Luận văn multifactorial evolutionary algorithms for clustered minimum routing cost tree problems in the multi domain network

IIANOI UNIVERSITY OF SCIENCE AND TECIINOLOGY MASTER THESIS Multifactorial Evolutionary Algorithms for Clustered Minimum Routing Cost Tree Problems in the Multi-domain Network TA BAO

IIANOI UNIVERSITY OF SCIENCE AND TECIINOLOGY

MASTER THESIS

Multifactorial Evolutionary Algorithms

for Clustered Minimum Routing Cost Tree

Problems in the Multi-domain Network

TA BAO TIIANG

Data science and Artificial intelligence

Supervisor: Assoc Prof Huynh Thỉ Thanh Binh

School: School of Information and Communication Technology

HA NOI, 2022

HANOI UNIVERSITY OF SCIENCE AND TECHNOLOGY

MASTER THESIS

Multifactorial Evolutionary Algorithms

for Clustered Minimum Routing Cost Tree Problems in the Multi-domain Network

TA BAO TIIANG

Data science and Artificial intelligence

Supervisor: Assoc Prof Huynh Thi Thanh Bính

Supervisor's Signalure

School: School of Tnformation and Communicatien Technnlowy

HA NOI, 2022

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Tiếng Anh: Multifactorial Evolutionary Algorithms for Clustered Minimum Routing Cost Tree Problems in the Mult-domain Network

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Declaration of Authorship and Topic Sentences

cost tree problems in the multi-domain network

Multifectorial evolutionary algorithms for clustered minimum routing

3 Contributions

» Develop a new encoding aud decoding scheme for wo clustered tree prob- lems: Clustered Mininum Routing Cost Tree (CluMRCT) ud Clustered Shortest, Path Tree (ClnSPT) The praposed method allows evolutionary algorithms to fimerion on complete and sparse graphs

¢ Design efficient multifactorial evolutionary algorithms to solve CluSPT and CInMRCT probleme simultaneously

Fivaluate the efficiency af the proposed algorithms and encoding methods

on varions instances The results proved that the proposed methods outperformed all existing approaches in terms of solution quality and

convergence trend

Declaration of Authorship

T declare that my thesis, titled ” Multifacterial Tivalutionary Algorithms for Clustered Minimum Routing Cost ‘Lree Problems in the Multi-omain Net work”, is the work of myself and my supervisor Associate Professor Huynh Thi Thanh Binh AB papers, sources, Lables used in this thesis huve been thoroughly cited

‘Thank you for this extensive instruction and your trust over such a long period Second, | would like to thank Dr ham Dinh 'I'hanb, Ph.D student Do ‘Tuan Anh, and Lb student ‘Iran Thì Huong for their unconditional support during the last fonr years at the Modelling, Simulation, and Optimization Laboratery (MSO Lab) I appreciate their speciolization end open-mindodness, which allowed me to discuss and talk about anything, whether rcscarch, programming, or anything clsc I Tearued u lot frou them Thope we will be uble Wo undertake :nuce collaborative work

in the near frture T am also grateful to my friends who assisted me in improving the quality of my thesis

Finally, I would like to thauk Vingroup JSC, the Vingroup Innovation Founda- tieu, and the Sdivul of Information and Couuuunication Technology (SoICT) fer supporting my studies during the Master's program I was funded by Vingroup ISC and supported by the Master, Ph.D Scholarship Programme of Vingroup Innova- tion Foundation (VINIF), Institute of Big Data, code VINIF.2020.ThS.BK.01 and VINIP.2021.ThS.BKK.O1 for two years, 2021 and 2022 These supports allow me to entirely focus on my scientific research

Abstract

Renl-world ncuvork architeevures seldom cxist in ischion, Muay of them are ci ther repetitive or share domain-specific similarities A good network architectnre for one system can also be helpful for another ‘Therefore, knowledge drawn from solving previous network design problems may he rensed to solve new problems more quickly and efficiently Meanwhile, traditional optimization algorithms often solve only one problem st a time from scratch and assume zero prior knowledge about these problems at hand It makes the capabilities of solvers not automat ieally grow with experience Thiy vhesis proposes multitasking evolutionary algo- rilluus Wo solve inulliple clustered tree problerns in unulli-dotain, nebworks simul- taneously ‘The thesis focuses an two clustered tree problems: Chustered Shortest Path Tree (CluSPT) and Clustered Minimum Ronting Cost Tree (CiMROT) Both are NP-Hard and representative cases of Client-Sever and Peer-to-Peer topologies

in multi-domain networks, respectively ‘I'he proposed algorithms help recluce the total time for optimization completion and facilitate online knowledge transfers be tween problems during the optimization process, thereby yielding superior results

to traditional single-task optimization methods

Keywords: Livolutionary Algorithms, Multitasking Evolutionary Algorithm, Mullifactorinl Evolutionary Algoritlin, Clustered Tree Problems

Author

Ta Tao Thang

Contents

a

Theoretical Basis

LL Overview of Mctu-heuristic Algoriliun

12 Multifectoriul Evolutionary Algoritin

Problem Formulation

2.1 Problem formulation

22 Related Works

Multitask Algorilhia for Clustered Shortest Path Tree

6

A

5.2) Resnits and Discnasions on Clustered Shortest Path Tree

5.3.1 Comprcheusive comparisons between the proposed algorithan

and guyoral state-of-the-art upprouches 5.32 Analyze the effect of the input graph size on the performance

of the proposed algorithm 5.3.3 Analyze the effectiveness of the proposed multi-parent crossover

5.4 Results ancl Discussions on Clustered Minimum Routing Cost ‘lree

5.4.1 Analyze the effentiveness of the proposed enending and decad-

ing method 5.4.2 Analyze the effectiveness of proposed hybridization stratesy 5.4.3 Comprehensive comparisons between the proposed algorithm

and several state-of-the-art: approaches

Solutions of the CluMRCT and CluS/"l problems

A ChuSPT encading example Devoding sicthod lor the CluSPT probleu 2

An invalid CluSPT encoding © 20.0200 00 ee

An example of the repairing process

An example of constructing the unified search space

An example of the proposed crossover peralor, os

An example of constructing specilic-lask representation from an ine dividual in the tmified search space

An CiMRCT encoding example

Proposed decoding method in the first level: a) An input graph:

b) A CluMRCT encoding: c-g) Steps to construct an intra-routing

Steps in the sveond level of the proposed decoding method 8) Âu incomplete solution obtained after the first level b) The priorilies of

clusters ¢-d) Steps to build mter-ronting spanning tree

An example of individual representation

Compare PT values of K-MFFA and state-of-the-art algorithms Compare RPD values of K-MFEA and state-of-the-art algorithms Couvergence treuds of K-MFEA and G-MFEA on Type 1, Type 5

Tumning time of K-MPT:A, G-MTTA and TIB-RGA Running time of K-MPEA in comparison with HB-RGA Relationship between the number of vertices and the performance of

Relationship between the improvement percentage (PT) and size of the input graph

5.8 Campare MEFEA-FA with PMEEA and PFA

3.9 Compue MFEA-FA with stateofEthe-art algoritluna

5.10 Compare the convergence trend 6

43

44

45

Summury of comparisons betweou K-MFEA and existing algorithins

The different revully obtained by K-MFEA when runuing with the uuober of parculs in range 21010 2

Parameter of MFEA-FA and comparison algorithms Summary results of PMIEA and L-MIEA Better, Equat, and Worse denote the number of tasks that PMFEA is better, cqual and worse

Ranking of algorithms given by Friedman test Wilcoxon signed rank test with a — 0.05

Results obtained Ly C-MFEA, HB-RGA und K-MFEA ou Type L

Results obtained hy G-MPEA, HB-RGA and K-MFEA on ‘lypes 3

Nesults obtained by G-MIMA, UB RGA and K-MFEA on ‘lype 5

Results obtained by G-MFIEA, IIB-RGA and K-MFEA on Type 6

Averaged Experimental resuks (unit: 104} on Type 1 where +, —, and — denote the number of tasks that each algorithm was better, equal, and worse than the MFEA-FA algorithm

Averaged Experimental resnits (unit: 10°} on Type 3 and Type 4

where —, —, and — denote the number of tasks thar each algorithm was better, equal, and worse than the MPTR:A-PA algorithm:

Averaged Experimental resulra (unit: 105} on Type 3 where +, =, and — denote the number of tasks that each algorithm was batter

equal, anc worse than the MFEA-FA algorithm

Averaged Experimental results (unit: 10%) an Type 6 Small where +,

=, and — denote the number of taska that each algorithm was better, equal, anc worse than the MLLA-I'A algorithm

Fxperimental results (amit: 108) on Type 6 Large where +, —, and

— denote the number of tasks that each algorithm was better, equal,

Preface

In the years following massive globalization efforts, multi-domain network architec- tures where end-users or devices are divided into clusters can be found in many real-world applications Some applications can be mentioned, such as agricultural

irrigation systems, transportation services, logistics, cable TV systems power net-

works, and distribution systems Therefore, network design problems on multi- domain networks have attracted much interest not only within the confinement of

research communities but even more so from members of governments and industry This th focuses on two network design problems in the multi-domain net-

work, which are Clustered Shortest-Path Tree (CluSPT)[16] and Clustered Minimum

Routing Cost Tree (CluMRCT)[27] The objective of the CluSPT problem is to find

a multi-domain client-server network architecture to minimize routing costs from a

central server to all other devices Meanwhile, the CliMRCT problem aims to find

a network des ign to minimize the total routing costs between any two devices in the multi-domain peer-to-peer network Additionally, these problems require communi-

cations within a cluster to be routed locally and not contain any vertex from other

clusters, and private information among nodes in each cluster needs to be circulated

internally and not transmitted through other clusters This allows network systems

to have high securit,

RCT thus brings a high economic efficiency not only for computer network system

y while reducing operational costs Solving CluSPT and CluM-

but also for various areas such as computational biology, product transportation,

logistics, as well as agricultural irrigation (11, 20]

However, both these problems are NP-hard Therefore, the preferred method to

tackle this problem is mainly through meta-heuristic algorithms, as solving a large instance of them using exact approaches is unfeasible and quite literally a waste of time A family of meta-heuristic algorithms that has found considerable success in dealing with NP-Hard problems are Evolutionary Algorithms (EAs) [, 18] ‘These

algorithms based their mechanisms on Darwin's evolution and natural selection the- ory Essentially, a multitude of solutions will first be randomized, encoded in a way

that the solutions are susceptible to change by evolution operators, namely muta-

tion and crossover While EAs themselves have been subjects of research since the

1990s, they still have many shortcomings, and the most important one is always

solving any new problem from scratch regardless of how similar they are to those

already solved in the past Knowledge drawn from past experience is not reused

to tackle new problems Meanwhile, network architectures in practice often share many common features and seldom exist in isolation A network architecture in

one system may be helpful in another, Therefore, a new multitasking variant of EA which is Multifactorial Evolutionary Algorithm (MFEA)|0, 17 21] is proposed to

handle these drawbacks The MFEA not only can solve multiple problems simul- taneously but also facilitate implicit knowledge transfers between problems during

the optimization process to obtain better solutions than solving them in isolation

There are many efforts using MFEA to solve CluSPT and CluMRCT [17, 49, 50,

52, 55] These methods, however, have a large computational time and only function

on complete graphs Meanwhile, in practical networks, each vertex only connects to

a certain number of other vertices Therefore, it makes algorithms hard to apply to real-world applications, Besides, these algorithms did not control negative trans fers

when solving low-similarity problems, leading to low-quality solutions jn some cases

The main contributions of this thesis include:

© Propose two novel encoding with significant consideration regarding, perfor- mance and memory usage for CluSPT and CluMRCT problems Notably, the proposed encodings are much smaller than existing approaches, allowing mul- titask and other meta-heuristic algorithms to effectively function on complete and sparse graphs Besides, these methods’ computational complexity is also analyzed carefully and thoroughly

e Examine the effectiveness of multi-parent crossover in multitasking algorithms

To the best of my knowledge, there is no similar approach in the multitasking

literature The results proved that it is a promising approach for large-scale

problems

@ Design a novel combination of multifactorial evolutionary and firefly algo- rithms to tackle low: imilarity tasks The proposed algorithm not only en-

hances the self-evolution of each task but also improves inter-task knowledge

transfers by delivering higher-quality solutions

e Examine the effectivenes of the proposed algorithm and encoding methods

‘on various types of instan The results proved that the proposed methods

outperformed all exi

ting approaches in terms of solution quality, ponivergence

trend and computational time

The thesis is organized as follows:

© Chapter 1 provides an overview of popular meta-heuristic algorithms, espe- cially the multifactorial evolutionary algorithm

e Chapter 2 describes problem formulations and outlines related works.

+ Chaprer 3 presenis a new encoding strategy and a novel multitasking algo- rithm using the multi-parent, crossover to solve multiple CluSPT problems simultaneously

© Chapter 4 presently u novel enoding strategy and u novel hybrid multitasking algarithm for the CluMRCT problem

Besides,

the effectiveness of wach component in Uke proposed algorithms and factors

© Chapter 5 provides experimental resulls on various types of iastane

allecting the qualily of the algeriluns are also cxurninod,

Chapter 6 concludes the thesis and outlines fulure works.

Meta-heuristic algorithms refer to a class of methods utilizing stochastic factors to

find the global or near-optimal solutions for complex optimization problems, Each meta-heuristic optimization algorithm holds two main features: exploration and

exploitation Exploration relates to the ability to search throughout the search

space to find an optimal solution and avoid local optima On the other hand, exploitation is the ability to locally search around elite solutions to improve their quality Such features are used for all meta-heuristic algorithms but with specific operators and mechanisms for each framework The advantages of these algorithms are simplicity, flexibility, and independence from the nature of the problem By using

stochastic factor meta-heuristic algorithms do not need to be concerned about the problem's derivative information and, therefore, become an effective method

for finding optimal solutions to a given optimization problem Another significant

advantage is the flexibility of this class of algorithms which allows them to solve any kind of optimization problem within a reasonable amount of time by following

s framework Therefore, meta-heuristies

a pre-defined structure of each algorithm’

are becoming more popular and received significant attention from the research

community,

Interestingly, most meta-heuristics come from familiar sources of inspiration

close to real life such as natural evolution, animal behaviors, human behaviors, or

physical phenomena For example, Swarm Intelligence (SI)_ [1] is one of the classes

in meta-heuristics, which imitates the social behavior of animal groups The

mecha-nism behind this method is based on sharing collective information of all individuals during the optimization process Particle Swarm Optimization (PSO) [1!), 25] de-

veloped by Kennedy and Eberhart [25], is one of the most representative algorithms

for this group PSO simulates the movement of organisms in a bird flock or fish school cooperating to find food During the optimization process, all candidates (particles) follow the best solutions in their path Each particle keeps track of

its coordinates in the search space and compares it with its own position of the best solution it has achieved (pbest), and the best solution obtained so far in the population (gbest) Some recent other algorithms belonging to this class are Grey Wolf Optimizer (GWO) [22], Galactic Swarm Optimization (GSO) [i], Whale Op- timization Algorithm (WOA) [20], Bat Algorithm (BA) [1], and Firefly Algorithm (FA) [54] Thanks to incorporating the best solutions’ information and successful history, these S-based algorithms have a fast convergence trend and strong exploita-

r [1, 20, 54]

Physical phenomena also inspire another class of meta-heuristies ‘These algo-

al laws in nature, characterizing the interaction of

tion abi

rithms are derived from phy

search agents through some rules based on physical processes One typical example

of this class is the Gravitational Search Algorithm [8] which is an optimization

in phy

method inspired by the theory of Newtonian gravity ics to update the po-

sition of a candidate toward the optimum point Recently, several meta-heuristics

based on human behaviors have been researched and developed An instance of this

type is the Teaching-Learning-Based Optimization (TLBO) [19] The framework is inspired by the influence of the teacher on his/her learners, with the optimization

: the “teacher phase” (learners learn

process being divided into two main phas

from the teacher) and the “learner phase” (learning by interacting with other learn- ers) ‘These two phases are repeated continuously until global convergence of the algorithm is obtained

Meanwhile, Evolutionary Algorithms (EA) [5] with one of the most famous rep- resentatives is Genetic Algorithm (GA) [15] is the most classic meta-heuristic cla

EA is inspired by natural evolution with its mechanism directly based on Darwin's

theory of evolution and natural selection, This algorithm reflects the natural selec-

tion process in which the fittest individuals are selected for reproduction to produce offspring of the next generation To obtain this, the algorithm uses two evolution- ary operators called crossover and mutation The functionality of mutation is to preserve the population diversity and local search around current individuals, while the crossover aims to combine good genes of parents to obtain offspring with better fitness Due to the simplicity and independence of the problem, EAs have shown remarkable success in dealing with NP-Hard problems [s]

Furthermore, recent years have also witnessed the strong development of hy-

brid methods, which combine meta-heuristic algorithms and different techniques for

complex optimization problems Lyu [2š] incorporated an EA-based algorithm and

an SI-based algorithm, which are GA and PSO, for periodic charging planning in the wireless rechargeable sensor network Aydilek [1 adopted two Sl-based algo- rithms together, which are FA and PSO, to tackle computationally expensive prob- lems, Bernal [26] proposed a combination of multiple meta-heuristies and Fuzzy logic-based initialization for autonomous robot navigation Besides, some studies that put meta-heuristics and machine learning together have outstanding results Sanchez [44] adopted the Firefly Algorithm (FA) and modular granular neural net-

works for human recognition Zivkovie [58] successfully proposed a hybrid method between machine learning and beetle antennae search [57] to predict COVID-19 cases Bacanin [1] adopted FA and convolutional neural network for magnetic reso-

nance image classification of glioma brain tumor grade Due to constitutive methods’

mutual complementarity, hybrid methods are considered as a promising and effective

approach to dealing with complex problems in many practical applications

These methods, however, still have some shortcomings [18] as follow

Firstly, existing meta-heuristics have not yet imitated the multitasking ability

of humans Each meta-heuristic only solves one problem at a time Meanwhile,

many real-world ystems, such as cloud computing, usually face many tasks

submitted by multiple users simultaneously

@ Secondly, meta-heuri ics always tackle new problems from scratch regardl

of how similar they are to those alread plved in the past Meta-heuristies as- sume zero prior knowledge about them at hand This makes the capabilities of

solvers not automatically grow with problem-solving experience Meanwhile

real-world problems seldom exist in isolation, Many of them are either repeti-

tive or share domain-specific similarities, and therefore, humans routinely use

a pool of knowledge drawn from past experiences when faced with a new task

It is necessary to design meta-heuristics that can imitate the human multitasking

ability and facilitate knowledge transfers between problems during the optimization

process, to reduce the total time for task completion and solve new problems more

efficiently and qui

Over the past few years, a novel research direction called Evolutionary Multitask Optimization (EMTO) [1S] has been inaugurated, and it immediately ushered in a

lassie EA, the EMTO paradigm focuses on ascertaining promising solutions for different problems simulta-

new age for the intelligent computing sphere Inspired by

neously Through optimizing multiple problems together, valuable genetic materials

3

can be transferred between tasks, which often plays a crucial part in improving the

quality of solutions, convergence characteristics, and resource usage compared to solving them separately

‘The emergence of the cloud service and its series of optimization challenges [7] prompted Gupta et al to devise Multifactorial Optimization (MFO), which

has currently matured into one of the most efficient and popular approach

s when

grappling with complex multitasking environments Besides, an algorithm coined MFEA was also introduced based on the concept of multifactorial inheritance from biology With its potentialities, MFEA have become a breakthrough in wide-ranging

sfully adopted the MEEA for tackling more than

applications Sagarna [43] succe

two software testing problems at the same time Chandra [!()] used the MFEA

to simultaneously search many neural network topologies Besides, many other

studies delved into the applications of MFEA, such as fuzzy cognitive map [17], data mining [56], big data [15], and wireless sensor networks [1 4, 24]

The driving force behind the methodology of MFEA is to utilize the implicit parallelism of population-based search with the omnidirectional transfer of knowl

edge across different tasks through crossover-based operators Therefore, not only

the MFEA can leverage the advantage of the traditional EA to allow the evolution

of the population towards optimality with fast convergence speed, but the algorithm can also implicitly acquire the similarity of the solution to each task and use this mutual information to improve the population in subsequent generations

In order to facilitate knowledge exchange in a multitask setting, all populations

of each constitutive task must be defined in a shared place, which is called the

Unified search space, The importance of unifying all factorial search spaces into

one with a single representation method is to create a common platform on which

the transfer of genetic materials among tasks can seamlessly occur Without loss of

Tc to be solved simultancously

DK respectively In such scenario, the unified search space X can be defined with its

., Di} Then, the MFEA employs a single

population P of individuals to solve K’ optimization tasks concurrently

generality, consider K minimization tasks T), T>,

with the search space dimensionality of each optimization task is D1 D:

dimensionality Dynisiead = mair{ Dy, D;

where each

task contributes an added factor influencing the evolution of the population Given

this background, several terminologies are defined to evaluate an individual p; in

population P as follows:

© Skill Factor; Skill factor 7; of p; is the one task on which the individual is most effective, This may be defined as 7) = argminy{r}}, where rj is the rank of individual p; on task T; (j = 1, ,K)

Scalar Fitness: Given the skill factor 7;, scalar fitness y; of p; is calculated

based on its best rank over all tasks, i.e., gi = 1/ri,

4

Algorithm 1: Psenudocode of basic MITA

5 | while stopping conditions ave not satisfied do

“ Ofspring populaLion 23:Ê) —

7 while |P.(2)| < N do

‘ Sample two individnals ja and ya random'y from P(f);

10 ay Tuirartask crossover on pq and pe;

a ssign offspring ie; and :e; skill factor 79;

Pra else if rand < rmp then

3e ú, #j ©- Tnter-raek crossover on pq and pp;

+ Randomly aasign offspring ars z; skill Factor ra or 79;

2 Update scalar fitness for cach individual in P,(2};

a PUL-+ 1) < gat the N biggest scalar fitness individuals in P(e:

as | end

ar end

The general framework of the MFEA is shown in Algorittun 1 During the

evolutionary process, besides the standard intra-tusk crossover between parcols of

the aame skill factor, the inter-task crossover can also occur between candidare

solutions associated with distinct ones This simple featnre apens up the opportunity

to transfer knowleclge through crossover-hased genetic exchange among tasks

Ag can be seen in Algorithm 1, the FFA ia under great control of a preser parameter that indicates the level of inter-task knowledge exchange, called ran dom mating probability {rmp) However, underlying inter-task similarities among different optimization tasks are herd to foresee in practice to select precisely this paramicler

tance into groups (clusters) was formed [| ; M0] Each cluster is considered a

domain in the network, and such a network divided into multiple domains is called a

multi-domain network Privacy issues are also warranted in multi-domain networks

when confidential, private information is circulated only internally in one domain and is not disclosed to competitors in another domain However, the division into

domains in the network poses new challenges that require optimizing routing costs

while maintaining intra-cluster and inter-cluster connectivity among devices

This thesis focuses on tackling two representative network design problems in

Clustered Shortest-Path Tree (CluSPT){10] and CluM-

RCT(27] The inputs and outputs of these problems are quite similar but differ-

the multi-domain networ!

ent in the objective function Both problems are defined on an undirected graph

G = (V, EB) where V is the set of nodes, E is the set of edges, and each edge is as- sociated with a positive weight The vertex set V is divided into m disjoint clusters

{C1,C2, ,Cm} For a cluster C,, the induced graph G[C\] is a maximum subgraph

of G that spans all vertices in C; However, the objective of the CluSPT problem

is to find a multi-domain network architecture to minimize routing costs from a

the CluMRCT problem aims to find

in the

central server to all other devices Meanwhil

a network design to minimize the total routing costs between any two device

multi-domain network Solving these problems is an urgent demand not only for computer network design but also for optimization of various areas such as compu- tational biology, product transportation, logistics, as well as agricultural irrigation

7

Input: - A weighted, undirected graph G = (V,£)

- V is partitioned into m clusters {C), C2

~ A source vertex 5 € V

Col:

Output: —_- A spanning tree T of G

Constraint: - The induced graph T[Ci](i = 1, ,m) is connected

Input: ~ A weighted, undirected graph G = (V, BE)

- V is partitioned into m clusters {C), Cy Cn}

Output: - A spanning tree T of G

Constraint: - The induced graph T[C,](i =

Objective: Minimize S* dr(u,v)

(a) An input graph (b) An invalid solution (6) A valid solution

Figure 2.1: Solutions of the CliuMRCT and CluSPT problems

The solution cases of both problems are illustrated in Figure 2.1 Figure 2.1(a) shows an input graph G whose vertex se is divided into three clusters The indueed

graph of Πon cluster 1 (denoted G[C\]) is composed of a vertex set {v1, v2 14,04} and

an edge set {(vi,02), (v1, us), (t1.va)}- Figure 2.1(b) deseribes an invalid solution in which the induced graph in cluster 2 is non-connected A valid solution is presented

in Figure 2.1(c), where the whole graph is a spanning tree, and the induced graph

8

in each cluster is also a tree Communications among vertices in each cluster are

routed locally and do not transmit through any vertex from other clusters

CluSPT and CluMRCT problems have attracted a lot of attention due to their

urgent demand Because of its NP-hardness, approximation algorithms, heuristics,

or meta-heuristic algorithms are effective methods to tackle these problems

In recent years, some multitasking meta-heuristic approaches were developed for solving multiple CluSPT problems at the same time The authors [50)] proposed

a Multifactorial Evolutionary Algorithm with new genetic operators The main

idea of these operators is that first comes the construction of a spanning tree for the

smallest tasks and afterward constructing spanning trees for larger tasks In [17, 5()], the authors took advantage of the Cayley code to encode the solution of CluSPT and proposed genetic operators The genetic operators introduced here are, conceptually,

similar to the genetic operator for binary and permutation representations However,

it limits its application to complete graphs only Therefore, the proposed MFE

suitable exclusively for complete graphs Binh at.el, [8] discussed a new algorithm

based on the EA and Dijkstr s Algorithm In a divide and conquer fashion, the

proposed algorithm decomposes the CluSPT problem into two sub-problems ‘The

first sub-problem’s solution is found by an EA, while Dijkstra’s Algorithm solves the

second subproblem The goal of the first sub-problem is to determine a spanning

tree that connects among the clusters, while that of the second sub-problem is to determine the best-spanning tree for each cluster The authors [51] proposed a

ic based on a randomized greedy algorithm and Dijkstra’s algorithm In [52], the authors described a method of applying MFEA based on deconstructing an

heuri

original problem into two problems In the proposed MFEA, the second task plays

a role as a local search method for improving the solutions determined in the first task

For the CliMRCT problem, both meta-heuri

were proposed An approximation method called R-Star was first proposed in [27]

s and approximation methods

It constructed a local star tree in each cluster and a global star tree to connect

clusters, However, this method works only on complete graphs It is considered a

2-approximation if the edges of the input graph obey the triangle inequality Be-

side ‘veral multitask methods based on the MFEA were proposed for the CluM- RCT In [55], the authors proposed an algorithm named E-MFEA for the CluMRCT

problem with two-level evolutionary operators The algorithm builds a solution for

the smallest dimensional tasks and then constructs it for higher dimensional tasks However, this algorithm does not control negative transfers when solving unrelated tasks, leading to low-quality solutions than single-task algorithms in some cases

9

The authors [19] proposed an adaptation of MFEA-IT (called aMFEA-II), which

is an improved MFEA to limit negative transfers by online learning the transfer rate in each generation The method outperformed E-MFEA in most cases, but

it was only more effective than R-Star in small-and-medium instances Moreover, both methods use the edge representation, making them suitable only for complete graphs that require any vertex (device) to connect to all other vertices (devices) Although a host of MFEA algorithms were proposed for solving CluSPT and CluMRCT problems in practice, they have revealed multiple drawbacks, i.e., only applicable on complete graphs, inefficient for finding the solution on large search

es, cannot avoid negative transfers between low-similarity problems, and have

significant computational time In the following chapters, the thesis introduces

novel encoding strategies for each problem to allow existing algorithms to function

on both complete and sparse graphs Besides, the th is also designs a novel transfer

mechanism based on multi-parent crossover and a hybrid approach to enhance the

knowledge transfer quality of existing algorithms

10

Chapter 3

Multitask Algorithm for Clustered

Shortest Path Tree

This chapter proposes a multitasking algorithm based on the MFEA algorithm for the CluSPT problem

titasking algorithms to function on both complete and sparse graphs while its size

Notably, a novel encoding strategy is proposed to allow mul-

only equals the number of clusters in the input graph This significantly reduces the search space dimension compared to existing encoding approaches Besides, in-

stead of only performing crossover between two parents, the proposed algorithm is

equipped with a novel multi-parent crossover to enhance knowledge transfers be-

tween tasks, To the best of my knowledge, this is the first effort to examine the

effectivene of the multi-parent crossover in multitasking algorithms

Although many encoding approaches are proposed for the CluSPT problem, such

Code [52

be applied to complete graphs Meanwhile, in practical networks

as Edge-set encoding[23, 50, 5l] and

53], these methods only can

each vertex only

connects to a certain number of other vertices, which makes algorithms hard to apply

to real-world applications In this work, an efficient and small-dimensional encoding

is proposed for the CluSPT to reduce computational resources and help multitasking algorithms can effectively function on both complete and sparse graphs

Given a CluSPT solution T, a local root r is defined as the first vertex to be

visited within its cluster when starting traversal from the source vertex s The cost

11

of T is can be calculated as follows:

to local roots rj(i = 1, m and rp = s) in clusters The second one is the routing

cost from each local root to all other vertices in each cluster

Besides, from Equation 3.1, it can observe that if a set of local roots (79,11 7m)

is given, the optimal solution for the CluSPT problem can be found in polynomial time by the Dijsktra’s algorithm Different sets of local roots result in different

CluSPT solutions, Therefore, instead of saving entire vertices and edges, this thesis

only uses information about local roots in each cluster to represent a CluSPT so-

lution Moreover, only a few nodes can be selected as local roots for each cluster

Because a local root needs to connect to another cluster, vertices selected as the

local root of a cluster must have direct connections to vertices in other clusters As

a result, the sparser the input graph, the smaller the search space needing to be

explored, Hence, the optimization process can be done more quickly

Figure 3.1: A CluSPT encoding example

For cluster Œ;, let Cj denote the set of vertices connecting directly to vertices

in other clusters Cj is called as the inter-vertex set of cluster Cj An example

shown in Figure 3.1 Figure 3 describes an input graph G with 14 vertices and

3 clusters Inter-vertex sets of the corresponding clusters in G are illustrated in

Figure 3.1b Figure 3.l¢ shows an individual encoding of the CluSPT problem, in which each element represents the selected local root of respective clusters

12

Notably, the length of the proposed encoding is equal to the number of clusters

in the input graph Meanwhile, im practical networks, the namber of clusters is always mmich smaller than the nimber of nodes Therefore, it can be concluded thar the proposed encoding strategy significantly reduces the search space dimension compared to existing encoding approaches

Step 1: Dijkstzu's ulgoritiun is used to build the shorlest path tece ía cách cluster

slarting [rom ils local root

Step 2: Comnections among the clnsters are built by a customized version of Dijkstra’s

algorithm First, the proposed method adds vertices of the cluster containing the source vertex 5 into a closed set V Among remaining clusters connected

to ¥ through their local roots, the proposed method chooses to add a cluster

Cy, in which the total routing cost froin the source vertex to its vertices ( C¡| x

dy(s,7;)) ts miniinum, into V

Step 3: Repeat step 2 until all clusters have been added to V’

The Dijkstru’s ulgurition in this work is inplemented using the Binury Heap struc

ture and ily complexity when runaiug ou the input graph G(¥,E) is O(E| |

[V| x log VI) So for ve runs on rn clusters (Cj, Hy) will cost OCD) 1 m(/EM |

|Œ| x fag(JŒ|))) Tn addition, the cost to hnild intercluster edges in T is O(m x

max”, |C;) ‘Therefore, the complexity of the decoding method is O(|f| —m x

max (Cr + Diam (Ge * fog |Ci])) Towever, compntational resources can he

further reduced in the decoding method Given a local root 73, the shortest path tree in each cluster is determined ‘Vherefore, during the optimization process, the algorithm only needs to compute shortest-path trees in Step 1 for every local root

ouve in gach cluster and reuses them for later computations

Figure 3.2 presents a process of constructing the CluSP'l’ solution Figure 3.2(a) shows (he input graph while an individual cneoding: is show in Figure 3.2(b) Vertex

1 is the souree vertex of Uke imput graph, Local roots of three dusters are 1, 12, and

8, respectively Firstly, shortest path (revs slurting from their loeal root are created

as shown in Figure 2.2(c) Next, the cluster containing sonrce vertex 1 is added inte V (red), as shown in Fignre 3.2(d) ‘The decoding method now considers all remaining clusters that connect directly to V, in this case, hoth cluster 2 (through

13

edge (1,12)) and cluster 3 (through edge (3,6)) Comparing these two routes, it can

be seen that routing from 12 to 1 is closer than from 3 to 1, and thus cluster 2 is

added to V (Figure 3.2(¢)) Cluster 3 now can connect to V through both 6 and 11,

Since the path 3— 6 — 1 is shorter than the path 3 — 11 — 1, cluster 3 is added

into V through edge (3,6)

Although the proposed encoding method has many advantages in storing, caleu- lating, and executing evolutionary operators, it contains a weakness in incomplete

graphs Because local roots of clusters are randomly selected, they may not guar-

antee connectivity between clusters in Step 2 of the decoding method

An example is depicted in Figure 3.3 Figure 3.3(a) describes the input graph

with 19 vertices and 3 clusters Figure 3.3(b) presents an invalid CluSPT encoding

As shown in Figure 3.3(b), vertex 12, 8, and 19 are assigned to be the local root

of clusters 2, 3, and 4, respectively, However, in the input graph, there is no path

from 1 (source vertex) to 8 and 19 such that they must be the first visited vertex

in its cluster when starting traversal from 1 Therefore, the local root property of

the selected vertices is not guaranteed A repairing method (RIM) is proposed to

fix these errors as follows:

Step 1: Add the cluster containing the source vertex to a closed set V’

Step 2: Among the remaining clusters, RIM adds into V’ all clusters whose local roots

directly connect to V’

Step 3: If there are still clusters outside of V’ after step 2 then do:

(a) Randomize an edge (u,v) with v € V/ and u¢ V’

(b) Determine the cluster containing u and change the local root of that cluster into u.

(c) Add vertices of that cluster to V’

Step 4: Repeat step 2 and step 3 until all clusters are added to V’

Cluster!) Clager? CSE! Chster2 Cluster! Chater 2 Cluster! Chuye2

An example of the repairing, process

n example of the repairing process

3.4 In Figure 3.4(a), Cluster

1 which contains source vertex, are added into V’ Then, only cluster 2 is directly connected to V’ through the edge (1,12) (highlighted in red) ‘Thus, cluster 2 is

shown in Figur

added into V', as shown in Figure 3.4(b) Next, since clusters 3 and 4 lack an explicit

connection to V’ from their local roots, RIM randomi an edge that satisfies this

condition It can be seen that the edges (3, 6), (3, 9) (6, 11), (18, 13), (14, 7), (14,

13) and (11, 14) are eligible for selection, Assume that the edge (11, 14) is chosen

As a result, the local root: of cluster 4 is changed to 14, and cluster 4 is added to V' (Figure 3.4(c)) Finally, the algorithm runs through the clusters outside of V’

again, in this case, only cluster 3, Because Cluster 3 can connect to V! through the edge (8, 19), it is then added to V’ The property of the resulting local roots is now guaranteed The computational complexity of the proposed repairing method

is O(m *|V|) where m and |V’| are the number of clusters and number of vertices in the input graph, respectively

Furthermore, to analyze the proposed repairing method's efficacy, the thesis has provided and proven a lemma about the maximum number of positions to be fixed for any invalid encoding, as shown in Lemma 3.3.1

Lemma 3.3.1 The maximum number of positions that must be fixed on the encoding

representation is |m/2|, with m being the number of clusters

Proof Consider each cluster C in the input graph @ as a vertex ¢ in new graph

G) An edge between two vertices c; and cy, exists in G; if cluster C; connects to

cluster C via its local root in G or vice versa, Because local roots are randomly

selected from the inter-vertex set, each cluster in G always has at least a direct

connection to another cluster through its local root Therefore, each vertex in G, is always connecting to another vertex In the worst case, G, isa forest with |m/2]

connected components To connect these connected components, it only needs to

16

modify one clnster’s local root in each component Therefore, the maximum mmber

Besides, from Lenrna 3.3.l, ïL can be coneluded that the simaller she uurober of

clusters, the smaller the probability af invalid encodings

In this section, the thesis introduces an approach based on MFEA (called K-MFEA) using the proposed encoding and decoding strategy to solve multiple CluSPT tasks simultaneously Besides, instead of only performing crossover between two parents, the proposed algorithm K-MI'EA is equipped with a novel multi-parent crossover

to enhance Imowledge transfers between tasks

The #® CIISTPT task is performed on an input graph Œ° — (VÌ, ữ*,s),¡ — 1, ,, where ¥'*, £",s! are set of vertices, set of edges, and source vertex, respec

tively V* is divided into m’ clusters ‘U'he j’ cluster of the i* task is denoted by

C2 = 1, mẺ and Ct = {CÍ Ơ§, Cj,} The loeal root of the j" cluster of

te #* task is denoted by +!

‘Lhe proposed algorithin’s structure is presented in Algorithm 2, and the imple-

mentation steps of the algorithm are discussed in detail in the subsections below

3.4.1 Unified search space

The USB for K CluSPT tasks is defined on a graph G,(V,C,m) as follows

@ The number of clnsters m2 = max(ml,m?, m/°) where mi ia the mumher af

"E task

clusters ín the ?#: CiuSi

« ‘The j" cluster of Gy is denoted by Cy, and Cy = Œj*"U C?*U ! CẾ" where CF} is the inter-vertox sel of the j cluster in the # (ask,

—Œ(UØ.J -U Ởu,

Vigure 3.5 illustrates steps to construct the USS for two tasks ‘f} andl 7> 'The iuput graphs C7 wud G? of wwe these tasks are described in Figure 3.5n aud Fig- ure 8.5¢, respectively Frou the input grup of cue tusk, all vertices Uhat do wot divceUy connect lo another cluster ure removed, leaving cach cluster ouly with inter- vertices and their corresponding edges, resulting in the graphs G"* and Œ?*, as shawn

in Figure 3.5h and Figure 3.5d, respectively The remaining vertices all fulfill the requirement of a Ineal root

Assign skill factor 32 = aN +1;

Construct specific representaion sr; af i

Algorithm 4;

Repair sv b Reler bo Scelivn 3.3;

Construct and evaluate the CluSPT solution from sr: for task x; only p Refer to Section 3.2;

idual TRER > Reter to

Update scalar fitness of each individual i P(0);

ile stopping conditions are not satisfied do Offspring populalion P(t) < 4:

while |P,{4)| < W do

jose & random individnals p(i 1 k} from P(8;

/* Perfor muti-parent crossover operator *

if (AN selectad indiniduaie have sine, a skill factor) or đnetui < remy) then s¡f#— 1, ,Ä) — Perform the ruulti-parent crossover on „(ý 1, È)

Assign randow!y the skill facvor of Lhe parents Ws offspring;

else

/* Perform mutation operator *⁄

ý Perform unulation on eaca pareut p;(i=1, 4} & Refer lo sulxoetion 344.4;

Assign respectively the skill factor of the parent ta offspring:

end Construct specific representation sr for each offspring > Refer to Algorithm 4 Repair sr > Refer ta Algorithm 3.3;

Construct and evaluate CluSPT so.ution Zor offspringsr Refer to Section 3.2; Pet) GPE for} B= Leys

end

22n() — the top 50% best individuals from #(¢);

Rit} Pet} UPaft):

Update scalar Ñtnens of each ind*vidua in RỤ)

Pet 1) + Get N fitrest individuals from (2);

+ +11,

18

Next, inter-vertices are merged in the respective clusters Figure 3.5e shows an

obtained USS where the red area in each cluster denotes the vertices used for the

first task while the orange area marks the vertices for the second task

3.4.2 Individual Initialization Method

Each clement in a unified individual representation is randomly selected from the respective cluster in G,,(V,C,m) The initialization method details are presented in Algorithm 3 with computational complexity of O(m)

Algorithm 3: Initialization Individual Method

help new offspring to inherit much more good characteristics from their parents It

iy milar to how farmers breed new crops by inheriting the excellent traits from

multiple trees: pest resistance of plant A, high yield of plant B, and fruit quality of plant C

Figure 3.6 shows offsprings obtained after performing N-MPCX ’s steps Fig-

ure 3.6(a) presents n (n = 4) input parents and n—1 random cut-points The it

offspring preserves two segments of genes which are from the (i — 1)" cut-point to

Parent [2 s|b[ninlrpsi 33 onspeing | [2 [ 5 |12[13|18|24|26]29|33

Pwent? [4 s [I10J14|18|24]2629J3L 0e? [1 J6 [I0]I4]IS]2225]50]31

s3 [16 ]I2JI3[I8]23]25J29J52 onserines [3] 7 [I]IS]20]23|25]29]32

rwen't [3 [7 ]I1[132022]250]34| @meis+ [4 | 5 [9 [16]17]21] 26)

Figure 3.6: An example of the proposed crossover operator

20

with Wand mm heing the nnmber oŸ parenis and the mmber of ciusters in the nnified search space, respectively

3.4.4 Mutation Opcrator

‘The mutation operator is performed by randomly selecting a cluster and replacing its local root with another inter-cluster vertex in the same cluster ‘I'he mutation iucthod’s computational complexity is O(L)

3.4.5 Mapping Individual Method

This part presents a method to construct a task-specific individual representation from a unified individual representation

Algorithm 4: Mapping Individual Methad

Input:

» Ax input graph of the i wask GEV BY CFC, stm)

» Unified search space G, iV £.C.m)

» Ar unified individual representation # = (rita, .tmb

Ontpnts A task-sparific individnal representation 1) = (rh ooh}

y+ find maximum index of vertex Jin 7?*, Và € [L, , K] smd (b g 73;

ae size, + the number of elements {a

Let { be the vertex corresponding to the #* element If f can be found within

the # cluster of the current task, ib will be selected to be the local rool Otherwise,

xf wmoug the i clusters of ull

the methed locates the maximun index of v

obher tasks, It lieu takes tie verlex whose index is the reminder of the division hetrween the maximum index and the current cinster’s size The method is deseribed

in Algorithm 4 with the computational complexity of O(m?}, where mis the namber

of cinsters in the nmified search space

al

Chser4 ‘A individual epesentation Any ivi represcaion

Figure 3.7; An example of constructing specific-task representation from an indi- vidual in the unified search space

Figure 3.7 illustrates how to construct specific individual representations for

tasks 1 and 2 from an unified representation Figure 3.7(a) describes the unified search space USS G, = (V,C,m) of two tasks Inter-vertex sets in Task 1 are Cl* = {1,6,5}, Cl* = {12,11,14}, and Cl* = {9,7,3} Inter-vertex sets in Task 2 are

CP = {3, 1,7}, CF = {11, 12, 13}, C3 = {6,8.9, 10}, and C? = {14, 15, 16, 18, 19},

respectively The mapping process for Task 1 is as follows, Due to Cluster 1 contain-

ing the source vertex 1, the local root of Cluster 1 is set as 1, Next, because vertex

14 exists within the inter-vertex set C}* of Cluster 2, it is selected to be the local

root of Cluster 2 However, vertex 6 does not appear in the inter-vertex set C}*, and

its maximum index in cluster 3 of the other task is 0 Therefore, vertex 9, having

the same index 0 (= 0 mod 3), will be chosen as the local root of Cluster 3 The

process for Task 2 is similar, Figure 3.7(b) demonstrates two obtained specific-task individual representations after applying the mapping method

Chapter summary

This chapter proposed a novel encoding strategy based on local roots for the CluSPT problem, Notably, the proposed encoding allows multitasking algorithms to fune- tion on both complete and sparse graphs and significantly reduces the search space dimension compared to existing encoding approaches Besides, a multitasking algo- rithm with a novel multi-parent crossover is designed to enhance knowledge transfers between tasks To the hest of my knowledge, this is the first effort to examine the effectiveness of the multi-parent crossover in multitasking algorithms

Chapter 4

Multitask Algorithm for Clustered

Minimum Routing Cost Tree

This chapter presents a new encoding and decoding method for the CluMRCT prob- lem, The encoding method encrypts each solution of the CliuMRCT problem by an array whose size is the number of vertices in the input graph, while the decoding

method takes advantage of a greedy algorithm and enables multitasking algorithms

to function efficiently on both complete and sparse graphs Besides, to reduce the

effect of blindness and randomness transfers, this chapter introduces a hybrid multi-

tasking algorithm named multifactorial firefly algorithm to tackle multiple Clustered Minimum Routing Cost Tree (CluMRCT) problems at the same time

There are many methods to represent a spanning tree in the literature A good

encoding strategy will offer many benefits to solve complex problems The most

prominent spanning-tree encoding schemes, namely Edge-set Encoding (ESE) [!1]

(uses a list of edges), Characteristic Vector Encoding (CSE) [I0] (utilizes a binary

structure whos ize is the number of edges), Predecessor Encoding (PE) [10] (stores the id of the parent node), Priifer Number Encoding (PNE) [17, 34] (employs an integer vector whose size is less than the number of vertices by 2), Network Ran- dom Keys Encoding (NRKE) [12] based on the priority of edges, and Node-Depth Encoding (NDE) [!), 15] based on the depth and degree of nodes, These methods, however, still have some shortcomings [!), 40] For example, they are highly redun- dant (¢.g., NRKE), or only work efficiently in the simple complete graph (e.g., PE, PNE, NDE);

operators (¢.g., ESE, CSE, PE, PNE, NDE); or cannot apply directly into the CluM- RCT problem Beside

encoding for the CluSPT problem in the previous chapter also cannot be applied to

asily generate unfeasible solutions, require the complex evolutionary

due to no information about source vertex, the proposed

23,

CluMRCT

In this part, a new encoding is designed based on the NRKE to adapt the CluMRCT problem’s constraint The proposed encoding method (PEM) assigns a priority value to each node instead of each edge as the original version to reduce redundancy The higher the priority, the earlier the node is visited, The proposed

ize than the NRKE An example of the PEM is shown

encoding has a much smaller

Cluster 1 Cluster 2 Cluster 3

Figure 4.1; An CluMRCT encoding example

The motivation of the proposed decoding method (PDM) is derived from decom-

posing the cost function of the CluMRCT solution T(V, £) as follows:

3” ara, v) Nr `, Anj+Д > 3” 5 aru) (41)

i=l uy i=l j=i+l ueŒ, ueC,

As can be obs rved in Equation 4,1, the cost of the solution can be decomposed

into two separate parts, The first one is the total cost of intra-routing between

any two vertices in each cluster Meanwhile, the second one represents the total

inter-routing cost between two vertices from two different clusters Based on this

idea, PDM is also implemented in two levels PDM traverses vertices according

to their priority to construct a minimum intra-routing cost spanning tree in each

cluster in the first level, while it builds a minimum inter-routing cost spanning tree

for connecting among clusters in the second level

In the first level, let T,(Vi, £1) be a subtree being under construction in a cluster

i whose set of vertices is Cy Initially, Vi = @ and E, = 0 The steps of PDM in

the first level are as follows:

© Step 1: PDM adds the highest priority vertex (called vy) in Cụ into Vj and removes vy from Cy

Step 2: Let y be the highe:

to Vj in Cy Among all edges connecting v to Vi, PDM chooses to add the edge causing the smallest increment in the routing cost of the tree T), Assume that

vp is connected to V through vertex v;(v; € Vj) The routing cost increment

priority vertex among vertices directly connected

24

(A) is calculated as in Equation 4.2

© Step 3: Go to step 2 until Cy is empty

The computational complexity of phase 1 in the decoding method is O(D" (CilogC+

C?)) where CjlogC; is a cost to sort the priorities of vertices in each cluster, and C?

is a cost to search and update connections in Equation 4.2

Figure 4.2: Proposed decoding method in the first level: a) An input graph; b)

A CluMRCT encoding; c-g) Steps to construct an intra-routing spanning tree in

Cluster 3

Figure 4.2 illustrates the construction process in the first level of the proposed decoding method Figure 4.2(a) and Figure 4.2(b) deseribe an input graph with 3

clusters and an individual encoding Figures 4.2(c)-4.2(g) illustrate steps to build

intra-connections among vertices in Cluster 3 Firstly, vertex 10 has the highest priority in Cluster 3, so it is added into a set Vj (red), as shown in Figure 4.2(c)

‘Then, vertex 11 is the highest priority vertex among the remaining ones adjacent

25,

to Vj Therefore, vertex 11 is added into Vj and connected ta Vi thrangh the edge (10, 11) due to only ane connection between these vertices, as shown in Figure 1.2(4) Next, vertex 14 is the highest priority vartex among vertices adjacent: to Vj The chosen edge connecting it and 1 is (14, 10) to minimize an increment in the routing cost, as shown in Figure 4.2(e) Similarly, vertex 12 is added and connected to 4 through edge (12, 14), as shown in Figure 4.2(f) Finally, vertex 18 is connected to

Vj Gough cdge (13.12) The fivul intru-routing spanuing tree in Cluster 3 is shown

in Figure 4.2(g) After the first phuse, intr-vonnectiouy in cuch duster huve been constructed, und the induced graph in euch cluster is ypanniug tree

In the sceond level, constructing the intcr-cluster cdgos Ìs similar to the first level

by ussiguing a priority to cuch eluster For simplicity und av more vost of iafor:nation storage, a cluster's priority is linked to vertices’ highest priovity in that cluster Ter

H (Vs, Fp) be the fñnal ChiMRCT solnien te be bnilt Ÿ — {C¡,Ca, , Cụ} is the set of unvisited clusters, and 7(Œ,, P3) is the subtree buïlt in the chster Œ in the first level ‘I'he steps of I'M in the second level are as follows:

« Step 1: Find the highest priority cluster {called Cg) and acd entire edges and vertices of the subtree 74(C, Sy), which are built in the first level, into (¥ £5) Ve = Cy and £y = 2, ‘Then, remove Cy from S$

@ Step 2: Determine the cluster C, that has the highest priority in S and at least one edge (u, v), u € Cy and v € Vy Next, PLM selects an edge (p,q) that connects T:(Ch, Bh) and Hyp © Cy and q C Vp, such that the increment (5) in routing cost of the troc H duc to new connections is minimized The extra cost(é) is calculated as in Equation 4.3

§ = anaes }” 2 (dn,(u,g) + 0(p,g) + dư{g, 0))

— [Wa] x YE du, (up) + [Me x [Ca] x psa) + [Cel x SE dala, 0)

(43)

‘Lhen, update Wy = WUC}, S=S\ Cy, and by = HU LS I {(p, gb

« Step 3: Repeat step 2 until 5 is empty

Figure 4.3 illustrates an example of the construction process in the second level

of the PDM Cluster 1 has the highest priority, so its entire vertices are added into aseu Vy (red) In Figure 4.3(e), Cluster 2 is wow the highest priority cluster umong the remaining oucs wdjuccnt to Vy Therefore, all vertieos of Cluster 2 are added inte Yj, aud Cluster 2 is connecicd te V; through the edge (1,6) Finally, Cluster 3

is added into Vy The edge (10,5) is chosen to minimize new connections’ additional cost The obtained spanning tree in Figure 4.3(e) is now the complete aolurion after the deending phase