Aj, Aj1) +bond(Aj, Aj+1)
J=1
45
0
45
0
0
80
5
75
45
5
53
3
0
75
3
78
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n
∑
AM =
j = 1
bond(Aj, Aj1) +bond(Aj, Aj+1)
J=2
45
0
45
0
0
80
5
75
45
5
53
3
0
75
3
78
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n
∑
AM =
j = 1
bond(Aj, Aj1) +bond(Aj, Aj+1)
J=4
45
0
45
0
0
80
5
75
45
5
53
3
0
75
3
78
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Approach for VF
• Find the combination that has the
maximum Affinity Measure
• It will represent best grouping among
attributes
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• We can find all possible AMs or
• We can use the BEA (Bond
Energy Algorithm)
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Bond Energy Algorithm
A1, A2, …, Ai-1, Ai, Aj, Aj+1, ……, An
AM1
AM2
AM3
AMold = AM1 + AM2+ AM3
AM2 involves Bond(Ai, Aj) twice
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BEA
A1, A2, …, Ai-1, Ai, Ak, Aj, Aj+1, ……, An
AM1
AM2
AM3
AMnew = AM1 + AM2+ AM3
AM2 involves Bond(Ai, Ak) and
Bond(Ak, Aj) twice
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BEA Contribution
cont(Ai, Ak, Aj) is the contribution in AM
when placing Ak between Ai and Aj
Cont(Ai, Ak, Aj) =
AMnew- AMold
= 2Bond(Ai,Ak) + 2Bond(Aj, Ak) –
2Bond(Ai, Aj)
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Steps in BEA
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• Input: The AA matrix
• Output: The clustered affinity matrix CA which is
a permutation of AA
• Initialization: Place and fix one of the columns of
AA in CA.
• Iteration: Place the remaining n-i columns in the
remaining i+1 positions in the CA matrix. For
each column, choose the placement that makes
the most contribution to the global affinity
measure.
• Row order: Order the rows according to the
column ordering.
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Bond Energy Algorithm
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•
•
•
Input AA
Output CA
Begin
CA (•, 1) ← AA (•, 1)
CA (•, 2) ← AA (•, 2)
Index ← 3
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While index ≤ n do
Begin
For I from 1 to index –1 by 1 do
calculate cont(Ai-1, Aindex, Ai) end-for
Calculate cont(Aindex-1, Aindex, Aindex+1)
Loc ← placement given by maximum cont value
For j from index to loc by –1 do
CA (•, j) ← AA (•, j-1)
end-for
CA (•, loc) ← AA (•, index)
Index ← index + 1
End-while
Order the rows according to relative order of columns
End {BEA}
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Thanks
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