2. GRAPH PARTITION PROBLEM
Input: G = (V, E, Ω1, Ω2, m) where
Z+
Z+
Ω1 :
Ω2:
m :
Notations:
Partition: π :
E —>
V —>
: weights on edges
: weights on nodes
number of subsets of G
V --> { π1, π2, π3,..., πm}
where
π = {v ε V |π(v)= πi,} i|1≤ i ≤ mi
such that πi n πj = for i = j
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3. NOTATIONS CONT…..
o Size of subset πi :
S(π i) = ΣΩ2(v) for all v in πi
W(π) = ∑|S(π i) – S(π j)| i,j| 1≤ i <j≤ m
(Imbalance weight)
o Cut size :
1. Ci = ΣΩ1(u,v) for all (u,v) ε E| u ε πi and v ε πjj
2. Cut(π) = ΣCi for all i,j| 1≤ i <j≤ m
(Cut Size Of π )
j
Objective: To find a partition of the nodes of G into m
disjoint, equal-sized subsets such that the sum
Cost = ω1*Cut(π)+ ω2*W(π)
is minimized
o It’s NP Complete problem 3
4. GENETIC ALGORITHM
Requirements
1. Np:
2.Ng :
Population Size at each iteration
Number of generations(iterations ) required
Cross Over rate (Pc% chromosomes undergo
cross over at each iteration)
mutationrate (Pm% of genes undergo
mutation)
3. Pc :
4. Pm:
5.
6.
α(alpha) : Scaling parameter
ω1 & ω2 : weights for cutcost & imbalance weight
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6. INTIAL POPULATION
(Pro)i = Ω2(vi) * Random(0,1) * N;
ȵ = μ = Maxi{(Pro)i}/
Where α is known as scaling factor
o (Pr )i = (Pro)i + Uniform(-ȵ, μ)
o Form Matrix X of above elements
as shown below
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8. COST ,MATES & CROSS OVER
For each chromosome form a partitioned graph & find the
corresponding Cut cost & fitness cost (Draw)
MATES Selection
Select a randomnumberr in (0,1) & select corresponding
chromosome for Mutation if r < Pc for cross over.
Cross Over
Select a number randomly from {1,2,3,…….ColX} & do
the cross over in the selected chromosomes
Replace old chromosomeswith their offsprings.
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9. MUTATION
Let Pm* RowX = # of genes undergo mutation = K
Generate K random numbers from {1,2,3 ………RowX}
Also K random numbers from {1,2,3 …….ColX}
For each element at intersectionreplaceit by a random number
generated between {MinX, , , , , , MaxX}
This way we get a new matrix X for next generation
computation.
Store the minimum value of cost at each iterations
At the end of finite number of iterations(Ng) find the
minimum among the stored cost values & corresponding
partition gives suboptimalpartition of G
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10. PARAMETER TUNNING
If we consider Np between 150 to 200
Ng between 80 to 100, Pc= 0.06 & Pm= 0.001
Above
GPP.
are sufficient to arrive at good solution to
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11. APPLICATIONS
VLSI circuits
Routing in distributed systems
Mapping parallel programming
Image segmentation in the field of computer
visions
Approximately computing suboptimal answers
most of the NP complete problems
to
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