Genetic algorithms for solving bounded diameter minimum spanning tree problem188

MINISTRY OF EDUCATION AND TRAINING HANOI UNIVERSITY OF SCIENCE AND TECHNOLOGY GENETIC ALGORITHMS FOR SOLVING BOUNDED DIAMETER MINIMUM SPANNING TREE PROBLEM By Huynh Thi Thanh Binh Supe

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MINISTRY OF EDUCATION AND TRAINING

HANOI UNIVERSITY OF SCIENCE AND TECHNOLOGY

GENETIC ALGORITHMS FOR SOLVING BOUNDED DIAMETER MINIMUM SPANNING TREE PROBLEM

By Huynh Thi Thanh Binh Supervisor: Associate Professor Nguyen Duc Nghia

A Dissertation submitted in partial fulfillment of the requirements

for the Degree of Doctor of Philosophy in Engineering

HaNoi, 2011

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Table of Contents

1.1 Motivation 7

1.2 Methodologies 8

1.3 Scope of research 14

1.4 Contributions 14

1.5 Outline 16

2 Bounded Diameter Minimum Spanning Tree and Related Works 18

2.1 Problem formulation 18

2.2 Related Optimization and Decision Problems 20

2.3 Related works 22

2.3.1 Exact approaches 23

2.3.2 Heuristic Methods 23

2.3.2.1 One Time Tree Construction Algorithm 24

2.3.2.2 Center-Based Tree Construction Algorithm 25

2.3.2.3 Randomized Greedy Heuristic Algorithm 25

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2.3.2.4 Improved Greedy Heurisitics (RGH I− and CBT C I− ) 27

2.3.2.5 Hierarchical clustering heuristic algorithm - HCH 28

2.3.2.6 Comments 29

2.3.3 Metaheuristic algorithms 30

2.3.4 Conclusion 39

3 Center-Based Recursive Clustering Heuristic Algorithm 41

3.1 The new greedy heuristic - Center-Based Recursive Clustering (CBRC) 41

3.2 The improvement of Centre-Based Recursive Clustering -CBRC I− 44

3.3 Experiments 45

3.3.1 Problem instances 45

3.3.2 Experiment setup 46

3.3.3 Result 46

3.4 Discussion 55

3.5 Conclusion 56

4 Genetic algorithm with multi-parent recombination operator 57

4.1 Individual representation and genetic operators 58

4.2 Experiments 61

4.2.3 System setting 62

4.2.4 Results and discussion 63

4.3 Conclusion 77

5 Multi-population Genetic Algorithm 79

5.1 Structure of the genetic algorithm 80

5.2 Experiments 83

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5.2.3 System setting 84

5.2.4 Result 85

5.2.5 Discussion 85

5.3 Conclusion 95

6 Steady-state genetic algorithm 97

6.1 Steady state genetic algorithm structure 97

6.1.1 Individual representation and initial population 97

6.1.2 Crossover 98

6.1.3 Mutation 98

6.1.4 Selection 99

6.2 Replacement policy 99

6.3 Experiments 100

6.3.3 Parameter 101

6.3.4 Result 101

6.4 Conclusion 106

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List of Figures

1.1 Scheme of genetic algorithm 11

2.1 TheBDST with 19 vertices and bounded diameter =4,D vis the center of

the tree 20

2.2 The BDST with 19 vertices and bounded diameter D=5, v1, v2 are the

centers of the tree 20

2.3 The best BDST found by OT T C algorithm on the Euclidean problem

instance with n= 100,D= 5 24

2.4 The best BDST found by CBT C algorithm on the Euclidean problem

instance with n= 100,D= 10 27

2.5 The best BDST found byCBT C I− algorithm on the Euclidean problem

instance with n= 100, D= 10 27

2.6 The best BDST found by RGH algorithm on the Euclidean problem

stance with n= 100,D= 10 28

2.7 The best BDST found by RGH −I algorithm on the Euclidean problem

2.8 A spanning tree on twelve nodes and an its edge-set representation 31

2.9 A spanning tree on eleven nodes and an its permutation-code representation 33

2.10 Center Move Mutation 34

2.11 Edge Delete Mutation 34

2.12 Subtree-Optimize Mutation 35

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2.13 The best BDST found by JR ESEA− algorithm on the Euclidean problem

2.14 The bestBDST found by JR P EA− algorithm on the Euclidean problem

2.15 The best BDST found by P EA I− algorithm on the Euclidean problem

3.1 A star-like structure of a typical solution to theBDMST problem 42

3.2 Greedy Edge Delete Local search 45

3.3 The best BDST found by CBRC heuristic on the Euclidean problem

stance with n= 100,D= 10 45

3.4 The bestBDST found byCBRC−I heuristic on the Euclidean problem

4.1 Comparison between the sum of the best solutions found by EA − xy 2

algorithm on all the problem instances 68

4.5 Comparison between the sum of the best solutions found by EA −xdk

algorithm on all the problem instances ( = x b, r, l k; = 2 5 7 9) ., , , 69

4.6 Comparison between the sum of the best solutions found by EA −xgk

4.7 Comparison between the sum of the best solutions found by EA−xmk

algorithm on all the problem instances ( = x b, r, l k; = 2 5 7 9) , , , 69

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4.8 Comparison between the sum of the average solutions found by EA xy− 2

algorithm on all the problem instances ( = x b, r, l) 69

algorithm on all the problem instances ( = x b, r, l) 70

4.12 Comparison between the sum of the average solutions found by EA−xdk

4.13 Comparison between the sum of the average solutions found by EA−xgk

4.14 Comparison between the sum of the average solutions found by EA xmk−

algorithm on all the problem instances ( = x b, r, l k; = 2 5 7 9) ., , , 714.15 Comparision between the best solution found by GA1, GA2, GA3, GA4,

GA5, GA6 on all the problem instance 71

4.16 Comparision between the standard deviation of the solution found by GA1,

GA2, GA3, GA4, GA5, GA6 on all the problem instance 71

5.1 Multi-population model 805.2 The comparision between the best results found by GA11, GA12, GA13,

GA14 andHGA on the instance with n= 250, D= 15, instance 1 865.3 The comparision between the mean results found by GA11, GA12, GA13,

GA14 andHGA on the instance with n= 250, D= 15, instance 1 915.4 The number of individuals from GA11, GA12, GA13, GA14 migrate to GA final 91

6.1 P EA I− algorithm 99

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List of Tables

3.1 Diameter Bound 46

3.2 Results of OT T C CBT C RGH CBRC CBRC, , , , −I, RGH −I on the

Non-Euclidean instances of the BDMST problem with n = 250 andD =

5 10 13 15 17 20 25., , , , , , 52

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on the 20 Euclidean problem instances 64

4.3 Comparision between the result found byEA xy− 5; x= d, g, m y; = l, r, b

on the 20 Euclidean problem instances 675.1 Comparision between the result with diﬀerent crossover probabily on the

Euclidean problem instance with number of vertices are 250,D=15 855.2 Comparision between the result with diﬀerent crossover probabily on the

Euclidean problem instance with number of vertices are 250,D=15 86

5.3 Comparision between the result found by RJ −ESEA, P EA I− , HGA,

MHGA on the 20 Euclidean problem instances 87

5.4 Comparision between the result found by RJ −ESEA, P EA I− , HGA,

MHGA on the 20 Non-Euclidean problem instances 88

5.5 Result of GA11, GA12, GA13, GA14 and HGA on 20 Euclidean BDMST

problem instances 89

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5.6 Result of GA11, GA12, GA13, GA14andHGA on 20 Non-EuclideanBDMST

5.7 Result of GA21, GA22, GA23, GA24 andMHGA on 20 Euclidean BDMST

5.8 Result of GA21, GA22, GA23, GA24 andMHGA on 20 Euclidean BDMST

6.1 Results ofP EA RGH− ,P EA RGHI− ,P EA CBRC− ,P EA CBRCI− ,

P EA CBRC P EA CBRCI P EA I− , − , − to reach the best solution on

the 20 Euclidean instances of BDMST problem of size 100, 250, 500 and 1,000 103

6.3 Results ofP EA RGH− ,P EA RGHI− ,P EA CBRC− ,P EA CBRCI− ,

P EA I− on the 20 Non-Euclidean instances of the BDMST problem of

size 100, 250, 500 and 1,000 104

6.4 Average number of iterations required by P EA RGH− , P EA RGHI− ,

P EA CBRC P EA CBRCI P EA I− , − , − to reach the best solution on

the 20 Non-Euclidean instances of BDMST problem of size 100, 250, 500

and 1,000 105

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The Bounded Diameter Minimum Spanning Tree (BDMST) problem is a combina-

torial optimization problem that arises in many applications such as design of wire-based

communication networks under quality of service requirements; design linear lightwave

networks, where it can minimize interference in the network by limiting the traﬃc in the

network lines Another practical application requiring a BDMST arises in data compres-sion, where some algorithms compress a ﬁle utilizing a tree data-structure, and decompress

a path in the tree to access a record in ad-hoc wireless networks distributed mutual clusion algorithms

problem has been shown to be also approximate-hard, in that there is no polynomial time

algorithm which could guarantee to ﬁnd a solution which has a cost within log(| |V ) of

the optimum, unlessP = NP Therefore, heuristic and meta-heuristic techniques are cur-

rently the only practical method for improving the solution quality in solving the BDM ST

problem, especially when | |V is large

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In this thesis, we survey the literature on the BDMST and then present new algorithmsfor solving this problem.

First, we propose a greedy heuristic algorithm called Center-Based Recursive Clustering

(CBRC) We extend the concept of center to each level of the partially constructed

spanning tree The algorithm can be seen as recursively clustering the vertices of the

graph: every internal node of the spanning tree is the center of the graph in the

tree rooted at this node and we recursive to ﬁnd the best center The new heuristic is

compared with other well-known heuristics for solving the BDMST problem, namely, the

One-Time-Tree-Construction (OT T C), the Randomized Greedy Heuristic (RGH) of Raidland Julstrom, the Center-Based Tree Construction (CBT C) of Julstrom, the RandomizedGreedy Heuristic with post-improvement (RGH I− ) and Center-Based Tree Construction

with post-improvement (CBT C I− ) of Singh and Gupta

And then, we introduce multi-parent recombination operator in Genetic Algorithms (GAs)

for solving the BDMST problem The proposed multi-parent recombination operator al-lows using more than two parents to create offspring We consider three different methodsfor choosing parents Three new methods for adding edges from the parents to the off-

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an-state genetic algorithm We present steady-an-state genetic algorithms which use diﬀerentheuristic algorithms for decoding We modify the decoder and the replacement policy

used in PEA−I so as to improve its performance We use four decoders by diﬀerent

well-known heuristic algorithms: RGH RGH I, − ,CBRC CBRC I, −

Experimental results are also reported to compare the eﬃciency of diﬀerent heuristic and

genetic algorithms for solving BDMST problem

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Dr Truong Thi Dieu Linh, Dr Le Minh Hoang, Dr Le Trong Vinh.

I would like to give special thanks to my parents, my husband and my daughters, who gave

me unconditional support and encouragement during the long time I needed to conductresearch and write this thesis

Also, thank to Ministry of Education and Training, Hanoi University of Science and Tech-nology, National Foundation for Science and Technology Development for their funding

for my research I would like to thank to my colleagues at School of Information andCommunication Technology, my friends, for their comments and encouragement

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wiring scheme that connects the sites using as little wire as possible It is the mother of

all network design problems This minimum spanning tree is a fundamental problem and

can be easy polynomial-time solved by using Prim or Kruskal algorithm

Another example concern with a traﬃc network whose nodes represent both origin and

destination areas for the vehicular traffic of a city and also intersections in the road work The arcs correspond to streets in the city, and the arc flows are the amount of traffictraversing the streets A typical network design problem would be to select a subset of the

possible road improvements subject to a budget constraint The design objective would

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charge design problem If all arc construction costs are set to zero, then the ﬁxed charge

be conﬁdent that it is very diﬃcult to solve If the arc construction costs are all equal

and totally dominate the routing costs (i.e., the optimal network design must be a tree),then the ﬁxed charge design problem becomes the optimum communication spanning treeproblem deﬁned by Hu [25]

Scott [4] has introduced another network synthesis problem, called the ”optimal network”

subject to the usual capacity and ﬂow routing constraints and the added constraint that

the total construction costs cannot exceed a given budget Optimal network problem is

NP −hard

In communication network design when requirements can be for example a limitation of

the maximum communication delay or the guarantee for a minimum signal-to-noise ratio,

thus, the number of relaying nodes on any path between two communication partners

decompress a path in the tree to access a record; in ad-hoc wireless networks distributed

mutual exclusion algorithms More detail about the applications of the BDMST are

presented in the next section

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consider several applications as bellow.

In communication network design, the requirements can be a limitation of the maximum

communication delay or the guarantee for a minimum signal-to-noise ratio Thus, the

processors requesting entrance to a critical section and processors grating the privilege to

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tures called bitmaps are used in compressing large ﬁles, see [9] It is required to compressthe ﬁles, so that they will occupy less memory space, while allowing reasonably fast access

In a ﬁrst step similar vectors are clustered To further increase the compression rate not

rithms are: Dynamic Programming (DP) [8], Constraint Programming (CP) [44], Branch-Bound, Linear Programming based BranchBound, BranchCut, BranchPrice, and Branch-

CutPrice [33] These approaches are only applicable to relatively small problem instances

due to run- time and sometime also memory restrictions while larger instances are solved

by heuristics Heuristics and especially metaheuristics can be seen as alternative when

large instances have to be solved in reasonable time, whereas these approaches are not

able to guarantee to reach the optimum

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There are a lot of heuristic algorithms based on diﬀerent approachs, such as: GreedyHeuristics, Local Search, Evolutionary Algorithms, These approaches can only applied

for speciﬁc problems Recently, researchers use metaheurisic algorithms to design a com-

candidate solutions However, metaheuristics do not guarantee an optimal solution is everfound Examples of metaheuristic algorithms are: Iterated Local Search [31], Tabu Search

[14], or Variable Neighborhood Search (V NS) [23], Simulated Annealing [30], Ant ColonyOptimization (ACO) [11], Evolutionary Algorithms (EA) [5], and Memetic Algorithms[32]

We will brieﬂy overview Greedy heuristic algorithms, Local search, Genetic algorithms

which we use for developping new algorithm for solvingBDMST

Greedy heuristic algorithm is an algorithm that follows the problem solving meta-

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• Greedy choice property: the choice made by a greedy algorithm may depend on

choices made so far but not on future choices or all the solutions to the subproblem

It iteratively makes one greedy choice after another, reducing each given problem

into a smaller one

• Optimal substructure: a problem has optimal substructure if the best next move

always leads to the optimal solution

Greedy algorithms mostly (but not always) fail to ﬁnd the globally optimal solution,because they usually do not operate exhaustively on all the data They can make com-mitments to certain choices too early which prevent them from ﬁnding the best overallsolution later For example, all known greedy coloring algorithms for the graph coloring

problem and all other NP −complete problems do not consistently ﬁnd optimum solu-tions Nevertheless, they are useful because they are quick to think up and often give good

approximations to the optimum

Local search is a metaheuristic for solving computationally hard optimization problems

Local search can be used on problems that can be formulated as ﬁnding a solution maxi-

mizing a criterion among a number of candidate solutions Local search algorithms move

from solution to solution in the space of candidate solutions (the search space) until a

solution deemed optimal is found or a time bound is elapsed

A local search algorithm starts from a candidate solution and then iteratively moves to a

neighbor solution This is only possible if a neighborhood relation is deﬁned on the search

space As an example, the neighborhood of a vertex cover is another vertex cover only

diﬀering by one node For boolean satisﬁability, the neighbors of a truth assignment are

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Figure 1.1: Scheme of genetic algorithm

The general scheme of a GA can be given in the ﬁgure 1.1

GAs are useful and eﬃcient when:

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• No mathematical analysis is available

• Traditional search methods fail

Representation: Objects forming possible solution within original problem context arecalled phenotypes, their encoding, the individuals within the GA, are called genotypes

The representation step speciﬁes the mapping from the phenotypes onto a set of genotypes

search could become too greedy and get stuck in a local optimum

Survivor Selection Mechanism: The role of survivor selection is to distinguish among

dividuals based on their quality In GA, the population size is (almost always) constant,

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thus a choice has to be made on which individuals will be allowed in the next

tion This decision is based on their ﬁtness values, favoring those with higher quality As

opposed to parent selection which is stochastic, survivor selection is often deterministic,

for instance, ranking the uniﬁed multiset of parents and oﬀspring and selecting the top

segment (ﬁtness biased), or selection only from the oﬀspring (age-biased)

Termination Condition: Notice that GAis stochastic and mostly there are no guarantees

to reach an optimum Commonly used conditions for terminations are the following:

1 The maximally allowed CPU times elapses

new algorithms for solving BDMST problem

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1.3 Scope of research

This thesis researches on the BDMST problem, propose new algorithms to ﬁndbetter solution, experiment on the problem instances which use by other research groups

Contributions will be presented in four chapters and can be summerized as follow:

1 We propose the Center-Based Recursive Clustering (CBRC) heuristic algorithm

CBRCis based onRGH (andCBT C) We extend the concept of center to each level

of the partially constructed spanning tree The algorithm can be seen as recursively

and others - RGH RGH, −I,CBT C OT T C CBT C, , −I - on the Euclidean andNon-Euclidean instances up to 1000 vertices On the Euclidean instances, the resultsshow the eﬀectiveness of our algorithms on the best, mean and deviation values On

the Non-Euclidean instances, the best results found by CBRC I− are the same with

the one found byOT T C

2 We also introduce three multi-parent recombination operators in genetic algorithmfor solving BDMST problem We consider three diﬀerent methods for choosing

parents: the ﬁrst one is based on Levenshtein distance between the parents, the sec-

ond one uses the best individual in the population and the last one uses randomly

chosen individual in the population We also experiment each method of choosing

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parents with three ways for adding edges from the parents into the oﬀspring: choosethe edge randomly, choose the edge which have minimum weight, choose the edgewhich have minimum weight in maximum sharing edge from the parents We exper-

iment on the Euclidean instances up to 1000 vertices We concentrate on analyzing

the recombination operator in genetic algorithms So we compare the results of ouralgorithms using, respectively, three mentioned multi-parent recombination opera-tors with another genetic algorithm using two-parent recombination operator on thesame problem

3 We propose a new hybrid genetic algorithm for solvingBDMST problem The new

genetic algorithm uses multi-population, where each population is initialized with adiﬀerent well known heuristic The individuals in each population will subsequently

compete for positions in a selection population, using a simulated annealing nism based on proportionate selection; in the selection population, they will combineand evolve toward the optimum Therefore, our research approaches employ diﬀer-

ent initial biases by using diﬀerent heuristics for initialization, and to hybridize the

individuals from these populations to promote the exploratory capacity of the GA

We compare our results with other genetic algorithms, namely, the genetic algorithm

in [40] of Raidl and Julstrom (called RJ −ESEA), the genetic algorithm of Alok

and Gupta in [46] (calledP EA I− ) and the genetic algorithm in each population onthe Euclidean and Non-Euclidean instances up to 1000 vertices The results show

the eﬀectiveness of our algorithm

4 We propose steady-state genetic algorithms which use diﬀerent heuristic algorithms

for decoding We modify the decoder and the replacement policy used in P EA−I

so as to improve its performance We use four decoders by diﬀerent well-known

heuristic algorithms: RGH RGH I, − ,CBRC CBRC I, − We experiment on the

Euclidean and Non-Euclidean instances up to 1000 vertices and the results show the

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outperform of our algorithms than the others.

1.5 Outline

This dissetation is organized as follow

of the algorithms for solvingBDMST only suitable for one kind of the problem instance:

Euclidean or Non-Euclidean instances So, in the remain chapters, we will present our

algorithms for solving BDMST We hope that our propose algorithms can be applied for

both Euclidean and Non-Euclidean instances to ﬁnd better solution

in the chapter 4, 5, 6, we present our genetic algorithms for solving BDMST

An EAs recombination operator should provide strong heritability This means that the

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that uses a multi-population, where each population is initialized with a diﬀerent well

known heuristic Chapter 5 presents new hybrid multi-population genetic algorithm in

which each population is initialized with a diﬀerent well know heuristic Chapter 6 will

introduce steady-state genetic algorithm for solving BDMST problem which uses diﬀer-ent heurisitics for decoding the tree

Finally, the conclusion summarizes the works

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Chapter 2

Bounded Diameter Minimum

Spanning Tree and Related Works

This chapter presents the formulation of BDMST and summarizes the related works in

the ﬁeld of the BDMST problem

Before introduce the approaches for solving BDMST, we state the problem

tices in ).T

Deﬁnition 3: (Center of tree) The center of a tree is the single vertex (if the diameter of

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If k is even then v [k] is called a center of the tree If k is odd then v [k] and v[k]+1 are

centers of the tree In the last case, the edge (v [k] ,v[k]+1) is called a center edge

Deﬁnition 4: (Radius) The radius of a tree is the minimum eccentricity among all nodes

of the tree

Deﬁnition 5: (Bounded Diameter Minimum Spanning Tree Problem - BDMST Let)

without lost of generality, we will assume that G is a complete graph

Thus, we can formulate the problem as:

and the bounded diameter is D= 4 and D = 5 respectively The number of edges in the

longest path is 4 and 5 respectively In ﬁgure 2.1, the bounded diameter is even number,

so the center of tree is only one vertex In ﬁgure 2.2, the bounded diameter is odd number,

so v1, v2 are the centers of tree and (v 1 ,v2) is center edge

Deﬁnition 6: (Decision BDMST problem) Let G = (V, E) be a connected undirected

graph with edge weights are 0 or 1 and two intergers D ≥ 2 and q ≥ 2 Does exit a

spanning tree with diameter less than or equal D and the weight of tree is ?q

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Figure 2.1: The BDST with 19 vertices and

bounded diameter =4, is the center of theD v

in O n( 2) (D = 2), respectively by iterating over all edges and connecting the remaining

nodes in time (O m.n D) ( = 3), which is bounded above by (O n 3) for complete graphs In

case, 4 ≤ | | −V 1, BDMST become NP hard− problem Detail about special cases with

D < 4 can be seen in [16] Reduction of BDMST is introduced in [13, 17]

2.2 Related Optimization and Decision Problems

Some of the well-known constrained minimum spanning tree problems require

imizing the weighted diameter of the spanning tree of a randomly-weighted graph These

problems are closely related to the problems that require optimizing the weighted radius

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problem, and vise versa In this section, we introduce some optimization and decisionproblems concern with BDMST.

Let = (G V, E) be a connected undirected graph with positive edge weights ( ) Supposew e

T = (V, ET) be a spanning tree of G

Problem 1: Bounded Weighted Diameter Minimum Spanning Tree problem (BW DMST)

Among all spanning trees ofGwhose weight of diameters do not exceed a given upper bound

D, ﬁnd the spanning tree with the minimal cost

Problem 2: Minimum Weighted Diameter Bounded Spanning Tree problem (MW DBST)

Among all spanning trees of G whose weight of tree do not exceed a given upper bound ,Sﬁnd the spanning tree with the minimal weighted diameter

Problem 3: Bounded Weighted Radius Minimum Spanning Tree problem (BWRMST)

Among all spanning trees of Gwhose weight of radius do not exceed a given upper bound

R, ﬁnd the spanning tree with the minimal cost

Problem 4: Minimum Weighted Radius Bounded Spanning Tree problem (MW RBST)

Among all spanning trees of G whose weight of tree do not exceed a given upper bound ,Sﬁnd the spanning tree with the minimal weighted radius

Problem 5: Bounded Weighted Diameter Bounded Spanning Tree problem (BW DBST)

Problem 7: Hop Constraint Minimum Spanning Tree Problem (HCMST Given a graph)

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connected undirected graph with positive edge weight w e( ) DCMST can be formulated as

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problem may be classiﬁed into two main categories: exact methods and inexact (heuristic)methods Exact algorithms are guaranteed to ﬁnd an optimal solution The run-timeincreases dramatically with the instance size, and often only apply for small instances.

algorithms for it and solved instances with up to 100 vertices Gouveia and Magnanti

[15] described a network ﬂow model that solved instances with up to 100 vertices and

graphs but take so much time and could not deal with large size problem instances

More recently, Gruber and Raidl suggested a branch and cut algorithm based on compact

0-1 integer linear programming [19] It is further strengthened by dynamically adding vi-olated connection and cycle elimination constraints within a branch-and-cut environment

They model BDMST problem into two cases: even diameter and odd diameter and solve

it seperately They experiment on the graph with maximum | |V = 40 and | |E = 200

However, being deterministic and exhaustive in nature, exact approaches could only be

used to solve small problem instances (e.g complete graphs with less than 100 nodes)

2.3.2 Heuristic Methods

Since exact algorithms are not able to solve the instances with thousands of nodes,heuristics have been developed We brieﬂy summerize some construction heuristic algo-

rithms which can solve for the instances up to thousands of nodes

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2.3.2.1 One Time Tree Construction Algorithm

diameter of the tree The algorithm time for appending each new edge, in the worst case,

is O n( 2) This step is repeated n −1 times, so the algorithm time is O n( 3) The quality

of the tree indentiﬁed by the algorithm depends heavily on the start vertex To identify

Figure 2.3 shows a smallest BDST found by OT T C, of diameter D = 5 on n = 100

Figure 2.3: The best BDST found byOTT C algorithm on the Euclidean problem instance

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2.3.2.2 Center-Based Tree Construction Algorithm

to bound each vertex eccentricity It suﬃces to bound each vertex’s depth by the number

of edges on the path from the tree’s center to the vertex No vertex can be more than

D

2 edges from the center, and the depth thus the eligibility of a vertex is ﬁxed when it

joins the tree Updating this algorithm data structures requires only linear time in the

worst case (constant time when a new vertex depth is D2 ), so the time complexity of

the algorithm is O n( 2) and O n( 3) if starting at each vertex

Julstrom also modiﬁed CBTC algorithm by choosing the starting vertex and all subse-

quent vertices at random from those not yet in the spanning tree The connection of

each new vertex v to the tree remains greedy It always uses the lowest-weight edge that

connects v to a vertex in the tree whose depth is less than D

2 The modiﬁed algorithmcalled Randomized center-based Tree Construction (RT C) The time complexity of RT C,

like that of CBT C, is O n( 2) Running the randomized heuristic n times and reporting

the best solution is thus O n( 3)

2.3.2.3 Randomized Greedy Heuristic Algorithm

Raidl and Julstrom proposed in [40] a modiﬁed version of OT T C, called Ran-

domized Greedy Heuristics (RGH) RGH starts from a centre by randomly selecting a

vertex and keeping it as the ﬁxed center during the search It then repeatedly extendsthe spanning tree from the center by adding a randomly chosen vertex from the remaining

vertices, and connecting it to a vertex that is already in the tree via an edge with thesmallest weight

The algorithm also diﬀer from OT T C in that it begin by ﬁxing the center of the tree.The starting vertex v0 is chosen randomly If D is even, v0 is the center If D is odd,

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another vertex v1 is chosen at random and v 0 ,v1 are the centers; the edge joining them

is the ﬁrst in the tree Instead of maintaining the eccentricity of vertex and path lengths

between vertices, the randomized heuristic stores the depth of each connected vertex: the

number of edges on the path from it to the center This value is set when a vertex joinsthe tree and does not subsequently change No vertex may have a depth greater than D2 ;

otherwise the diameter constraint is violated or v0(v1) is displaced from the center

Sketch of RGH algorithm can be presented in the algorithm 1

Identifying the vertex u ∈ C that is nearest to v requires time O | |(C) = O n( ) This

O n( 2), which have a factor of less than that of n OT T C Running the randomized greedy

heuristic n times and taking the best solution, as withOT T C, require O n( 3) time

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2.3.2.4 Improved Greedy Heurisitics (RGH−I and CBT C I− )

Singh and Gupta [46] extended greedy constructive heuristic with a local searchstep that reevaluate previous vertex connections after appending each new vertex

They check for each vertex v if it can be connected to a better parent vertex other than

the one to which it is currently connected without violating the diameter constraint The

vertex, which oﬀers the maximum reduction in the cost of BDST is selected and whole

subtree rooted at vertex is deleted from its current location and reconnected to the treevvia the vertex selected

This improvement is applicable to CBT C also and the obtained algorithm will be denoted

byCBT C I−

Figure 2.4 and 2.5 show the best BDST found by CBT C algorithm on the Euclidean

Figure 2.4: The bestBDST found by CBTC

algorithm on the Euclidean problem instance

instance with the number of vertices is 100 and D = 10 respectively The tree on the

ﬁgure 2.5 found by apply the local search on the best tree found by CBT C algorithm(ﬁgure 2.4) and can be seen on the circle mark

Figure 2.6 and 2.7 show the best BD ST found by RGH on the Euclidean problem

instance with n = 100, D = 10 respectively The tree on the ﬁgure 2.7 found by applythe local search on the best tree found by RGH algorithm (ﬁgure 2.6) and can be seen

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Figure 2.6: The best BDST found byRGH

algorithm on the Euclidean problem instance

on the circle mark Singh and Gupta [46] experiment on the Euclidean instances with the

number of vertices are 50, 100, 250, 500 and 1000 diameter bound is set to 5, 10, 15, 20,

25 respectively

2.3.2.5 Hierarchical clustering heuristic algorithm - HCH

In [21], Gruber and Raidl propose a constructive heuristic that exploits a

chical clustering to guide the process of building a backbone The clustering heuristic

constructs diameter constrained trees within three steps: determining a hierarchical clus-tering, reducing the height of this clustering according to the diameter bound, and ﬁnally

deriving a BDMST from this height-restricted clustering

They experiment on the Euclidean instances from Beasley’s OR-Library [7] | |V = 1000

and 15 ﬁrst instances are used On large Euclidean instances the BDMST s obtained

by the HCH outperforms other construction heuristics signiﬁcantly, especially when the

diameter bound is tight and it takes only few seconds but it can not apply to the Euclidean instances

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which the number of vertices are 50, 100, 250, 500, 1000 and the diameter bound is set to

5, 10, 15, 20, 25 respectively The experimental results show that:

OT T C CBT C, and CBT C − On almost instances,I OT T C gives the best results onthe min, mean value

In [28], Julstrom experimsent on 240 graphs, 120 Euclidean and an equal number withedge weights chosen at random The Euclidean graphs consisted of points randomly placed

15, 20 for | |V = 250, 10, 15, 20, 30 for | |V = 500, 10, 20, 30, 50 for | |V = 1000

The experimental results on [28, 46, 40] show that:

On the Euclidean instances, the best and average results found by RGH −I are better

than RGH OTT C CBT C, , andCBT C I− When Dis small, CBT C identiﬁesBDST s

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that are slightly shorter than those OT T C ﬁnds, but RT C trees are much shorter than

those ofOT T C andCBT C WhenOT T C CBT C, are applied to problem instances whose

vertices are points in Euclidean space and whose edge weights are the distances between

the points, the weight of BDMST found by the heuristic are much larger than minimum,

In EA, representation methods are important role and decide all the operator in thealgorithm

Representation methods: There are a lot of methods for representing individuals, especiallyspanning tree: Characteristic vectors, Predecessor coding, Prufer number, Link and node,Edge-set-encoding, Permutation code In this thesis, we will use Edge-set-encoding andPermutation code

BDMST

Tiêu đề	Genetic Algorithms For Solving Bounded Diameter Minimum Spanning Tree Problem
Tác giả	Huynh Thi Thanh Binh
Người hướng dẫn	Associate Professor Nguyen Duc Nghia
Trường học	Hanoi University of Science and Technology
Thể loại	dissertation
Năm xuất bản	2011
Thành phố	Hanoi

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