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RESOURCE MANAGEMENT AND ALLOCATION IN COMPUTER AND COMMUNICATION NETWORKS Technical Report Department of Computer Science and Engineering University of Minnesota 4-192 EECS Building 200 Union Street SE Minneapolis, MN 55455-0159 USA TR 03-024 RESOURCE MANAGEMENT AND ALLOCATION IN COMPUTER AND COMMUNICATION NETWORKS Zhigang Gong May 22, 2003
Transcript
Page 1: RESOURCE MANAGEMENT AND ALLOCATION IN COMPUTER AND COMMUNICATION NETWORKS Technical Report ·  · 2015-02-12RESOURCE MANAGEMENT AND ALLOCATION IN COMPUTER AND COMMUNICATION NETWORKS

RESOURCE MANAGEMENT AND ALLOCATION IN COMPUTER AND COMMUNICATION NETWORKS

Technical Report

Department of Computer Science

and Engineering

University of Minnesota

4-192 EECS Building

200 Union Street SE

Minneapolis, MN 55455-0159 USA

TR 03-024

RESOURCE MANAGEMENT AND ALLOCATION IN COMPUTER

AND COMMUNICATION NETWORKS

Zhigang Gong

May 22, 2003

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UNIVERSITY OF MINNESOTA

This is to certify that I have examined this copy of a doctoral thesis by

Zhigang Gong

and have found that it is complete and satisfactory in all respects, and that any and all

revisions required by the final examining committee have been made.

Dingzhu Du

Name of Faculty Adviser(s)

Signature of Faculty Adviser(s)

Date

GRADUATE SCHOOL

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RESOURCE MANAGEMENT AND ALLOCATION IN

COMPUTER AND COMMUNICATION NETWORKS

A THESIS

SUBMITTED TO THE FACULTY OF THE GRADUATE SCHOOL OF

THE UNIVERSITY OF MINNESOTA

BY

ZHIGANG GONG

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

DING ZHU DU, Adviser

April 2003

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ACKNOWLEDGEMENTS

First and foremost, I want to thank my advisor, Dr. D. Z. Du, for encouraging and

helping me take this path which was beyond my own expectation. It is impossible to get

this far without you, Dr. Du.

I want to thank Dr. Shituo Han for conversations that revealed the beauty of many

classical complexity theory problems. I also benefited from many discussions with Dr.

Lu Ran, Dr. Xiuzhen Cheng and other members of the wireless network and mobile

computing group.

Thanks also goes to Dr. Kerry R. Kelts, who believed in me when I was right out

of college and supported me financially and spiritually. Kerry, thanks, I made it, finally,

although from a different side of the mountain. I also want to thank my middle school

math teacher, Mr. Raosheng Zhang, for influence that lasted a long time.

Finally, last but not least, thanks to my wife, my parents and my in-laws for

insisting and supporting me in the last few years to take this challenge.

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DEDICATION

For my mother, Fenglan, an extraordinary woman.

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ABSTRACT

This thesis focuses on the design and application of approximation algorithms in

computer and communication systems. Many problems in these computer and

communication systems are known as NP-hard in combinatorial computation complexity

theory. Several of them can be solved approximately using a common approximation

theory approach: the combination of greedy, linear programming and randomization. We

first extended the known approximation techniques and then applied the technique to

several chosen problems.

Multifiber wavelength assignment problem is in optical communication

infrastructure area, we applied the combined technique above and ran simulation based

on the NSFNet and a randomly generated topology. Criticality driven session admission

in Enterprise Resource Management Software is another example in computer software

system that benefits from the proposed technique. Further, we applied the technique in

communication network protocol design area and obtained best know result so far in

balancing the cost and performance in minimum cost QoS multicasting.

Finally, we proposed a new polynomial time approximation algorithm to solve the

operational version of Wavelength Assignment Problem. The new algorithm is in graph

coloring form but is based on the core of Dantzig’s simplex method: optimal is within the

extreme points. We show that the computation complexity of this approach is O(N2W),

which is as good as the existing sequential greedy algorithm in theory. Further, the

simulation on randomly generated data set as well as NSFNet data set support the

conclusion that the approximation factor of the proposed algorithm is close to that of the

existing one.

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TABLE OF CONTENTS

ABSTRACT .................................................................................................................................................. 6

CHAPTER 1 INTRODUCTION............................................................................................................ 9

1. RESOURCE MANAGEMENT PROBLEM ....................................................................................................11

2 GENERAL COVER PROBLEM ...................................................................................................................11

3 RELATIONSHIPS BETWEEN RMP AND GCP.............................................................................................12

3.1 Transformation ..............................................................................................................................12

3.2 Relaxation ......................................................................................................................................14

3.3 Duality ...........................................................................................................................................15

4. IMPROVEMENT WITH RANDOMIZATION .................................................................................................16

4.1 Basic Idea ......................................................................................................................................16

4.2 Analysis ..........................................................................................................................................16

CHAPTER 2 ENTERPRISE RESOURCE PLANNING SYSTEM SESSION MANAGEMENT..18

1. RESOURCE MANAGEMENT PROBLEM AND FORMULATION ......................................................................18

1.1 What is resource management in session based ERP system?.......................................................18

1.2 Formulate Resource Management Problem ..................................................................................20

1.3 Formulate the General Cover Problem .........................................................................................22

1.4 Related work ..................................................................................................................................23

2. SOLUTION..............................................................................................................................................23

2.1 Greedy Algorithm...........................................................................................................................23

2.2 Linear Programming Algorithm ....................................................................................................26

3. CONCLUSION .........................................................................................................................................28

CHAPTER 3 MULTI-FIBER WAVELENGTH ASSIGNMENT .....................................................29

1. INTRODUCTION .....................................................................................................................................29

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2. RELATED WORKS .................................................................................................................................31

3. NETWORK MODEL AND PROBLEM SPECIFICATION ...............................................................................32

4. OUR APPROACHES................................................................................................................................35

4.1 A Greedy Algorithm Based on LP Relaxation................................................................................36

4.2 The Primal-dual Based Adjustment Heuristic................................................................................38

5. SIMULATIONS ......................................................................................................................................41

6. CONCLUSION ........................................................................................................................................48

CHAPTER 4 WAVELENGTH ASSIGNMENT USING GRAPHED SIMPLEX METHOD.........49

1. INTRODUCTION ......................................................................................................................................49

2. FOUNDATION .........................................................................................................................................53

3. OUR APPROACH .....................................................................................................................................60

GRAPHED SIMPLEX ALGORITHM ...............................................................................................................61

4. SIMULATION ..........................................................................................................................................63

4.1 Random generated path graph.......................................................................................................63

5. DISCUSSION ...........................................................................................................................................67

CHAPTER 5 CONCLUSION ...............................................................................................................70

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CHAPTER 1 INTRODUCTION

This thesis deals with designing and applying polynomial time approximation

algorithms to solve NP-hard optimization problems, particularly the optimization of

resource utilization problems in computer and communication systems. Typically, the

decision versions of these problems are in NP, and are therefore NP-complete. From the

viewpoint of exact solutions, all NP-complete problems are equally hard, since they are

reducible to each other via polynomial time reductions. Such a reduction maps optimal

solutions of the given instance to optimal solutions of the transformed instance and

preserves the number of solutions.

On the other hand these NP-hard problems are very unlikely to be solved by

polynomial time algorithms since P � NP. Despite the invention of new high speed

computers, obtaining the exact optimum solution for any of these optimization problems

will require too much computing time: years, decades, centuries or even longer, for input

of only moderate size. However, the practical values of the solutions to these types of

optimization problems drive the scientists to find the optimal or near-optimal solutions in

relatively short times even when the sizes of the problems are substantially large. The

polynomial time algorithms that are designed to output near-optimal feasible solutions for

NP-hard optimization problems are referred to as approximation algorithms in

combinatorial complexity theory.

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We have many new terms to clarify in this section. An optimization problem is a

problem with a series of constraint and an objective, usually the objective is either

maximize or minimize something such as cost, length or weight. We categorize the two

types into maximization problems or minimization problems accordingly. A feasible

solution is a solution that satisfies the problem’s constraints. An objective function is a

polynomial time computable function that outputs a rational number after adopting a

feasible solution. An optimal solution is the best among all feasible solutions that

produces the maximum or minimum value on objective function. The approximation

solution is the feasible solution close to the optimal solution for the NP-hard optimization

problem when the exact solution is hard to find through polynomial time computation.

We measure how close the approximation solution APP(I) for a specific optimization

problem instance I is compared to the optimal solution OPT(I) by approximation factor:

the ratio between APP(I) and OPT(I) for maximization problem or OPT(I)/APP(I) for

minimization problem.

In this thesis we will give the introduction in Chapter 1; then develop and apply

the techniques to computer and communication system problems in Chapter 2, 3, 4, and

5. In Chapter 2, we will apply the approximation algorithms to session admission control

problem in enterprise resource planning software area. In Chapter 3, we will apply the

algorithms to solve multifiber wavelength assignment problem in optical network

communication infrastructure area. In Chapter 4, we developed a new technique to solve

the NP-complete Wavelength Assignment Problem and compared the technique to the

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existing sequential greedy algorithm using randomly generated data set and NSFNet

traffic data set. Finally, we designate Chapter 5 for our conclusion.

1. Resource Management Problem

Resource-Management-Problem: Given certain resources R = { mrrr ,...,, 21 },

maximize the finished jobs X = { nxxx ,...,, 21 }, under the resource restrictions described

as following. C = { njmicij ..1,..1, �� } is the cost factor. W = { nwww ,...,, 21 } is the

weight factor.

)1(

}1,0{,...,,:

...

...

...

...

..

...:

21

2211

22222121

11212111

2211

�����

��������

����

n

mnmnmm

nn

nn

nn

xxxGiven

rxcxcxc

rxcxcxc

rxcxcxc

TS

xwxwxwUMAX

2 General Cover Problem

General-Cover-Problem: Given certain amount of "jobs" Y = { nyyy ,...,, 21 },

minimize the certain resources used to cover all "jobs". C = { njmicij ..1,..1, �� } is the

cost factor. W = { nwww ,...,, 21 } is the weight factor. B = { mbbb ,...,, 21 } is some

constraint factor.

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)2(

}1,0{,...,,:

...

...

...

...

..

...:

21

2211

22222121

11212111

2211

�����

��������

����

n

mnmnmm

nn

nn

nn

yyyGiven

bycycyc

bycycyc

bycycyc

ts

ywywywVMin

3 Relationships between RMP and GCP

3.1 Transformation

If we set ii yx ��1 , then expression (1) becomes:

)3(

}1,0{,...,,:

)...()...(

...

)...()...(

)...()...(

..

)...()...(:

21

221121

2222212122221

1121211111211

221121

���������

����������������

��������

n

mnmnmmmnmm

nnn

nnn

nnn

yyyGiven

rycycycccc

rycycycccc

rycycycccc

ts

ywywywwwwUMAX

Let nn ywywywV ���� ...2211 , maximize U is equivalent to minimize V because

nwww ��� ...21 is constant. Therefore (3) can be written as:

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)4(

}1,0{,...,,:

)...()...(

...

)...()...(

)...()...(

..

...:

21

212211

2222212222121

1112111212111

2211

���������

����������������

����

n

mmnmmnmnmm

nnn

nnn

nn

yyyGiven

rcccycycyc

rcccycycyc

rcccycycyc

ts

ywywywVMIN

(4) is (2) if we let 1111211 )...( brccc n ����� and so on. Therefore, we proved that

Resource-Management-Problem can be transformed into General-Cover-Problem.

Next, we prove that General-Cover-Problem can be transformed into Resource-

Management-Problem. Let ii xy ��1 , then expression (1) becomes:

)5(

}1,0{,...,,:

)...()...(

...

)...()...(

)...()...(

..

)...()...(:

21

221121

2222212122221

1121211111211

221121

���������

����������������

��������

n

mnmnmmmnmm

nnn

nnn

nnn

xxxGiven

bxcxcxcccc

bxcxcxcccc

bxcxcxcccc

ts

xwxwxwwwwVMIN

.

Same as above, minimize V is maximize U = ( nn xwxwxw ��� ...2211 ) since

( nwww ��� ...21 ) is constant. (5) becomes:

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)6(

}1,0{,...,,:

)...()...(

...

)...()...(

)...()...(

..

...:

21

212211

2222212222121

1112111212111

2211

���������

����������������

����

n

mmnmmnmnmm

nnn

nnn

nn

xxxwhere

bcccxcxcxc

bcccxcxcxc

bcccxcxcxc

ts

xwxwxwUMAX

(6) is the same as (1) as long as 1111211 )...( rbccc n ����� and so on so forth.

3.2 Relaxation

Solving (1) or (2) directly to obtain an exact solution is NP-hard because the

constraint on X and Y being binary. To obtain an approximate solution, a technique

called relaxation can be used. Relaxation is an important technique for designing

approximation algorithms. By relaxation, we mean to relax some constraints on feasible

solutions, so that the feasible domain is enlarged. The purpose of relaxation on selected

feasible domain is to change the intractability of the optimization problem. For example,

GCP is NP-hard and its feasible domain consists of 0 or 1. If we relax the feasible

domain to contain all rational numbers between 0 and 1, the problem is not NP-hard

anymore and an exact solution to the relaxed version of the optimization problem is

possible within polynomial computation time.

However, the optimal solution obtained from relaxation is often not a feasible

solution of the original problem. In case of GCP, it is more than likely the solutions fall

between 0 and 1, instead of being either 0 or 1. Thus, one has to modify it to construct a

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feasible solution for the original problem. For GCP, we use greedy approach to bridge

this modification.

From the idea of relaxation, we see that an approximation uses relaxation

technique usually in two steps. In the first step, one analyzes the feasible solutions to get

a new relaxed feasible domain and an optimal solution in the new feasible domain. This

solution usually is not a feasible solution for the original problem. Thus, one needs to

modify it to obtain a feasible solution in the second step.

3.3 Duality

The linear-programming-based approximation is good for large-scale problems

because there are a lot of software packages for solving linear programming problems.

However, solving linear programming problems are slow in general. This limits the

applications of the approximation technique. In this section, we introduce a technique

called duality to speed up computation time of the approximation. An interesting

technique about rounding is to use the duality theory in linear programming. From the

duality theory, we know that every dual feasible solution (i.e., a feasible solution of dual

linear programming) provides a lower bound for minimum value of primal linear

programming. Thus, we can use this lower bound to establish the performance ratio of

approximation. This means that a “good enough” dual feasible solution may do the same

job as the optimal solution of the linear programming. This would save running time

dramatically in some cases.

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4. Improvement with Randomization

4.1 Basic Idea

The basic idea of randomization is to randomly round the fractional value of the

optimal solution to either 0 or 1 with a fixed probability. The probabilities have to be

chosen in such a way that the output of this randomized algorithm has a non-zero

probability of being a feasible solution to the original integer program with the value of

the objective function not too far from the optimal fractional (hence integer) solution.

This randomized algorithm can be turned into a deterministic algorithm by a de-

randomization procedure using a so-called pessimistic estimator which is simply an upper

bound on certain conditional probabilities. The interested reader can find out more about

these techniques in [5].

4.2 Analysis

For simplicity, we only analyze the Set Cover Problem with randomized

rounding.

nixSSxtosubject

xwMinimize

ij

Svi

j

n

i

ii

ji

������

��

1},1,0{,;1:

1

We relax this integer program and allow each variable xi to assume values in the inetrval

[0, 1]. Let ix̂ be the value assigned to xi in an optimal fractional solution of the relaxed

program. We can supply randomized rounding and set each variable xi to 1 with

probability ix̂ . Clearly, the expected weight of the elements chosen to the cover is equal

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to � �

n

i ii xw1

ˆ . What is the probability that subset Si is covered? Suppose that Si contains

elements v1, v2, …, vk; we know that � �

k

j jx1

. The probability that Si is covered is:

ekx k

k

j

j

11)

11(1)ˆ1(1

1

���������

That is, the probability that subset Si is covered is a constant. To increase the probability

of covering the family of subsets S, randomized rounding can be repeatedly applied to the

set of variables that were not set to 1. For example, by repeating the randomized

rounding procedure t=O(log m) times, we can guarantee that the probability that a subset

Sj is not covered is at most m2

1. Thus, the probability that S is not covered after t

repetitions is at most 2

1. The expected weight of the cover after t repetitions is at most

� ��

k

i ii xwt1

ˆ , i.e., t times the weight of an optimal fractional cover. Thus the conclusion

is that for every instance {V, S} of the set cover problem, randomized rounding finds an

O(log m)-approximate cover, with probability at least ½.

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CHAPTER 2 ENTERPRISE RESOURCE PLANNING SYSTEM

SESSION MANAGEMENT

1. Resource management problem and formulation

1.1 What is resource management in session based ERP system?

The computer system today has experienced exponential growth in its

computation power with the innovation in memory and hard disk. However, this growth

also induced exponential growth in resource demand because high power made high

resource-demanding applications possible. These applications range from office

productivity, to multimedia consumer software, to enterprise resource management and

supply chain management system.

For example, providing a web-based enterprise wide business solution is the

major theme of the next phase of information technology evolution. With a web-based

system that links customers, suppliers, and enterprise users, the resource management

problem has suddenly added a new flavor. The new characteristics are criticality based,

Quality of Service bounded, and being session specific.

In general, the available computer and network resources are not enough to satisfy

all applications at their best QoS at all time. Considering the following scenario: a cable

company had a sudden shortage in an area, people started log in to see what’s going on.

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How do you allocate the resource such that your high premium customers can logged on

while the low premium users does not get resources or being preempted if necessary.

Different users and sessions has different criticality. This is the criticality driven

application.

Another aspect is the quality of services. One of the major characteristics of the

Internet activity is that the load is not predictable. When there is breaking news, the

system load can be many times of magnitude more than the regular load. Today the

common three tier architecture is trying to resolve the problem by adding the middle

business logic tier to solve the work load balance problem. However, few of the software

system considered the Quality of Services factor in admission control. When a request of

services arrive, the system admits it without making any quality of service judgment

despite the system response time is already long. Therefore, the whole system has to be

shut down and restarted with more instances add, this further brings the problem of lost

session data or added delay if persistent session is required.

The third aspect of the ERP resource management problem is that they are session

based. The session based characteristic is the major difference from per processor based

resource management. A session is an instance of application executing in the system

that uses CPU, memory, disk IO and network.

A Resource Management System is responsible for dividing system resources

among resource competing applications and gives precedence to applications with higher

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criticality. Before an application starts execution, it submits its resource requirements in

a range of minimum and maximum and also criticality level to the resource manager.

The resource manager (RM) determines availability and assigns resources to the new

application that would satisfy at least the minimum QoS. When there are not enough

available resources, the RM takes resources away from lower criticality applications and

allocates them to the new higher criticality application. With resources allocated, the

new application is ready to be scheduled for execution.

The method to allocate and distribute multiple resources to many applications is

not trivial. It can have a big impact on overall system utilization and application QoS.

This chapter defines a mathematical model for the problem of resource management for

multiple resources (RMP) and describes two solutions based on the greedy method and

on linear programming approximation of the ERP General Cover Problem that are close

to the optimal solution.

This chapter continues with a description of the application model in the

remaining of this section. Then, the problem definition and the two solutions are listed in

section 2. Section 3 presents some related work and section 4 concludes the paper with

some final remarks.

1.2 Formulate Resource Management Problem

We model each resource of type i as a bucket with capacity ),...,1(, miri � , where

m is the number of resource types in the system. Each session j, j �(1 … n), then

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demands ija amount of resource i. The resource constraints on Resource i (such as

memory) for all the admitted sessions on the system can be described as:

),...,1(,1

mirsan

j

ijij ����

,

where js = 1 if session j is admitted and js = 0 if session j is not.

Usually the resource management system is designed to meet a specific set of

goals, such as maximizing the overall system utilization, thus reduce the initial equipment

investment, or maximizing QoS for critical sessions. In this thesis, we focus on

optimizing the Quality of Services for critical sessions. This optimization goal can be

mathematically stated as the following:

nnscscscUMaximize ���� ...2211

where jc > 0 is a weighted factor, representing the criticality of session j.

Thus maximizing U actually maximizing the overall criticality of the admitted

sessions.

Without loss of generality, the Resource Management Problem (RMP) can be

formulated as:

��

��

����

n

j

ijij

nn

mirsa

TS

scscscUMaximize

1

2211

),...,1(,

..

...:

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where jc is the criticality associated with each session j, ija is the weight factor

of session j’s demand on resource i, and is = 1 if session i is admitted, 0 otherwise.

This problem is known to be a NP hard optimization problem [5]. Thus we need

to take approaches such as approximation scheme to solve the problem. Note that [5] is

special case of this problem where i = 4 for four types of resources.

1.3 Formulate the General Cover Problem

Another category of the problems, in resource-demand management, is General-

Cover-Problem (GCP): Given certain amount of "jobs", minimize the certain resource

used to cover all "jobs".

��

��

����

n

j

ijij

nn

mibsa

TS

scscscVMinimize

1

2211

),...,1(,

..

...:

where jc is the criticality associated with each session j, ija is the weight factor

of session j’s demand on resource i, and is = 0 if session i is admitted, 1 otherwise.

This problem is also a NP hard optimization problem.

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1.4 Related work

Extensive research can be found in the area of resource management optimization.

This work is best understood in the context of [5][4]. [5] is based on linear programming

and [4] is based on primal-dual theorem. The simulation in [4] suggested that the linear

programming approach is as good as the primal-dual solution in performance ratio.

Further APP � OPT-4, which means that the approximation schedules at most 4 session

less than the optimal.

2. Solution

With the understanding that GCP is NP hard, we present two approximation

algorithms. The first one is a greedy algorithm with performance ratio of

)),1(,ln(max11

��

��m

i

ij nja and running time of O( mn2 ). The second one is linear

programming approach that has a performance ratio of )),1(,ln(max1

��

�n

j

ij mia and

running time of O( mn2 ).

2.1 Greedy Algorithm

In this section we present a greedy algorithm for solving the GCP and then we

find its performance ratio and running time. This algorithm is also presented in [4] and

[5].

General Cover Problem Greedy Algorithm (GCPG):

I = {1, 2, ... , m}

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24

J = {1, 2, ... , n}

while I �� do

find j* � J such that

)/(max/ ''** ��

Ii

jijJj

Iijij

caca

where );,min(' ��

��

Jj

ijiijij abaa

I � I - }|{ '* �

��

Jj

ijiijabai ;

J � J – {j*};

endwhile;

output the feasible solution X G as following:

.10 JforjJandXforjX G

j

G

j ����

Obviously, the running time of this algorithm is O(mn2). Let’s prove the

performance ratio of this algorithm is )ln(max11

1 ��

���m

i

ijnj a .

Theorem 2.1 The greedy algorithm of general cover problem GCPG produce an

approximation solution that is within a factor of )ln(max11

1 ��

���m

i

ijnj a from the optimal.

Proof: We will give an overview of the proof that was given by [4].

Suppose the algorithm select 1, 2, …, k as j* in the computation,

For each j � k, let i-weight of the xj defined by:

��

�m

i

ij

jij

a

cajiw

1

"

"

),(

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25

where

)),0max(,min(1

'"

'

��

��

j

j

ijijij abaa

for each j > k and i with aij > 0, let’s define the i-weight of xj by:

)1()),,(max(),( '' kjjiwjiw ���

for each i and j with aij = 0, let’s define the i weight of xj by:

0),( �jiw

The first observation is that .),(1 111

����� ���

��

k

j

m

i

k

j

j

n

j

G

jj jiwcxc

Therefore for each j, with m’ such that aij, …, am’j > 0 and am’+1j = amj = 0,

)ln1()...

...(),('

1'1

'

21

2

1

1

1

����

����

���

��m

i

ijj

jmj

jm

jj

j

j

j

j

m

i

acaa

a

aa

a

a

acjiw , see [4]

Suppose x* is the optimal solution of the GCP. Then

����

�11 *

),(),(

jx

k

j

jiwjiw

This follows directly from the definition of w(i,j) for j > k. Therefore

OPTaca

acjiw

jiwjiwjiwcAPP

m

i

ijnj

x

j

m

i

ijnj

x

m

i

ijj

x

m

i

m

i x

k

j

m

i

m

i

k

j

k

j

j

j

jj

j

))ln(max1()ln1(max

)ln1(),(

),(),(),(

1

1

11

1

1

'

11 1

1 11 1 1 11

*

**

*

���� ���

���� ���

���

����

� �� �

� �� � � ��

����

���

����

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26

2.2 Linear Programming Algorithm

In this section we will present a linear programming solution for the ERP resource

management problem. We will also present the performance ratio and running time.

This is also seen in [4] and [5]. The expanded IP form for this problem is:

)..1,..1(,0,,

},1,0{:

...

...

...

...

:

...min

2211

22222121

11212111

2211

njmicba

andxwhere

bxaxaxa

bxaxaxa

bxaxaxa

subjestto

xcxcxc

jiij

j

mnmnmm

nn

nn

nn

���

�����

��������

���

This IP problem can be relaxed to an LP problem by expand the solution domain

from xj �{0,1} to 0 � xj �1. First we notice that bi and cj can be assumed to be positive.

If bi = 0, we can remove the i-th constraint. If cj = 0, we may set xj = I and remove the

corresponding column. Suppose ),...,,( **

2

*

1

*

nxxxx � is an optimal solution of the listed

LP, we can use the following technique to obtain an approximation solution

),..,,( 21

A

n

AAA xxxx � :

Let A

jx =1 if ax j �* and A

jx =0 if ax j �* .

Therefore, we can summarize the algorithm as:

Find an optimal solution x* for the LP

Output the approximation x A as following:

Let A

jx =1 if ax j �* and A

jx =0 if ax j �* .

How do we determine the value of a? Let’s look at:

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27

����������

���������ax

jj

ax

jj

n

j

jj

ax

jj

ax

j

A

nn

AA

jjjj

xabxaxaxaaxaxaxa****

*

11

*

1

1

*

1

*

111212111 ...

Therefore the constraint has been transformed to make sure that:

1

*

11*

bxabax

jj

j

����

Because b1 is integer, thus we only need to make sure that 1*

*

1 ���ax

jj

j

xa . Because we can

choose a to make: 11

1

*

1*

�� ����

n

j

j

ax

jj aaxa

j

, therefore we can say the following:

If we choose ��

�n

j

jaa1

1/1 , then 11212111 ... bxaxaxa A

nn

AA ���� holds. Further, if we

choose a = 1/f, where ��

���

n

j

ijmi af1

1max , then i

A

nin

A

i

A

i bxaxaxa ���� ...2211 holds for

any i = 1..m.

Now we will prove that the performance ratio of this polynomial time

approximation is ��

���

n

j

ijmi af1

1max .

)(

)...(...

,,

**

22

*

112211

*

optimalf

xcxcxcfxcxcxc

fxxj

nn

A

nn

AA

j

A

j

���������

���

So xA is an f-optimal solution of the general cover problem.

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The running time of this solution is the sum of the running time of obtaining x*

and the running time of obtaining xA. The random algorithm solves LP problem with m

constraints and n variables in O(mn2). The running time to find f is O(mn). The running

time to get xA is O(n). Therefore the total running time is O(mn

2) + O(mn) + O(n) =

O(mn2).

3. Conclusion

Many session-based ERP systems have been developed and installed without

proper QoS integration in recent years. In this chapter we presented a mathematical

model and solution for this problem which belongs to resource management problem

category. The solution is obtained by reducing the problem to general cover problem

then obtained by using greedy algorithm and linear programming approach. Further we

presented performance ratio and running time of both solutions. For a specific case one

of the approach can be adopted based on the problem size.

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CHAPTER 3 MULTI-FIBER WAVELENGTH ASSIGNMENT

1. Introduction

With fast growth of the Internet and World Wide Web, the network bandwidth

requirements have increased dramatically in recent years. The theoretical research and

technology development in wavelength-division multiplexing (WDM) networks are now

evolving at a staggering pace [21] to fulfill the increasing bandwidth requirement and the

deployment of new network services. Wavelength routed all-optical WDM networks

provides a viable solution for future wide-area networks (WANs) and metropolitan-area

networks (MANs).

In wavelength-routed all-optical WDM networks, a lightpath is an optical

communication channel between two nodes. The same wavelength should be used by a

lightpath throughout all the links in its route [18], which is known as wavelength

continuity constraint. Two lightpaths must use different wavelengths if they share a

common fiber link. This is known as wavelength conflict constraint. In all-optical

networks, any pair of communication end ports must have a lightpath, where there is no

need of wavelength conversion. Since all-optical wavelength conversion is still not a

matured technology, the all-optical networks would remain to be the main stream of

WDM networks.

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Most of the previous research on routing and wavelength assignment (RWA) was

focused on single-fiber systems, in which it’s assumed that every fiber link contains only

a single fiber. Due to the constraints of wavelength continuing and wavelength conflict, a

single fiber WDM systems either requires a large number of available wavelengths, or

faces the problem of high probability of blockings. In this paper, we discuss the routing

and wavelength assignment problem in multi-fiber systems. We call it mf-RWA

problem. In multi-fiber WDM networks, each link contains a number of parallel fibers.

And each fiber is able to carry optical channels on a set of wavelengths. A wavelength

that cannot continue on the next hop on the same fiber can be switched to another fiber at

an optical cross-connect (OXC) if the wavelength is free on that fiber. Thus, multi-fiber

WDM networks can reduce the blockings of the system dramatically [18].

The traditional RWA problem is to find routes for a set of connections and to

assign wavelengths to them, such that the total number of wavelengths used is

minimized. The RWA problem was proved to be NP-complete [5,6].

In this paper we address RWA problem in a different way. We assume that the

number of wavelengths available on each fiber is given, and the routes of connections are

known. Our goal is to minimize the system blockings. The problem can be defined

formally as below: given a network G with F fibers per link and a set of connections C,

assuming that each fiber can support W wavelengths and the routes of C are known,

assign wavelengths to the connections in C, such that the number of connections blocked

is minimized. That is, we aim at maximizing the number of connections in C that can

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be assigned with wavelengths. This problem is NP-complete for single fiber network

[22]. Hence it is NP-complete for multi-fiber network.

We first formulate the mf-RWA problem as an integer linear programming (ILP),

and then propose two heuristics using some techniques for solving ILP. Our approach is

different from traditional methods for the wavelength assignment problem that uses

algorithms for coloring graphs. Our simulation study shows that increasing only a small

number of fibers of a link can considerably improve the quality of network performance.

2. Related Works

RWA problem in multifiber optical networks is a relatively new research topic,

where few results have been obtained. Jeong and Ayanoglu [18] extended Barry and

Humblet’s blocking probability analysis model [22] to the multifiber case. With their

stochastic approach, they found that multiple fibers could improve wavelength utilization

and network performance. A similar stochastic approach was proposed in [19]. Li and

Simha [21] considered RWA problem in multifiber WDM star and ring networks. Their

results showed that the ability to switch between fibers increases wavelength utilization.

They derived sharper per-fiber bounds on the number of required wavelengths for the

multifiber version of the wavelength assignment problem in star and ring networks.

Nomikos and Zachos [22] considered a different variant of RWA problem in multifiber

networks: how to use a minimal number of active links to satisfy a given set of requests.

They proposed an algorithm that can find an optimal solution to this problem in chains,

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and for the problem in stars and rings, which are NP-complete, they proposed some

approximation algorithms.

Nomikos and Zachos [21] considered the mf-RWA problem in single-fiber

networks with some special topologies. They presented a polynomial-time algorithm for

chain networks, and a 2/3-approximation algorithm for ring networks. In this paper, we

consider mf-RWA problem in general network (i.e., with arbitrary topology). Two

heuristics are proposed to solve it.

3. Network Model and Problem Specification

The network is modeled by a graph G=(V, E), where V represents the set of nodes

of the network, E represents the set of links available in the network. For any i, j �V, if

there is a link between i and j, then there are F parallel fibers between i and j. There are

W wavelengths available on every fiber in the network G. A connection is denoted by a

pair of nodes (s, d). A set C is a collection of t connections denoted by C={(s1, d1), … (st,

dt)}. We assume that traffic of each connection can be supported by one lightpath (i.e.,

one wavelength). If the traffic demand between a pair of nodes is more than what a

lightpath can support, it is split into multiple connections, each of which is supported by a

lightpath. Therefore, some connections in C may have the same node pair (s, d). A route

for a connection in C is a simple path in G, which is assumed to be known. Note that due

to the limit of number of wavelengths available, not every connection in C can be

assigned a wavelength without causing wavelength conflict.

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Since wavelength conversion is not allowed, each connection will be assigned

with a wavelength (throughout all links in its route). Our problem is to assign W

wavelengths to as many connections as possible under the wavelength conflict constraint.

A connection, which cannot be assigned with a wavelength, is said to be blocked.

Therefore, our objective can also be considered as minimizing the number of connections

to be blocked.

Since a wavelength on a fiber can at most support one connection, the number of

connections that pass through a link containing F parallel fibers by a wavelength is at

most F.

For the simplicity of presentation, we introduce the following notations.

i i-th connection in C , when used as a subscript;

ci the route for connection i;

w w-th wavelength number, when used as a subscript;

l, m endpoints of a link in G;

t the number of connections in C;

W the number of wavelengths available on a fiber;

F the number of fibers on a link;

),( ml

i� indicator, ),( ml

i� =1 if connection i uses link (l, m); otherwise ),( ml

i� =0;

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wix , variables, wix , =1 if connection i is assigned wavelength w; otherwise wix , =0;

),(

,

ml

wix variables, ),(

,

ml

wix =1 if connection i goes through link (l, m) and

assigned wavelength w; otherwise ),(

,

ml

wix =0;

The problem then can be formulated as 0-1 programming as following.

(6) ....,,2,1,...,,2,1,),( 1,or 0

)5(;,....,2,1,,...,2,1 ,1or 0

)4( ;,...,2,1,...,,2,1,),( ,

(3) ;,...,2,1,),( ,

(2) ;,...,2,1 ,1

)1( )max(

),(

,

,

,

),(),(

,

1

),(

,

1

,

1 1

,

WwtiEmlx

Wwtix

WwtiEmlxx

WwEmlFx

tixtoSubject

xILP

ml

wi

wi

wi

ml

i

ml

wi

t

i

ml

wi

W

w

wi

t

i

W

w

wi

�����

��������

����

��

��

��

� �

The objective function (1) is to maximize the number of connections that are

assigned the wavelengths. Since for connection i that is not assigned any wavelength,

we have wix , = 0 for w = 1, 2, … , W, so it contributes nothing to the objective function.

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The inequality (2) is to ensure that at most one wavelength can be assigned to

connection i, i=1, 2, … , t.

The inequality (3) represents the wavelength conflict constraints. It ensures that

at most F connections that pass link (l, m) can be assigned with wavelength w.

a) The equality (4) represents the wavelength continuity constraint. It ensures that

if connection i is assigned a wavelength w, then every link in the route of the

connection shall be assigned the same wavelength.

Since inequality (3) and equality (4) together mean the number of connections

that pass through a link containing F parallel fibers by a wavelength is at most F, they

can be merged into the following inequality, and as a result, ),(

,

ml

wix will disappear in the

ILP.

.,...,2,1,),(

}),|({

, WwEmlFxicmli

wi ������

4. Our Approaches

We have formulated the mf-RWA problem as integer linear programming, which

was known as NP-hard. So in this section, we propose two heuristics to solve it.

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4.1 A Greedy Algorithm Based on LP Relaxation

This heuristic is a two-stage algorithm. At the first stage, an optimal solution for

a linear programming (LP) relaxation of the ILP is computed. The obtained solution to

LP may be fractional, so it may not satisfy the integer constraint (5) and (6). At the

second stage, a greedy algorithm is employed to find an integral solution based on the

optimal solution obtained in the first stage.

At the first stage, we obtain the LP relaxation of the ILP by removing the integer

constraint on variables wix , for all i and w, and then solve it. The LP formulation

becomes:

.,....,2,1,,...,2,1 ,10

;,...,2,1,),( ,

;...,,2,1 ,1

)max(

,

}),|({

,

1

,

1 1

,

Wwtix

WwEmlFx

tixtoSubject

xLP

wi

cmli

wi

W

w

wi

t

i

W

w

wi

i

����

����

��

��

��

� �

At the second stage, after we get the optimal solution { *

,wix } to the LP, we sort the

components of { *

,wix } in descending order. Without loss of generality, we denote them

by nxxx ��� ....21 after relabelling the subscripts. Now we round those fractional

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37

components in { *

,wix } and obtain an integer solution to the ILP. Here we adopt a greedy

strategy that tries to set variables mostly close to 1 to 1. Let A

ix be a solution to the ILP.

First, let 11 �Ax and 0�

A

jx , for j�1. If this does not violate the constraints of the ILP,

then set 11 �Ax , otherwise set 01 �

Ax . Second, we consider the value of Ax2 . Let Ax2 =1

and 0�A

jx , j �1, 2, and Ax1 unchanged. If this does not violate the constraints of the

ILP, then set Ax2 = 1,otherwise set Ax2 = 0. This process is repeated until A

nx is set either

1 or 0. The heuristic is more formally presented as follows.

The Greedy algorithm for mf-RWA Problem

Formulate the ILP of mf-RWA problem.

Solve the LP of mf-RWA problem.

Sort the components of obtained optimal solution { *

,wix } to LP,

Get and denote the resulting order, nxxx ��� ....21 .

For i=1 to n do

set 1�A

ix ,

set 0�A

jx , for j � i.

If {A

ix } satisfies the constraints of the ILP

Then set 1�A

ix else set 0�A

ix .

set i = i+1.

End-for

Return solution { A

ix }.

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We have seen that in the above heuristic, the optimal solution to a linear

programming relaxation is employed to find out the priority of variables being assigned

with 1. There are two disadvantages with this approach:

(1) Computing the optimal solution takes O(n3.5

) time [22] for LP of n variables. It is the

main portion of the total computation time for this heuristic.

(2) After some variables are rounded to 1, the priority of the remaining variables to be

assigned to 1 may be changed. However, the current heuristic follows the predetermined

priority throughout the whole rounding process. This may affect the quality of the output

solution.

In order to reduce the computational time and improve the quality of output

solution, we will combine the two stages in this heuristic into one by applying the primal-

dual techniques for linear programming.

4.2 The Primal-dual Based Adjustment Heuristic

This heuristic combines the two stages in the first heuristic into one. When the

optimal solution of the LP relaxation is approached, the approximation for the ILP is also

improved. Therefore, there is no need to produce an optimal solution to the LP. This

will reduce the computational time significantly. The idea is as follows. Initially, set

every *

,wix to 1, and then adjust some *

,wix to 0 in a greedy order until all constraints of the

ILP are satisfied. The key to this heuristic is to find a good greedy order, in which *

,wix

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39

can be adjusted. The selection of *

,wix is based on the primal-dual theorem for linear

programming [22]. To simplify the description of this heuristic, we consider the primal

form of linear programming (PLP).

.1 ,10

|;|i1 ,

max

1

1

Wtnjx

EWtmbxatoSubject

xPLP

j

n

j

jij

n

j

j

������

��������

Where jia , is 0 or 1 and TFFb ),....,,1,...,1,1(� , in which there are t’s 1 and t�W’s F.

The dual of the above linear programming (DLP) is as follows.

.1 ,0

;1 ,0

;1 ,1

min

1

11

njz

miy

njyaztoSubject

zybDLP

j

i

i

m

i

ijj

n

j

j

m

i

ii

������

����

���

��

According to the primal-dual theorem, for any primal feasible solution jx

( nj ��1 ) and any dual feasible solution iy ( mi ��1 ), jz ( nj ��1 ), 0�jz will

imply 1�jx for any nj ��1 . Consider the dual feasible solution,

)1(1),1(0 njzmiy ji ������ ,

and the corresponding primal solution,

)1(1 njx j ��� .

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40

If the primary solution is a primal feasible solution, then the connection with

corresponding value of 1 is assigned a wavelength. Otherwise, we have to select some j

by setting xj to 0. The xj will be selected to minimize the dual objective function by

maximizing some yi while keeping others to be 0. To keep the dual solution feasible, the

zj’s should be correspondingly decreased and some zj will vanish because yi is maximized.

Suppose yi is chosen to be increased, then yi can be increased by at most:

)max/(1/1min11

ijnj

ijnj

i aay����

���

For each nj ��1 , jz will decrease by iij ya � and jz will become 0 if

ija achieves ijnj a��1max . Therefore, the dual objective function will be reduced by

)/(max)( 1

11

ijnji

n

j

ijii

n

j

iij abaybya ����

����� ��

To maximize this reduction, we will increase iy such that )/(max)( 1� ��� ijnjiij aba

achieves ))/(max)((max 1

11

ijnji

n

j

ijmi

aba �����

�� . Once such iy is found, we will set xj

to 0, where ija achieves ijnj a��1max because the corresponding jz now becomes 0.

This leads to our second greedy algorithm.

Primal Dual based Algorithm

Step 1. set .1for ,1 njx j ���

set },...,2,1,1|{ njxjS j ��� .

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41

Step 2. While { jx } is not a feasible solution to the ILP do

Find i, such that ijSj

Sj

iij aba �

� � max/)( = ))/(max)((max 1

1

1 ijnji

n

j

ijmi aba ���

�� ��

Find Sj� such that ija = ijsj a�max .

set 0�jx ,

set S = S -{ j }.

End-while.

In the above heuristic, set S records the set of variables whose values are 1, the

corresponding connections are assigned wavelengths.

5. Simulations

The objectives of the simulation work are three-fold:

(a) analyze the relationship between the success rate of connections and the number

of wavelengths on a fiber.

(b) analyze the relationship between the success rate of connections and the number

of fibers in a link.

(c) analyze and compare the performance of the two heuristics we proposed.

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42

We first simulate the proposed algorithms on the NSFNET and then work on

randomly generated networks.

Fig. 1 shows the physical topology of NSFNET [20] which has 14 nodes and 21

edges. The routes between two nodes are the shortest paths between them, in terms of

number of hops. The traffic matrix is given in table 1 in appendix which is obtained by

modifying the traffic matrix given in [20]. The entries are used here as the numbers of

connections between nodes.

The topology of the general network G(V, E) is constructed by using the approach

introduced in [21]. The nodes are distributed randomly over a rectangular coordinate

grid. Each node is placed at a location with integer coordinates. A link between two

nodes u and v is added by using the probability function F(u,v)=�exp (-d(u,v)/��), where

d(u,v) is the distance between u and v, � is the maximum distance between any two

nodes, and 0<�, � <1. Large values of � produce graphs with higher link densities, in this

Fig. 1 NSFNET

1

2

3

4

5

6

7

8

9

10

1112

13

14

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case the network G(V, E) consists of links. While small values of � increase the density

of short links relative to longer ones.

Fig. 2 includes the simulation results on NSFNET. Fig. 2 (a) shows the success

rate of connections versus the number of multiple fibers on a link while the number of

wavelengths is fixed at 7, 8, and 12. Fig. 2 (b) shows the success rate of connections

versus the number of wavelengths available on a fiber while the number of multiple

fibers on a link is fixed at 2, 4, and 16. Fig. 2 (c) shows how the success rate of

connections setup is affected by both parameters of W and F.

0

10

20

30

40

50

60

70

80

90

100

0 2 4 6 8 10 12 14 16 18

Number of fibers F

Su

cc

es

s r

ate

of

co

nn

ec

tio

ns

(%)

W=7(Alg.2)

W=7(Alg.1)

W=8(Alg.2)

W=8(Alg.1)

W=12(Alg.2)

W=12(Alg.1)

(a)

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0

10

20

30

40

50

60

70

80

90

100

5 7 9 11 13 15 17 19

Number of wavelengths W

Su

cc

es

s r

ate

of

co

nn

ec

tio

ns

(%)

F=2(Alg.1)

F=2(Alg.2)

F=4(Alg.1)

F=4(Alg.2)

F=16(Alg.1)

F=16(Alg.2)

(b)

10 12 14 16 18 20

2 (Alg.2)

4 ( Alg.1)0

20

40

60

80

100

Success rate of

connections(%)

W

F

2 (Alg.2)

2 (Alg.1)

4 (Alg.2)

4 ( Alg.1)

8 (Alg.2)

8( Alg.1)

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Fig. 2. Simulation results in the NSFNET.

For general networks, the number of nodes in the network is set to 40. Graphs are

generated and tested until a connected graph is found. And get 165 total edges. To

generate traffics in the simulated network, we randomly select the number of connections

between 0 and 4 for every node pair. The total number of connections is 1555. The route

for a connection is also the shortest path in terms of number of hops.

Fig. 3 includes the simulation results on general networks. Fig. 3 (a) shows the

success rate of connections versus the number of multiple fibers on a link while the

number of wavelengths is fixed at 7, 8, and 12. Fig. 3 (b) shows the success rate of

connections versus the number of wavelengths available on a fiber while the number of

multiple fibers on a link is fixed at 1, 2, and 4. Fig. 3 (c) shows how the success rate of

connections setup is affected by both parameters of W and F.

0

10

20

30

40

50

60

70

80

90

100

0 2 4 6 8 10 12 14 16 18

Number of fibers F

Su

ccess r

ate

of

co

nn

ecti

on

s(%

)

W=7(Alg.1)

W=7(Alg.2)

W=8(Alg.1)

W=8( Alg.2)

W=12(Alg.1)

W=12(Alg.2)

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(a)

0

10

20

30

40

50

60

70

80

6 8 10 12 14

Number of wavelengths W

Su

cc

es

s r

ate

of

co

nn

ec

tio

ns

(%)

F=1(Alg.1)

F=1(Alg.2)

F=2(Alg.1)

F=2(Alg.2)

F=4(Alg.1)

F=4(Alg.2)

(b)

7 8 9 10

1(Alg.2)

2(Alg.1)

4(Alg.2)

0

10

20

30

40

50

60

70

Success rate of

connections(%)

W

F

1(Alg.2)

1(Alg.1)

2(Alg.1)

2(Alg.1)

4(Alg.2)

4(Alg.1)

(c )

Fig. 3. Simulation result in general networks.

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From these simulation results we draw the following conclusions.

1. For those two important network parameters, the number of wavelengths on a

fiber W and the number of multiple fibers F of a link, increasing one of them with

the other fixed will increase the success rate of connections. This is simply

because when there are more wavelengths or fibers, there will be more freedom

to choose and assign proper wavelengths to connections. This will increase the

number of connections that can be supported, and the success rate as the number

of connections is fixed.

2. The increasing ratio of success rate versus the number of fibers is bigger than the

ratio of success rate versus the number of wavelengths. This can be observed by

comparing Fig. 2(a) with Fig. 2(b), and Fig. 3(a) with Fig. 3(b), respectively.

This is because that the wavelength can be assigned to F connections whose

routes share a physical link.

3. When the number of wavelengths is fixed, the increasing ratio of success rate

versus small number of fibers is bigger than the ratio of success rate versus larger

number of fibers. This can be observed by comparing the case of F � 5 and the

case of F � 5 in Fig. 2(a) and Fig. 3(a). It implies that adding as much as

possible fibers into the network is not very effective in improving the quality of

network performance.

4. When the number of fibers is fixed, the increasing ratio of success rate versus the

number of wavelengths is linear. This can be observed from Fig. 2(b) and Fig.

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3(b). This means that in order to improve the quality of network performance

wavelengths should be used as much as possible.

5. The performance of the first heuristic is better than the performance of the second

heuristic. Since the second heuristic does not solve the LP exactly as the first

heuristic. However, the difference in their performances is not very big

especially in the case that the number of wavelengths is fixed, which can be

observed from Fig. 2(a) and Fig. 3(a). This means that the second heuristic is

not only more efficient than the first heuristic but also effective.

6. Conclusion

In this paper, we formulated the wavelength assignment problem (mf-RAW) in

multifiber WDM networks as linear programming problem. Two heuristic approximation

algorithms were presented based on the linear relaxation of the binary integer linear

programming and primal dual technique, respectively. Simulations have been conducted

to show, under a certain traffic situation, how the number of connections that can be

supported by the system is affected by the number of wavelengths on a fiber and the

number of fibers in a link.

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CHAPTER 4 WAVELENGTH ASSIGNMENT USING GRAPHED

SIMPLEX METHOD

Abstract

The NP-complete routing and wavelength assignment problems in wavelength-

routed optical networks are typically solved using either integer linear programming or

graph coloring heuristically. In this paper we propose a new polynomial time

approximation algorithm to solve the operational version of Wavelength Assignment

Problem. The hybrid algorithm is in graph coloring form but based on the core of

Dantzig’s simplex method: optimal is within the extreme points. We show that the

computation complexity of this approach is O(N2W), which is as good as the existing

sequential greedy algorithm in theory. Further, the simulation on randomly generated

data set as well as NSFNet data set support the conclusion that the approximation factor

of the proposed algorithm is close to that of the existing one.

1. Introduction

Wavelength Division Multiplexing (WDM) technology is inevitably the solution

to meet the emerging high bandwidth demand of next generation distributed computing

applications such as video-conferencing, Internet game, and Video-on-demand.

However, with the current reduced investment to the Internet and telecommunication

industries, optimizing the use of existing infrastructure becomes more important than

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adding new gears such as converters, routers and fibers. The paradigm has shifted from

construction phase to operation phase. With this shift, a few changes can be predicted and

have been observed. In stead of focusing on Lightpath Topology Design (LTD)

problems, which is vital in an infrastructure phase, much of the attention has been shifted

into Routing and Wavelength Assignment (RWA) problems.

Although M. Shiva Kumar and P. Sreenivasa Kumar 2002 [69] proposed an ILP

algorithm to solve RWA as a whole, RWA typically was solved in two parts: lightpath

routing problem and wavelength assignment problem (WAP). Lightpath routing is to

determine the route of the lightpath in the physical topology; wavelength assignment is to

assign wavelengths to the route such that no two lightpaths are assigned the same

wavelength in the same physical links. In this paper we focus on wavelength assignment

part of the RWA problem. This implies that a request is the same as a connection

because the route is predetermined. The optimization objective of WAP has changed

from traditionally minimizing the number of wavelengths needed to maximizing the use

of existing wavelengths. The operational version of WAP with maximum load (WAPmax)

and no converter can be stated as: given a WDM network G, a request set C, and the

number of available wavelength W, how many requests can be satisfied at most? While

the optimization objective is different from the traditional WAP the problem is the same.

For ease of distinguishing, we term the later one WAPmin. WAPmin has been shown to be

NP-complete by Imrich Chlamtac et al., 1992 [70].

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The polynomial time approximation solutions to this problem can be summarized

to three categories: linear programming (LP) approach [69][71][72], graph approach

[73][74] and statistical blocking model approach [75]. None of them to this date can

declare the end of the chase for the best solution to WAP. The LP approach was first

seen in literature by Nico Wauters and Piet Demeester 1996 [71], where the network was

modeled as a multi-variant traffic flow system. The system is then formulated as an

Integer Programming (IP) problem, which is solved using a standard commercial package

after the IP being relaxed to an LP problem. Finally, the solution in real number domain

is tightened into a feasible solution in integer number domain using a greedy algorithm.

Further, Deying Li et. al., 2002 [76] applied the LP technique and developed a new

primal-dual technique to WAPmax in multi-fiber optical network. Although the

computation complexity is reasonably at O(n3.5

) and O(n2) respectively, the problem is

that it is quite wasteful in computation for relaxed LP calculation and misrepresenting the

optimal in tightening back to integer domain.

The graph approach to find a polynomial time approximation solution to the NP-

complete WAP problem was first introduced by Imrich Chlamtac, 1992 [70]. The actual

heuristic to graph coloring was discussed by Norman Biggs, 1990 [77] and D. de Werra

1990 [78]. Despite early successes like Brook’s theorem, there are very few general

results in graph coloring. The greedy algorithm appeared in Biggs has a computation

complexity of O(n2k) and it is shown that the optimal solution is largely affected by the

order of vertices chosen to be colored. Recently, Pallavi Manohar et. al. 2002 proposes

an alternative method using maximum edge disjoint paths to solve the problem [79].

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The blocking statistic model approach is focused on the overall state of the

network capacity after satisfying every request (Suresh Subramaniam et al, 1997) [75].

In other words, it assigns a wavelength to a request in a greedy fashion such that the

remaining path capacity is maximized in the network. While the method can deal with

dynamic request (on-line WAP) and works well for simple topology, maximization over

all the possible paths in a general network for each request is heavy in computation.

This paper proposes a new graph heuristic based on the core of Dantzig’s simplex

algorithm in solving Linear Programming problem [80]. The hybrid approach avoids the

wasteful computation in integer constraint relaxation and the misrepresentation in the

final greedy step in pure LP approach. Further, this approach provides insightful

approach to WAPmax compared to the existing sequential algorithm in the graph coloring

area.

In the rest of this paper, we will recap the classical simplex method and the NP-

completeness of WAPmax in Section 2. In Section 3, we will present the hybrid graph

algorithm based on LP’s simplex method and give the running time analysis. In section

4, we will present simulation result based on hypothetical randomly generated data set

and real world NSFNet data set. Finally, we will discuss the variation of this algorithm

under different scenarios in Section 4.

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2. Foundation

Let’s first define the network model and the LP problem description for the ease

of discussion. The network is modeled by a graph G=(V, E), where V represents the set

of nodes of the network, E represents the set of links available in the network. For any i, j

�V, if there is a link between i and j, then there are W wavelengths available between i

and j. A connection is denoted by a pair of nodes (s, d). A set C is a collection of t

connections denoted by C={(s1, d1), …, (st, dt)}. We assume that traffic of each

connection can be supported by one lightpath (i.e., one wavelength). If the traffic

demand between a pair of nodes is more than what a lightpath can support, it is split into

multiple connections, each of which is supported by a lightpath. Therefore, some

connections in C may have the same node pair (s, d), but using different wavelength. A

route for a connection in C is a simple path in G, which is assumed to be known. Note

that due to the limit of number of wavelengths available, not every connection in C can

be assigned a wavelength without causing wavelength conflict.

Since wavelength conversion is not allowed, each connection will be assigned

with a wavelength (throughout all links in its route). Our objective is to assign W

wavelengths to as many connections as possible under the wavelength conflict constraint.

We introduce the following notations.

i i-th connection in C, when used as a subscript;

l, m endpoints of a link in G;

t the number of connections in C;

W the number of wavelengths available on a fiber;

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),( ml

i� indicator, ),( ml

i� =1 if connection i uses link (l, m); otherwise ),( ml

i� =0;

ix variables, ix =1 if connection i is assigned a wavelength; otherwise ix =0;

),( ml

ix variables, ),( ml

ix =1 if connection i goes through link (l, m) and assigned

a wavelength; otherwise ),( ml

ix =0;

The problem then can be formulated as 0-1 programming as following.

The objective function (1) is to maximize the number of connections that are

assigned the wavelengths. Since for connection i that is not assigned any wavelength, we

have ix = 0 so it contributes nothing to the objective function. The equality (2) ensures

that only W wavelength being used on any link. The equality (3) represents the

wavelength continuity constraint. It ensures that if connection i is assigned a wavelength

w, then every link in the route of the connection shall be assigned the same wavelength.

The inequality (4) and (5) are to ensure that at most one wavelength can be assigned to

connection i, i=1, 2, … , t.

(5) ;...,,2,1,),( 1,or 0

(4) ;,...,2,1 ,1or 0

)3( ;...,,2,1,),( ,

(2) ),( ,

)1( )max(

),(

),(),(

1

),(

1

tiEmlx

tix

tiEmlxx

EmlWxtoSubject

xILP

ml

i

i

i

ml

i

ml

i

t

i

ml

i

t

i

i

������

����

�����

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At this point, the traditional solution is to relax the integer solution constraint and

then solve the problem using classical LP algorithms such as Karmarkar’s Interior Path

Method or Dantzig’s simplex method. Dantzig’s simplex algorithm is given in Figure 1

[80]. However, a detailed look of the simplex method reveals that the core of simplex

method is to move from one extreme point to another, pushing the slack variable to 0 or

as small as possible while increase the objective value [81]. This implies that perhaps a

heuristic can be drawn for WAPmax based on the network graph.

Table 1. Dantzig’s Simplex Algorithm

Input: values for the entries of the matrix A, and the vectors b and c in the LP-

model max{ 0,| �� xbAxxcT }.

Output: Either

(i) the message: no solution; or

(ii) the message: the model is unbounded, or

(iii) an optimal solution of the model.

Step 0: Initialization.

Determine an initial basic feasible solution for the LP-model with slack

variables. If none exists, then stop: the model has no feasible solution.

Go to Step 1.

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Step 1: Gaussian elimination.

Suppose the current basic feasible solution is �����

�NI

BI

x

x with respect to the

nonsingular basis matrix B in [A �]mI [B N]. Using Gaussian elimination,

transform [B N b] into [ bBNBI BI

m

11 �� ], and [ T

NI

T

BI cc ] into the

current objective vector [ NBcc T

BI

T

NI

T 10 �

� ]. The current objective

value is bBcT

BI

1� .

Go to Step 2.

Step 2. Choosing a new basic variable; optimality test.

If all current objective coefficients are nonpositive then stop: An optimum

has been reached. Otherwise, select a positive entry in the current

objective vector; say � is the index of that objective coefficient. Then �x

will become basic variable.

Go to Step 3.

Step 3. Minimum-ratio Test

Determine in the column of NB 1� with index � an index },...,1{ mk � for

which the quotient �kk NBbB )/()( 11 �� is minimal and 0)( 1 ��

�kNB ;

kbB )( 1� is the current right-hand side. Let � be the column index for

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which 1)( ��k

BI

mI .

If 0)( 1 ��

�kNB for each },,...,1{ mk � then stop: The model is unbounded.

Go to Step 4.

Step 4. Exchanging.

Exchange the column with index � by the column with index

� to obtain a

new basis matrix B; i. e. BI := (BI \ {�})� {

�} and NI := (NI \ {

�}) � {

�}.

As in step 1, apply Gaussian elimination to obtain again

[ bBNBI BI

m

11 �� ] for this new basis matrix B; moreover, z := z +

�� xNBcc T

BI

T

NI )( 1� with �� kk NBbBx )/()( 11 ��

� .

Return to Step 2.

Now let’s look at the graph side. Let’s first prove that WAPmax problem is NP-

complete using the help of the classical n-graph-colorability problem. Although it is

intuitive to relate the edges in n-graph(edge)-colorability problem to coloring the paths of

the network graph for WAPmax, the actual proof is done through n-graph(vertex)-

colorability. Here is how to transform the original graph from coloring of paths to

coloring of vertices. Consider a graph representation of the network G, where the

vertices of the graph represent nodes in the network, with an undirected edge between

two vertices corresponding to an optical fiber link between the corresponding nodes.

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Since we only consider the wavelength assignment part of the RWA problem, the

request, therefore, the route, for each lightpath corresponds to a path in G, thus the set of

routes that have been specified for the lightpaths correspond to a set of paths, say C.

Now consider another graph G’, the path graph of G, constructed using the following

method. Each path in C corresponds to a node in G’, and two nodes in G’ are connected

by an undirected edge if the corresponding paths in C share a common edge in G [82].

Solving the WAP problem is then equivalent to solving the classical graph coloring

problem on G’, that is, we have to assign an available color to nodes of G’ such that

adjacent nodes are assigned distinct colors and the total number of colored nodes are

maximized.

Theorem 1. WAPmax is NP-complete.

Proof: First we show that solving the n-graph(vertex)-colorability problem would also

solve WAPmax. We create the path graph G’ for the network and request as described

above. The maximum number of nodes colored with K (W=K) color such that no two

adjacent nodes are colored the same, would produce the maximum number of paths

assigned with W (or K) wavelengths such that no two interconnected paths share the

same wavelengths. Thus finding the most colored nodes would also yield the most paths

with wavelength assigned.

Second, we show that solving WAPmax will also solve the n-graph(vertex)-colorability

problem, thus proving that it is unlikely a polynomial time solution can be found to

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WAPmax problem since n-graph(vertex)-colorability is NP-complete. To do that, we give

a polynomial time algorithm that translates any graph coloring problem to a network and

an appropriate set of lightpath requests. Given a graph ),( ccc EVG we translate the

coloring of cG into a wavelength assignment problem as follows: we create a node 0

iv for

every node cvi� ; and for every edge cEjie ��� , we create four new nodes x, y, k

iv ,

l

jv and directed edges .,,,, 11 l

j

i

k

l

j

k

i vyvyyxxvxv ����� �� Attach the mark i to

edges going from/to iv ’s and yx � . Repeat for the mark j. The designation of a node

in the new graph k

iv stands for the k’th replication of the node corresponding to node i in

the original network, k = 0…d( i ) where d( i ) is the node degree of i . The path demand

set L is defined by the | cV | paths where path i requires use of all links having i as a

mark. The computation complexity of the algorithm is O(| cE |).

The proof of the NP-completeness directs us to focus on finding polynomial time

approximation solution for WAPmax. The existing sequential algorithm is: Go through

the vertices in order, assign the first color to every vertex for which it is available, repeat

for the second color, and so on, until the number of available colors is exhausted. This

algorithm in general works well for some classes of graphs, but badly for others [83].

This algorithm has a running time of O(N2W) for the worst case when a node has links to

every other nodes in the graph. N is the number of nodes in G’, N � |C|. Let’s say the

maximum degree of path graph G’ is D, then apparently, if W � D+1, all the paths can

be satisfied. The proof can be seen in [16]. Further, if there are only a few large degree

nodes in the graph, it makes sense to satisfy those nodes first. This is the largest-first

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vertex ordering. In D. W. Matula et. al. 1972 [16], the smallest-last vertex ordering was

shown to be the most economical among N! possible orders.

3. Our approach

With the understanding that the coloring is in the node at the path graph G’, we

examine Dantzig’s simplex method in its origination rather than its format, or tableaus.

The classic introduction to Simplex method in LP usually starts with graphical solution

method and reveals that what’s behind the simplex method is moving from one vertex to

another to finally reach optimal. We try to find out what is the equivalent of basic and

nonbasic variables, how will pivoting be done in the graph, and finally after we get the

basic feasible solution, what does it mean in WAP.

We need to explain an important concept first. The slack variable in this problem

is the number of free, unassigned colors, or wavelengths, in each node. In LP equation

(2), a slack variable sx can be added as EmlWxx ml

s

t

i

ml

i ������

),( ,),(

1

),( . The basic

variables are the slack variables, the nonbasic variables are the nodes in the network.

Basis is the vector composed of all the slack variables. Pivot is walking from one

extreme point to the adjacent one in the feasible region. In addition, an extreme point

corresponds to a point where a basic variable (a slack variable) or a nonbasic variable is

zero. Pivoting is then a process that greedily use up all the free colors (assign slack

variable = 0 where it is possible) of each node and move to the next one. A basic feasible

solution is a valid allocation of colors on the network graph, satisfying all the constraints.

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These include that the number of colors does not exceed W and no two adjacent nodes

have the same color. However, with many ways of assigning the color, a basic feasible

solution does not necessary yield the most number of colored nodes, or in another word,

the most number of satisfied requests. Further, it may not guarantee the fairness. The

basic feasible solution shall specify which node is used in what color, it shall also specify

how many colors are used (or are still free) in each node. Based on above understanding,

an algorithm can be drawn:

Graphed Simplex Algorithm

Step 1. Transformation: transform the original graph G to path graph G’ in the

following ways: each request path in G becomes a node in G’, if two paths

share at least a link, then an edges will be drawn between the two nodes in

the new path graph. If a node has no edges connect to it in G’, it can be

ignored in this concern because we know that we can satisfy the

corresponding request in G.

Step 2. Initialization: sort the vertices in G’ based on their degree in descending

order, v1, v2, v3, ..., vn;

Assign an initial color to all the nodes.

Verify_color().

Step 3. For i =1 to N,

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Assign_Color_To_Node(vi)

End For

Step 4. Find_Max()

Assign_Color_To_Node(v)

Visit v’s node list and find all the available colors to this node, assign a

random available color to v.

Verify _color();

Verify_Color()

For i=1 to N

Verify v[i]’s node list, if there is same color, set neighbor’s color to 0.

End For

Find_Max()

For i=1 to W

For j=1 to N

count colored node

End For

Record the maximum

End For

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As far as the running time analysis, let’s say that the original request set is C,

number of vertices in G’ is N, where N � |C|, Step 1, takes O(|C|), step 2 takes O(NlgN),

step 3 in worst case scenario, when there is a node in G’ has a degree of (N-1), the

running time is O(N2W). Therefore, the total running time for the worst case is: O(N

2W).

4. Simulation

The goal of the simulation is to compare the optimality of the graphed simplex

method with the existing greedy algorithm. The simulation is divided into two sections.

Section 4.1 is on randomly generated path graphs. Section 4.2 is on NSFNet data set

obtained from [76]. We first present how we construct the simulation then we will show

the results in each section.

4.1 Random generated path graph

To compare the optimality of graphed simplex method and the greedy method on

randomly generated graph. First we generate a random path graph based on a preset

number of path nodes and a preset number of colors. The experiments show that 15

nodes in the path graph and 2 or 3 colors worked the best. We then run all three

algorithms on this randomly generated data set to find the maximum number of colored

nodes for comparison. The results are shown in Table 1 and Table 2. We can see from

the data that both graphed simplex method and greedy method have good optimality,

their results in general are very close to the optimal solution obtained by brutal force

method. Further, the optimality of graphed simplex method is very close to the

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optimality of greedy method. The average approximation factor of the graphed simplex

method is slightly lower than the average optimal factor of the greedy method, however,

the difference is reducing as the number of colors increase. Another point worth to state

is that the greedy algorithm is not always better than the graphed simplex algorithm, in

some cases, the graphed simplex algorithm is better.

Table 1. Optimality comparison on random generated graph with 3 wavelength.

Number GSM Greedy Brutal GSM OPT Factor Greedy OPT Factor

1 10 10 13 0.77 0.77

2 9 8 11 0.82 0.73

3 9 11 11 0.82 1.00

4 11 12 13 0.85 0.92

5 13 13 13 1.00 1.00

Average: 10 11 12 0.85 0.88

* 15 nodes in the path graph..

Table 2. Optimality comparison on randomly generated graph with 2 wavelength.

Number GSM Greedy Brutal GSM OPT Factor Greedy OPT Factor

1 10 11 11 0.91 1.00

2 11 13 13 0.85 1.00

3 11 10 12 0.92 0.83

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4 8 8 10 0.80 0.80

5 5 6 9 0.56 0.67

Average: 9 10 11 0.81 0.86

* 15 nodes in the path graph.

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4.2 NSFNet data set

To compare the optimality of graphed simplex method and the greedy method on

a real case scenario, we used the NSFNet traffic data set to simulate the application of the

algorithm. The objective is to compare the optimal solutions obtained by using graphed

simplex method and greedy method on the same data set. NSFNet has 14 network nodes

and 86 unique request. We assume that the cost between each adjacent node is the same

and found shortest path for each request. We then transferred the network graph into path

graph as described in section 2. The graph is expressed using node list data structure.

We run both algorithms on the path graph and find the maximum satisfied paths. The

results are shown in Table 3. Again, we can see that the results of both methods are very

Figure 1. Comparison of 3 methods on random graph with 2 colors

4

5

6

7

8

9

10

11

12

13

14

1 2 3 4 5 6

Running time

Sati

sfi

ed

Req

uests

GSM Greedy Brutal

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close. Greedy algorithm is slightly better than the graphed simplex method in general

although sometime graphed simplex method is better than the greedy algorithm. Because

of the number of nodes in the NSF path graph, it is impractical to compute the optimal

solution using a personal computer.

5. Discussion

The order in which the nodes are assigned color, therefore, the requests are

satisfied, determines the optimality of the solution. However, there are |C|! number of

possibilities if we want to use brutal force to find the maximum. Our algorithm greedily

Figure 2. Comparison of 3 methods on random path graph with 3

colors

6

7

8

9

10

11

12

13

14

1 2 3 4 5 6

Running time

Satisfied R

equests

GSM Greedy Brutal

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assigns color to nodes with large degrees first, such that after the coloring, the remaining

state is healthiest.

There are variations for the two techniques that reduce the blocking rate of the

network: wavelength conversion and multi-fiber. A path with wavelength conversion can

be viewed as two paths with one joint, therefore, it is two nodes with an edges connected

in the path graph. Again, we only consider the wavelength assignment portion of the

problem, not considering the routing part. Further it was argued by Nico Wauters and

Piet Demeester 1996 [71] that the wavelength converter does not make a significant

difference in wavelength requirement. The graphed simplex algorithm can also be

applied to multi-fiber case easily with the slight modification of the number of colors

Figure 3. Comparison of GSM & Greedy at 2 & 6 colors

50

55

60

65

70

75

80

85

90

1 2 3 4 5 6

Running Number

Sati

sfi

ed

Req

uests

GSM 2 Greedy 2 GSM 6 Greedy 6

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available in each node in the path graph. Further, for the semi lightpath scenario, the

color needs to be grouped based on the fiber. The fiber switch for semi lightpath can be

treated as two nodes connected by an edge but uses two different colors: color A in fiber

1 and color A in fiber 2.

Table 3. Comparison of GSM and Greedy on NSFNet with 2, 3, 4, 5, 6 wavelengths.

Color 2 3 4 5 6

Number GSM Greedy GSM Greedy GSM Greedy GSM Greedy GSM Greedy

1 58 60 76 76 81 82 84 84 84 84

2 56 60 82 76 84 82 82 84 83 84

3 58 60 82 76 78 82 84 84 84 84

4 58 60 75 76 83 82 81 84 82 84

5 58 60 78 76 81 82 83 84 84 84

AVG 58 60 79 76 81 82 83 84 83 84

* 86 nodes in the path graph.

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CHAPTER 5 CONCLUSION

In this thesis, we focused on the design and application of approximation

algorithms in computer and communication systems, particularly on the problems that are

known as NP-hard in combinatorial computation complexity theory. We obtained

approximation solution of several such problems using the common approximation

theory approach: the combination of greedy, linear programming and randomization.

Multi-fiber wavelength assignment problem is an example NP-hard problem in

optical communication infrastructure area. We first formulated the wavelength

assignment problem (mf-RAW) in multifiber WDM networks as linear programming

problem. We then presented two heuristic approximation algorithms based on the linear

relaxation of the binary integer linear programming and primal dual technique,

respectively. Further we conduct simulations to show how the number of connections

that can be supported by the system is affected by the number of wavelengths on a fiber

and the number of fibers in a link under a certain traffic situation

We further developed a new hybrid algorithm to solve the NP-hard wavelength

assignment problem using the combination of simplex method and graph colorability.

The running time analysis show that the new graphed simplex method is as good as the

existing sequential greedy algorithm. The simulations on randomly generated graph and

NSFNet data set also reveal that the new algorithm is as good as the existing sequential

greedy algorithm.

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Criticality driven session admission in Enterprise Resource Management Software

is another example NP-hard problem in computer software system that benefits from the

proposed technique. It is a resource management problem in essence. We first presented

a mathematical model for this problem, which is first time to be seen in the literature.

The solution is obtained by reducing the problem to general cover problem then obtained

by using greedy algorithm and linear programming approach. Further we presented

performance ratio and running time of both solutions. The performance ratio is

��

���

n

j

ijmi af1

1max

and the running time is O(mn2). We recommend adopting one of the

approaches for a specific case based on the problem size.

Future research directions include extending the current approximation algorithm

to more realistic computer and communication systems, further developing the proposed

approximation techniques in theory and by simulation using computation devices with

larger computation power, and applying the further developed techniques to new

problems in computer and communication networks.

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Appendix 1. The traffic matrix for NFSNET

Nodes1 2 3 4 5 6 7 8 9 10 11 12 13 14

1 0 23 20 10 19 8 13 50 7 4 26 33 25 18

2 0 4 1 6 0 2 8 1 1 10 3 6 1

3 0 15 3 12 3 28 1 3 17 2 7 4

4 0 1 0 0 1 1 1 4 4 2 1

5 0 1 3 20 6 8 23 38 7 4

6 0 1 1 1 0 2 2 2 0

7 0 37 7 6 50 30 53 8

8 0 11 14 29 23 29 6

9 0 2 12 8 5 4

10 0 36 47 20 15

11 0 19 12 10

12 0 21 9

13 0 20

14 0


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