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Fundamentals of Algorithms
CS502-Spring2015
ASSIGNMENT 4
Deadline
Your assignment must be uploaded/submitted at or before 7th August 2015.
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Submit your solution as Microsoft word file in .doc/.docx format.
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Objectives:
The research paper given will help you to better understand the shortest path algorithms. And also you will get to know how to write summaries in your own words which will help you in your research/thesis writing in higher degrees.
Question (15 marks):
The research paper “A Review and Evaluations of Shortest Path Algorithms” has been attached. You are required to read and understand the paper and write comprehensive summary of any three of the discussed shortest path algorithms in your own words.
Note:
PDF FILE MUST READ
A Review And Evaluations Of Shortest Path
Algorithms
Kairanbay Magzhan, Hajar Mat Jani
Abstract: Nowadays, in computer networks, the routing is based on the shortest path problem. This will help in minimizing the overall costs of setting up
computer networks. New technologies such as map-related systems are also applying the shortest path problem. This paper’s main objective is to
evaluate the Dijkstra’s Algorithm, Floyd-Warshall Algorithm, Bellman-Ford Algorithm, and Genetic Algorithm (GA) in solving the shortest path problem. A
short review is performed on the various types of shortest path algorithms. Further explanations and implementations of the algorithms are illustrated in
graphical forms to show how each of the algorithms works. A framework of the GA for finding optimal solutions to the shortest path problem is
presented. The results of evaluating the Dijkstra’s, Floyd-Warshall and Bellman-Ford algorithms along with their time complexity conclude the paper.
Index Terms: Bellman-Ford Algorithm, Computer Networks, Dijkstra’s Algorithm, Floyd-Warshall Algorithm, Genetic Algorithm (GA), Shortest Path
Problem.
————————————————————
1 INTRODUCTION
THE shortest path problem is a problem of finding the shortest
path or route from a starting point to a final destination.
Generally, in order to represent the shortest path problem we
use graphs. A graph is a mathematical abstract object, which
contains sets of vertices and edges. Edges connect pairs of
vertices. Along the edges of a graph it is possible to walk by
moving from one vertex to other vertices. Depending on
whether or not one can walk along the edges by both sides or
by only one side determines if the graph is a directed graph or
an undirected graph. In addition, lengths of edges are often
called weights, and the weights are normally used for
calculating the shortest path from one point to another point. In
the real world it is possible to apply the graph theory to
different types of scenarios. For example, in order to
represent a map we can use a graph, where vertices
represent cities and edges represent routes that connect the
cities. If routes are one-way then the graph will be directed;
otherwise, it will be undirected. There exist different types of
algorithms that solve the shortest path problem. However,
only several of the most popular conventional shortest path
algorithms along with one that uses genetic algorithm are
going to be discussed in this paper, and they are as follows:
1. Dijkstra’s Algorithm
2. Floyd-Warshall Algorithm
3. Bellman-Ford Algorithm
4. Genetic Algorithm (GA)
Other than GA, nowadays, there are also many intelligent
shortest path algorithms that have been introduced in several
past research papers. For example, the authors in [1] used a
heuristic method for computing the shortest path from one
point to another point within traffic networks.
They proposed a ―new dynamic direction restricted algorithm
obtained by extending the Dijkstra’s algorithm [1].‖ In another
paper [2], a heuristic GA was used for solving the single
source shortest path (SSSP) problem. Its main goal was to
investigate the SSSP problem within the Internet routing
setting, particularly when considering the cost of transmitting
messages/packets is significantly high, and the search space
is normally very large. In a paper by Li, Qi, and Ruan [3], an
efficient algorithm named Li-Qi (LQ) was proposed for the
SSSP problem with the objective of finding a simple path of
the smallest total weights from a specific initial or source
vertex to every other vertex within the graph. The ideas of the
queue and the relaxation form the basis of this newly
introduced algorithm; the vertices may be queued several
times, and furthermore, only the source vertex and relaxed
vertices are being queued [3].
2 RESEARCH OBJECTIVES
The following list gives the objectives of this research paper:
To determine and identify the concepts of the shortest
path problem.
To determine the representation of graphs in computer in
order to solve the shortest path problem, as well as to
understand the different basic terms of a graph.
To explain the general concepts and the implementations
of Dijkstra’s Algorithm, Floyd-Warshall Algorithm,
Bellman-Ford Algorithm, and Genetic Algorithm.
To evaluate each algorithm, and presents the
evaluations’ results.
3 LITERATURE REVIEW
As mentioned earlier, a graph can be used to represent a map
where the cities are represented by vertices and the routes or
roads are represented by edges within the graph. In this
section, a graph representation of a map is explained further,
and brief descriptions and implementations of the four shortest
path algorithms being studied are presented.
3.1 Representation of the Graph
In order to represent a graph in a computer we will use
adjacency matrix a. The dimension of the matrix will be equal
to (n x n), where n is number of vertices in graph. The
element of matrix a[i][j] is identified by an edge that connects
the i-th and j-th vertices; the value here represents the weight
of the corresponding edge. However, if there is no edge
————————————————
Kairanbay Magzhan is currently pursuing bachelor’s
degree program in Computer Science and Software
Engineering at International IT University, Kazakhstan.
E-mail: magzhan.kairanbay@gmail.com
Dr. Hajar Mat Jani is currently a Senior Lecturer at
Universiti Tenaga Nasional, Malaysia.
E-mails: hajar@uniten.edu.my, hajarmj@gmail.com.
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between vertices i and j, the value in (a[i][j]) will be equal to
infinity. An array of edges is another common representation of
the graph. If m is the number of edges in a graph, then in
order to represent the graph we have to use m x 3 two-dimensional arrays; in each row, the first vertex, the second
vertex, and the edge that connects them are also stored. The
benefit of using an array of edges in comparison to adjacency
matrix is when there is more than one edge that connects two
vertices we cannot use adjacency matrix in order to represent
graph.
3.2 Dijkstra’s Algorithm: Explanation and
Implementation
For each vertex within a graph we assign a label that determines
the minimal length from the starting point s to other vertices v of
the graph. In a computer we can do it by declaring an array d[].
The algorithm works sequentially, and in each step it tries to
decrease the value of the label of the vertices. The algorithm
stops when all vertices have been visited. The label at the starting
point s is equal to zero (d[s]=0); however , labels in other vertices
v are equal to infinity (d[v]=∞), which means that the length from
the starting point s to other vertices is unknown. In a computer
we can just use a very big number in order to represent infinity . In
addition, for each vertex v we have to identify whether it has been
visited or not. In order to do that, we declare an array of Boolean
type called u[v], where initially , all vertices are assigned as
unvisited (u[v] = false ). The Dijkstra’s algorithm consists of n
iterations. If all vertices have been visited, then the algorithm
finishes; otherwise, from the list of unvisited vertices we have to
choose the vertex which has the minimum (smallest) value at its
label (At the beginning, we will choose a starting point s). After
that, we will consider all neighbors of this vertex (Neighbors of a
vertex are those vertices that have common edges with the initial
vertex). For each unvisited neighbor we will consider a new
length , which is equal to the sum of the label’s value at the initial
vertex v (d[v]) and the length of edge l that connects them. If the
resulting value is less than the value at the label, then we have to
change the value in that label with the newly obtained value [4].
d [ neighbors ] = min ( d [ neighbors ] , d[ v ] + l ) (1)
After considering all of the neighbors, we will assign the initial
vertex as visited (u[v] = true). After repeating this step n times,
all vertices of the graph will be visited and the algorithm
finishes or terminates. The vertices that are not connected with
the starting point will remain by being assigned to infinity. In
order to restore the shortest path from the starting point to
other vertices, we need to identify array p [], where for each
vertex, where v ≠ s, we will store the number of vertex p[v],
which penultimate vertices in the shortest path. In other
words, a complete path from s to v is equal to the following
statement [5]
P = ( s , … , p [ p [ p [ v ] ] ] , p [ p [ v ] ] , p [ v ] , v ) (2)
Fig. 1 shows an excerpt of the Dijkstra’s algorithm, which is
written in Java.
3.3 Floyd-Warshall Algorithm: Explanation and
Implementation
Consider the graph G, where vertices were numbered from 1
to n. The notation d
ijk
means the shortest path from i to j, which
also passes through vertex k. Obviously if there is exists edge
between vertices i and j it will be equal to d
ij0
, otherwise it can
assigned as infinity. However , for other values of d
ijk
there can
be two choices: (1) If the shortest path from i to j does not
pass through the vertex k then value of d
ijk
will be equal to d
ijk-1. (2) If the shortest path from i to j passes through the vertex k
then first it goes from i to k, after that goes from k to j. In this
case the value of d
ijk
will be equal to d
ikk-1
+ d
kjk-1. And in order
to determine the shortest path we just need to find the
minimum of these two statements [6]:
d
ij0
= the length of edge between vertices i and j (3)
d
ijk
= min (d
ijk-1
, dikk-1 + d
kjk-1) (4)
Fig. 2 shows an excerpt of the Floyd-Warshall algorithm, which
is written in Java.
3.4 Bellman-Ford Algorithm: Explanation and
Implementation
In comparison to Dijkstra’s algorithm, the Bellman-Ford
algorithm admits or acknowledges the edges with negative
weights. That is why, a graph can contain cycles of negative
weights, which will generate numerous number of paths from
Fig. 1. Implementation in Java: An Excerpt
Fig. 2. Implementation in Java: An Excerpt
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the starting point to the final destination, where each cycle will
minimize the length of the shortest path. Taking into
consideration this fact let’s assume that our graph does not
contain cycles with negative weights. The array d[] will store
the minimal length from the starting point s to other vertices.
The algorithm consists of several phases, where in each
phase it needs to minimize the value of all edges by replacing
d[b] to following statement d[a] + c; a and b are vertices of the
graph, and c is the corresponding edge that connects them.
And in order to calculate the length of all shortest paths in a
graph it requires n – 1 phases, but for those vertices of a
graph that are unreachable, the value of elements of the array
will remain by being assigned to infinity [7]. Fig. 3 shows an
excerpt of the Bellman-Ford algorithm, which is written in Java.
3.5 Genetic Algorithm (GA)
Intelligent algorithms have been introduced in finding optimal
shortest paths in many situations that require the systems to
search through a very large search space within limited time
frame and also in accommodating an ever-changing
environment. One of these algorithms is GA. By definition,
genetic algorithms are a class or group of ―stochastic search
algorithms‖ that are based on biological evolution [8]. GA is
mostly used for optimization problems. It uses several genetic
operations such as selection, crossover, and mutation in order
to generate a new generation of population, which represents
a set of solutions (chromosomes) to the current problem. In
addition, on average, this new generation is supposed to be
better in terms of their overall fitness value as compared to the
previous population. Each individual or chromosome within
the population will be assigned a fitness value, which is
calculated based on a pre-determined fitness function that
measures how optimal its solution is in solving the current
problem. In order to solve the shortest path problem using the
GA [9], we need to generate a number of solutions, and then
choose the most optimal one among the provided set of
possible solutions. In order to solve the problem, an initial
population that forms the first set of chromosomes to be used
in the GA is randomly created. Each chromosome represents
one possible solution to the current problem at hand. After
that, they (chromosomes) are estimated using certain fitness
function, which determines how well the solutions are. T aking
into account the fitness value of each solution or chromosome,
some chromosomes or individuals will be selected (selection
operation), and the basic genetic operations such as
crossover and mutation are applied on these chromosomes.
Then, the fitness value of each chromosome is re-calculated,
and the best solutions are selected to be considered for the
next generation. This process continues until the criteria of the
given problem will not be achieved. Thus we can identify the
following stages of a GA:
Step 1: Determine the fitness function; in our case we
need to maximize the following function f(Chk) = (∑edge)-1, where Chk is k-th chromosome and ∑edge is the sum
of edges from starting point to final destination.
Step 2: Create initial population – a population that
contains n individuals. At this stage we do not need to
create fittest individuals, because it is probable that GA
will transfer them into viable population. In order to create
chromosomes for initial population, we will produce
random paths from the starting point to final destination.
Step 3: Selection – the stage of GA that is used to select
two chromosomes for genetic operations such as
crossover and mutation. There are different types of
selection methods; however, the Roulette Wheel
selection method is chosen in order to solve the shortest
path problem.
Step 4: Crossover – the process of reproduction where
descendants are inherit traits of both parents mixing
them in some way. Individuals for reproduction will be
chosen from whole population (not from the survivors in
the first iteration), because we need to keep diversity of
individuals, otherwise entire population will be hammered
with single copies of one individual. There exist different
types of crossover methods; however, for our problem
we will use the simplest method, which is called single
point crossover.
Step 5: Mutation – the act of changing the value of some
gene. Mutation keeps the genetic diversity of the
population by changing genes of selected chromosome.
If the fittest chromosome does not change after a specific
number of iterations, which was described above, then the
algorithm will terminate; the most optimal solution is
automatically the fittest chromosome among the whole
population. Fig. 4 illustrates the flowchart of the shortest path
problem’s solution using GA. This diagram is a framework that
will be used for the implementation of GA in finding the
shortest path or route in a given map of a city named Almaty in
Kazakhstan, which is part of our current and future works. The
GA used here is quite general in nature except for the part
where loops might be introduced while executing GA in finding
the most optimal solution; all loops must be removed because
loops must not exist in a path.
Fig. 3. Implementation in Java: An Excerpt
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4 RESULTS
4.1 Test Results
Dijkstra’s, Floyd-Warshall and Bellman-Ford algorithms were
tested using pre-defined test cases and automated checking
system available in the websites [10][11][12]. In the following
figures (Fig. 5, Fig. 6, Fig. 7) some information such as the
number of test cases, results, author , date of submission, and
programming language used for each algorithm are provided.
Fig. 4. A Framework: Genetic Algorithm for the Shortest Path Problem
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4.2 The Time Complexity
The time complexity for each algorithm is illustrated in Table I;
n represents the total number of vertices, and m is the total
number of edges.
TABLE 1
TIME COMPLEXITY
5 CONCLUSION AND FUTURE WORK
The computed time complexity for each of the Dijkstra’s,
Floyd-Warshall and Bellman-Ford algorithms show that these
algorithms are acceptable in terms of their overall performance
in solving the shortest path problem. All of these algorithms
produce only one solution. However, the main advantage of
GA over these algorithms is that it may produce a number of
different optimal solutions since the result can differ every time
the GA is executed. In the future, the proposed GA framework
will be extended and improved in finding the shortest path or
distance between two places in a map that represents any
types of networks. In addition, other artificial intelligence
techniques such as fuzzy logic and neural networks can also
be implemented in improving existing shortest path algorithms
in order to make them more intelligent and more efficient.
ACKNOWLEDGMENTS
This research study was partially funded by the Fundamental
Research Grant Scheme (FRGS), Ministry of Higher Education
(MOHE), Malaysia. This paper would not have been possible
without the assistance, support and patience of my supervisor,
Dr. Hajar Mat Jani. I would like to thank Dr. Hajar for her
invaluable advice and unsurpassed knowledge. Finally, I would
like to thank my parents for giving birth to me and supporting
me throughout my life.
REFERENCES
[1] C. Xi, F . Qi, and L. Wei, ―A New Shortest Path
Algorithm based on Heuristic Strategy,‖ Proc. of the
6th World Congress on Intelligent Control and
Automation, Vol. 1, pp. 2531–2536, 2006.
[2] B.S. Hasan, M.A. Khamees, and A.S.H. Mahmoud,
―A Heuristic Genetic Algorithm for the Single Source
Shortest Path Problem, ‖ Proc. of International
Conference on Computer Systems and Applications,
Fig. 5. Results of T esting for Dijkstra’s Algorithm
Fig. 6. Results of T esting for Floyd-Warshall Algorithm
Fig. 7. Results of T esting for Bellman-Ford Algorithm
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pp. 187-194, 2007.
[3] T . Li, L. Qi, and D. Ruan, ―An Efficient Algorithm for
the Single-Source Shortest Path Problem in Graph
Theory‖, Proc. of 3rd International Conference on
Intelligent System and Knowledge Engineering, Vol.
1, pp. 152-157, 2008.
[4] M. Jordan, ―Notes 7 for CS170‖, UC Berkeley, 2005.
[5] J. Chamero, ―Dijkstra’s Algorithm‖ Discrete
Structures & Algorithms, 2006.
[6] S. Skiena, A. Revilla, ―Programming Challenges, The
Programming Contest Training Manual‖ pp. 248 –
250.
[7] S. Hougardy, The Floyd-Warshall, ―Algorithm on
Graphs with Negative Cycles‖, University of Bonn,
2010.
[8] M. Negnevitsky , Artificial Intelligence: A Guide to
Intelligent Systems, Third Edition, Addison-Wesley,
2011.
[9] I. Rakip, U. Atila, ―A Genetic Algorithm Approach for
Finding the Shortest Driving Time on Mobile
Devices‖, Scientific Research and Essays, Dept. of
Computer Engineering, 2011.
[10] Dijkstra’s Algorithm, Available at
http://informatics.mccme.ru/moodle/mod/statements/
view.php?id=193#1. 2012.
[11] Floyd-Warshall Algorithm, Available at
http://informatics.mccme.ru/moodle/mod/statements/
view.php?id=218#1. 2012.
[12] Bellman-Ford Algorithm, Available at
http://informatics.mccme.ru/moodle/mod/statements/
view.php?id=260#1. 2012.
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