1. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print),
INTERNATIONAL JOURNAL OF COMPUTER ENGINEERING &
ISSN 0976 - 6375(Online), Volume 4, Issue 6, November - December (2013), © IAEME
TECHNOLOGY (IJCET)
ISSN 0976 – 6367(Print)
ISSN 0976 – 6375(Online)
Volume 4, Issue 6, November - December (2013), pp. 204-211
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AN APPROACH TO SOLVE GRAPH PARTITIONING PROBLEM USING
GENETIC ALGORITHM
Prof. Sharadindu Roy,
Prof. Samar sen Sarma
University of Calcutta
ABSTRACT
This paper addresses the problem of partitioning a graph and particular the use of the genetic
algorithms for graph partitioning. It is a NP hard problem. The solution of a graph (circuit)
partitioning problem is global optimum. In this practical paper solution is easy and we can easily
apply genetic operator in this type of problem. One of the application of Graph partitioning in chip
designing. The target of our approach is to design modular chip which can be realized with self
standard IC chip if VLSI circuit can be converted into a graph and obviously we have taken into
account minimum power consumption.
Our approach to partition the Graph into sub Graphs and minimization of the number of cuts
is our main objective. Our approach with Genetic algorithm calculates the fitness (number of cuts)
and discard the solution with low fitness value. Multi way partitioning can be easily implemented if
we increase number of crossover point. Fitness value achievement depends on crossover and
mutation probability. We can change crossover boundary when fitness value is low in previous
generation and continue until optimum result is found.
keywords: Genetic Algorithm, NP-Hard, Net List, Crossover, Mutation, Cut Size, Partitioning.
INTRODUCTION
Graph partitioning consist of dividing a graph into parts(sub graph) each of which can be
implemented as a separate component(e.g. a chip) that satisfies certain design constraints one search
constraints is the area of the component. The limited area of a component forces the designer to lay
out a graph (circuit) on several components. There has been a larger amount of work done in the area
of graph partitioning and clustering. In graph partitioning, the graph is divided into two
(Bi-partitioning) or more (Multi-way partitioning) parts.
A chip may contain several million transistors. Due to the limitations of memory space and
computation power available it may not be possible to lay out the entire chip in the same step.
Therefore the chip is normally partitioned into sub-chips. These sub-portioned are called blocks. The
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2. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print),
ISSN 0976 - 6375(Online), Volume 4, Issue 6, November - December (2013), © IAEME
actual partitioning process considers many factors such as the size and of the blocks and number of
interconnection between blocks. The output of partitioning is a set of blocks and the interconnections
required between blocks. Minimization of delay due to inter connection between partitions and ratiocut minimization. Efficient easily applied algorithms for optimal clustering to minimize delay in
digital networks were developed by Lawler at al.kernighan and lin proposed heuristic for two way
partitioning which was the first iterative algorithm based on swapping of vertices.
PARTITIONING IS REQUIRED TO MINIMIZE THE NUMBER OF CUTS
The number of inter-connection among partitioning has to be minimized. Reducing the interconnection not only reduce the delay but also reduce the interface between the partitioned making it
easier for independent design and fabrication. It is also called min-cut problem.
Area of each partition is used as a constraint to reduce the fabrication cost with minimum
area or as a balance constraint so that partitions are of all most equal size. Various researchers have
achieved varying level of success using various optimization techniques. The current work attempts
to use simple genetic algorithm for multi way graph partitioning. Because genetic algorithm is an
emerging technique. This technique has been applied to several problems, most of which are graph
related because the genetic metaphor can be most easily applied to these types of problems. Genetic
requires more memory but take less time.
PARTITIONING MEANS WE CAN
Divides the circuit into smaller partition that can be efficiently handled Start with initial
partition Count cut size Find best partitioning (minimum cut size).
THE GENETIC ALGORITHM INVOLVES THE FOLLOWING BASIC STEPS
Evaluation( evaluating the fitness value) Crossover(Part of the first parent copied rest is
copied from second parent. Two crossover points are selected, binary or octal string from beginning
of chromosome to the first crossover point is copied from parent 1. The part from the first to the
second crossover point is copied from the second parent and the rest is copied from the first parent.
crossover point is copied from parent 1. The part from the first to the second crossover point is
copied from the second parent and the rest is copied from the first parent Mutation rate is the
probability of mutation which is used calculates number of bits to be muted. The mutation operator
preserves the diversity among the population which is also very important for the search. Mutation
probabilities are smaller in natural population leading us to conclude that mutation is appropriately
considered a secondary mechanism of genetic algorithm adoption.
Individual
1101011101110100
and mutation probability, Pm = 0.375
Then we say, 6 bit will be inverted.
Cut size
Cut size is number of edge between partitions
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3. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976
0976-6367(Print),
ISSN 0976 - 6375(Online), Volume 4, Issue 6, November - December (2013), © IAEME
SOLUTION METHODOLOGY
Graph will be partitioned initially with the cut. It is pointless to consider the cut cost of such
nitially
nets, since no matter how the graph circuit is partitioned and the net-list is traversed in a breadth
list
breadthfirst-search order and the nodes are assigned to the chromosome in this order. If two nodes are
directly connected to each other, there is a high probability that there partition bits will be closed to
part
each other in the chromosome.
The BFS sequence of the net-list will be parent 1 and any sequence of the net
net list
net-list will be
parent. The genetic algorithm are not guar
guaranteed to find the minimum cost (optimum), but able to
find very good solutions for a wide range of problem. Each generation choice to best chromosome
(Best fitness) from previous generation for parent.
Best
parent
Chromosome representation
For binary encoding
B
A
Total node 6
Total edge 7
Cut size 5
D
C
F
E
Parent 1
A
E B
Parent 2
E B A
F D C (BFS sequence)
D C
F (Any order)
Encoded Parent 1
1 1 0 0 0 1
Encoded Parent 2
1 0 1 0 1 0
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4. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print),
ISSN 0976 - 6375(Online), Volume 4, Issue 6, November - December (2013), © IAEME
Algorithm
Step1:
Start
step2:
Input the graph (with Initial partition)
step3:
Traversed the graph (In BFS and Any sequence)
Step4
Encode sequence(traversing) as partition number.
Step5:
Select parent1 and parent 2
Step6:
Define crossover type and mutation probability
Step7:
Apply crossover operator on parents
(Created child1 and child2)
Step8:
Apply mutation operator on
offspring (on child1 and child2)
(created muted1 and muted 2)
Step9:
Count fitness for each individual (calculate cut size)
Step10:
If optimum result is found then go to step14 else go to step11
Step 11:
Chose two best fitness chromosomes for next generation
(choice1 for 1st choice, choice 2 for 2nd choice)
Step12:
If choice1 & choice2 are not equal with previous generation then
go to step 7 else go to step 13
Step13:
Change crossover boundary and mutation probability and go to step 7
Step 14:
stop
Consider the graph with block number is assigned as in the following.
A
B
E
G
C
D
H
F
0
1
2
3
O
P
K
L
Q
R
M
N
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5. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print),
ISSN 0976 - 6375(Online), Volume 4, Issue 6, November - December (2013), © IAEME
Some sequence with cutsize(fitness) and average cut is given as tabular form in the following.
BFS sequence: A C D B O E Q R P H F G K M N L
INDIVIDUAL
CUT
0011020112332233
19
0001123233112031
19
0011123233112233
19
0 0 0 1 0 20 1 1 2 3 3 2 0 3 1
18
0011231233112233
19
0001 200112332031
AVG_CUT
23
19
EXPERIMENTAL RESULT AND DISCUSSION
The above algorithm has been coded and run in MATLAB(7.3 version). The following graph
contain 16 vertex and apply algorithm on the graph. This method has been used to minimize the
interconnections that are the min cut problem with a balance constraint.
Consider the graph with block number is assigned as in the following.
A
B
E
G
C
D
H
F
0
1
2
3
O
P
K
L
Q
R
M
N
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6. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print),
ISSN 0976 - 6375(Online), Volume 4, Issue 6, November - December (2013), © IAEME
In this code we calculate minimum cut, maximum cut and average cut(mean cut).Some
iteration wise result is given in tabular form in the following.
Number of Generation
MIN
FITNESS
1
10
18
14
2
10
16
12
21
10
14
11
40
10
13
11
80
10
13
11
91
10
14
11
100
MAX
FITNESS
10
13
We get mean fitness value = 11
AVG_CUT
11
This code also produce the following graph. This is “GENERATION VS FITNESS” graph.
“GENERATION” is plot in the X-axis(0-100) and “MEAN FITNESS” plot in Y-axis(0-20).
According to the graph we see that mean fitness value is 14 in generation 1 and reduced to 11 in
generation 10 and continues to 21.If we see all the generation ,Fitness value is 11 in most of the
generation. The optimum result is heavily depends on cross over and mutation probability.
From the graph We see that mean fitness value = 11
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7. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print),
ISSN 0976 - 6375(Online), Volume 4, Issue 6, November - December (2013), © IAEME
CONCLUSION
“Genetic Algorithms are good at taking large, potentially huge search spaces and navigating
them, looking for optimal combinations of things, solutions you might not otherwise find in a
lifetime.”
According to the Charles Darwin’s theory –Choose the best ,Discard the rest is followed in this
paper. We have followed to Holland regarding Genetic algorithm. The algorithm can partition graph
into a number of sub-graph. Our method calculates the fitness value and discards solutions with low
fitness value. Increasing the number of crossover points does not necessarily increase the fitness, as
the fitness achieved depends on crossover and mutation probability. If the graph size will increase
,Binary encoding is less promising method .In those case Other method like octal encoding is used.In
the above case we use octal encoding.
We have followed survival of fittest of Charles Darwin’s theory. Main philosophy of genetic
algorithm is followed to Holland. Circuit partition is one of the key areas in chip designing. The
algorithm can partition circuit into a number of sub-circuits. Our method calculates the fitness value
and discards solutions with low fitness value. The increase in number of crossover points does not
necessarily the fitness, as the fitness achieved depends on crossover as well as mutation probability.
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