The graph colouring problem can be solved using methods based on Satisfiability (SAT). An instance of the problem is defined by a Boolean expression written using Boolean variables and the logical connectives AND, OR and NOT. It has to be determined whether there is an assignment of TRUE and FALSE values to the variables that makes the entire expression true.A SAT problem is syntactically and semantically quite simple. Many Constraint Satisfaction Problems (CSPs)in AI and OR can be formulated in SAT. These make use of two kinds of
searchalgorithms: Deterministic and Randomized.It has been found that deterministic methods when run on hard CSP instances are frequently very slow in execution.A deterministic method always outputs a solution in the end, but it can take an enormous amount of time to do so.This has led to the development of randomized search algorithms like GSAT, which are typically based on local (i.e., neighbourhood) search. Such methodshave been applied very successfully to find good solutions to hard decision problems
Cs6702 graph theory and applications question bankappasami
This document contains questions and answers related to the course CS6702 Graph Theory and Applications. It is divided into 6 units which cover topics like introduction to graphs, trees, connectivity, planarity, graph colorings, directed graphs, permutations and combinations, and generating functions. For each unit, it lists short answer questions in Part A and longer proof or explanation questions in Part B. The questions test concepts fundamental to graph theory such as definitions of graphs, trees, connectivity, colorings, isomorphism, and counting methods applied to graph theory.
Cs6702 graph theory and applications Anna University question paper apr may 2...appasami
This document contains a past exam paper for the subject "Graph Theory and Applications". It includes 15 multiple choice and long answer questions covering topics such as Euler graphs, planar graphs, spanning trees, matchings, counting principles, and graph algorithms. The key provided contains concise definitions and explanations for the concepts and steps to solve problems in 3 sentences or less.
Cs6402 design and analysis of algorithms impartant part b questions appasamiappasami
This document outlines important questions from 5 units that could appear on the CS64O2 DESIGN AND ANALYSIS OF ALGORITHMS exam. Unit I covers algorithm problem types, problem solving fundamentals, asymptotic notations, and analyzing recursive/non-recursive algorithms. Unit II discusses closest pair, convex hull, TSP, KP, sorting algorithms, and binary search. Unit III is about dynamic programming, graph algorithms, OBST, and HMT. Unit IV is linear programming problems. Unit V covers complexity classes, backtracking, branch and bound, and approximation algorithms for NP-hard problems like TSP and KP. The document provides guidance on questions to study for the exam.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
This document contains a past exam paper for the subject "Design and Analysis of Algorithms". It has 2 parts with a total of 15 questions. Part A covers basic algorithm concepts like recurrence relations, efficiency classes, minimum spanning trees, and more. Part B involves solving algorithm problems using techniques like dynamic programming, Huffman coding, shortest paths, and more. It also tests concepts like P vs NP, approximation algorithms, and analysis of algorithm efficiency.
A linear algorithm for the grundy number of a treeijcsit
A coloring of a graph G = (V ,E) is a partition {V1, V2, . . . , Vk} of V into independent sets or color classes.
A vertex v
Vi is a Grundy vertex if it is adjacent to at least one vertex in each color class Vj for every j <i.
A coloring is a Grundy coloring if every color class contains at least one Grundy vertex, and the Grundy
number of a graph is the maximum number of colors in a Grundy coloring. We derive a natural upper
bound on this parameter and show that graphs with sufficiently large girth achieve equality in the bound.
In particular, this gives a linear-time algorithm to determine the Grundy number of a tree.
A New Hendecagonal Fuzzy Number For Optimization Problemsijtsrd
A new fuzzy number called Hendecagonal fuzzy number and its membership function is introduced, which is used to represent the uncertainty with eleven points. The fuzzy numbers with ten ordinates exists in literature. The aim of this paper is to define Hendecagonal fuzzy number and its arithmetic operations. Also a direct approach is proposed to solve fuzzy assignment problem (FAP) and fuzzy travelling salesman (FTSP) in which the cost and distance are represented by Hendecagonal fuzzy numbers. Numerical example shows the effectiveness of the proposed method and the Hendecagonal fuzzy number M. Revathi | Dr. M. Valliathal | R. Saravanan | Dr. K. Rathi"A New Hendecagonal Fuzzy Number For Optimization Problems" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-1 | Issue-5 , August 2017, URL: http://www.ijtsrd.com/papers/ijtsrd2258.pdf http://www.ijtsrd.com/mathemetics/applied-mathamatics/2258/a-new-hendecagonal-fuzzy-number-for-optimization-problems/m-revathi
The document proposes a new "bound and decomposition method" to find an optimal fuzzy solution for fully fuzzy linear programming problems. The method decomposes the given fully fuzzy linear programming problem into three crisp linear programming problems: a middle level problem, upper level problem, and lower level problem. The three crisp linear programming problems are then solved separately. Using the optimal solutions from these problems, an optimal fuzzy solution for the original fully fuzzy linear programming problem is obtained. The proposed method does not require fuzzy ranking functions or restrictions on the coefficients matrix elements.
Cs6702 graph theory and applications question bankappasami
This document contains questions and answers related to the course CS6702 Graph Theory and Applications. It is divided into 6 units which cover topics like introduction to graphs, trees, connectivity, planarity, graph colorings, directed graphs, permutations and combinations, and generating functions. For each unit, it lists short answer questions in Part A and longer proof or explanation questions in Part B. The questions test concepts fundamental to graph theory such as definitions of graphs, trees, connectivity, colorings, isomorphism, and counting methods applied to graph theory.
Cs6702 graph theory and applications Anna University question paper apr may 2...appasami
This document contains a past exam paper for the subject "Graph Theory and Applications". It includes 15 multiple choice and long answer questions covering topics such as Euler graphs, planar graphs, spanning trees, matchings, counting principles, and graph algorithms. The key provided contains concise definitions and explanations for the concepts and steps to solve problems in 3 sentences or less.
Cs6402 design and analysis of algorithms impartant part b questions appasamiappasami
This document outlines important questions from 5 units that could appear on the CS64O2 DESIGN AND ANALYSIS OF ALGORITHMS exam. Unit I covers algorithm problem types, problem solving fundamentals, asymptotic notations, and analyzing recursive/non-recursive algorithms. Unit II discusses closest pair, convex hull, TSP, KP, sorting algorithms, and binary search. Unit III is about dynamic programming, graph algorithms, OBST, and HMT. Unit IV is linear programming problems. Unit V covers complexity classes, backtracking, branch and bound, and approximation algorithms for NP-hard problems like TSP and KP. The document provides guidance on questions to study for the exam.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
This document contains a past exam paper for the subject "Design and Analysis of Algorithms". It has 2 parts with a total of 15 questions. Part A covers basic algorithm concepts like recurrence relations, efficiency classes, minimum spanning trees, and more. Part B involves solving algorithm problems using techniques like dynamic programming, Huffman coding, shortest paths, and more. It also tests concepts like P vs NP, approximation algorithms, and analysis of algorithm efficiency.
A linear algorithm for the grundy number of a treeijcsit
A coloring of a graph G = (V ,E) is a partition {V1, V2, . . . , Vk} of V into independent sets or color classes.
A vertex v
Vi is a Grundy vertex if it is adjacent to at least one vertex in each color class Vj for every j <i.
A coloring is a Grundy coloring if every color class contains at least one Grundy vertex, and the Grundy
number of a graph is the maximum number of colors in a Grundy coloring. We derive a natural upper
bound on this parameter and show that graphs with sufficiently large girth achieve equality in the bound.
In particular, this gives a linear-time algorithm to determine the Grundy number of a tree.
A New Hendecagonal Fuzzy Number For Optimization Problemsijtsrd
A new fuzzy number called Hendecagonal fuzzy number and its membership function is introduced, which is used to represent the uncertainty with eleven points. The fuzzy numbers with ten ordinates exists in literature. The aim of this paper is to define Hendecagonal fuzzy number and its arithmetic operations. Also a direct approach is proposed to solve fuzzy assignment problem (FAP) and fuzzy travelling salesman (FTSP) in which the cost and distance are represented by Hendecagonal fuzzy numbers. Numerical example shows the effectiveness of the proposed method and the Hendecagonal fuzzy number M. Revathi | Dr. M. Valliathal | R. Saravanan | Dr. K. Rathi"A New Hendecagonal Fuzzy Number For Optimization Problems" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-1 | Issue-5 , August 2017, URL: http://www.ijtsrd.com/papers/ijtsrd2258.pdf http://www.ijtsrd.com/mathemetics/applied-mathamatics/2258/a-new-hendecagonal-fuzzy-number-for-optimization-problems/m-revathi
The document proposes a new "bound and decomposition method" to find an optimal fuzzy solution for fully fuzzy linear programming problems. The method decomposes the given fully fuzzy linear programming problem into three crisp linear programming problems: a middle level problem, upper level problem, and lower level problem. The three crisp linear programming problems are then solved separately. Using the optimal solutions from these problems, an optimal fuzzy solution for the original fully fuzzy linear programming problem is obtained. The proposed method does not require fuzzy ranking functions or restrictions on the coefficients matrix elements.
This document contains 71 multiple choice questions about regression analysis from the textbook "Quantitative Analysis for Management, 11e". The questions cover topics such as simple and multiple linear regression, assumptions of regression models, measuring model fit, and testing models for significance. Correct answers are provided along with a difficulty rating and topic for each question.
This document contains 26 multiple choice questions and their answers related to graph theory. It begins by defining key graph theory terms like graphs, vertices, edges, simple graphs, and applications of graph theory. It then discusses incidence, adjacency, degrees, finite and infinite graphs, isolated and pendant vertices, null graphs, and multigraphs. The document also defines complete graphs, regular graphs, cycles, isomorphism, subgraphs, walks, paths, circuits, connectivity, components, Euler graphs, Hamiltonian circuits/paths, trees, properties of trees, distance in trees, eccentricity, center, distance metric, radius, diameter, rooted trees, and binary rooted trees.
The document presents a method for solving fuzzy assignment problems using triangular and trapezoidal fuzzy numbers. It formulates the fuzzy assignment problem into a crisp linear programming problem that can be solved using the Hungarian method. The paper also uses Robust's ranking method to transform fuzzy costs into crisp values, allowing conventional solution methods to be applied. It aims to provide a more realistic approach to assignment problems by considering costs as fuzzy numbers rather than deterministic values.
International Refereed Journal of Engineering and Science (IRJES)irjes
International Refereed Journal of Engineering and Science (IRJES) is a leading international journal for publication of new ideas, the state of the art research results and fundamental advances in all aspects of Engineering and Science. IRJES is a open access, peer reviewed international journal with a primary objective to provide the academic community and industry for the submission of half of original research and applications
Graph theory is a branch of mathematics that studies networks and their structures. It began in 1736 when Euler solved the Konigsberg bridges problem. Graphs represent objects called nodes connected by relationships called edges. They are used to model many real-world applications. Similarity searching uses graph representations of molecules to identify structurally or pharmacologically similar compounds. Various molecular descriptors and similarity coefficients are used with substructure, reduced graph, and 3D methods. Data fusion combines results from multiple search methods to improve accuracy. Graph theory continues to be important for modeling complex systems and networks in science.
This document contains exercises, hints, and solutions for Chapter 1 of the book "Introduction to the Design and Analysis of Algorithms." It includes 11 exercises related to algorithms for computing greatest common divisors, square roots, binary representations, and other topics. The document also provides hints for each exercise to help students solve them and includes the solutions.
SMU BSC IT FALL / SUMMER 2013 SOLVED ASSIGNMENTSsmumbahelp
This document provides sample questions for various subjects for different semesters. It begins by providing contact information for getting fully solved assignments. It then lists sample questions in 3 sentences or less for subjects like Computer Organization and Architecture, Discrete Mathematics, Operating Systems, Technical Communication, Computer Networks, Mobile Computing, WML and WAP Programming, Cryptography and Network Security, C# Programming, Data Mining, Software Project Management, Computer Graphics, Artificial Intelligence, Fundamentals of IT, Mathematics for IT, Logic Design, C Programming and Data Structures, Database Management Systems, Fundamentals of Algorithms, Software Engineering, and Visual Basic. Students can check answers by contacting provided details or checking blog archives.
Different Solutions to a Mathematical Problem: A Case Study of Calculus 12theijes
An important duty of the mathematics teacher is to train and to develop thinking for students. To accomplish this duty, teachers can organize creative activities for students through activities of solving problems. In particular, there is an effective way to train students to think is that teachers can organize activities of solving problems in many different ways. Based on this idea, we implement an experiment for students in grade 12 to calculate integrals in various ways. The results of the study showed that students were active to find out different solutions to the given problem.
The document provides a sample paper for the Common Admission Test (CAT) with 20 multiple-choice questions covering quantitative ability, logical reasoning, and data interpretation. It also provides solutions and explanations for the questions. The sample paper is intended to help CAT exam preparation by providing practice questions.
IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
Role Model of Graph Coloring Application in Labeled 2D Line Drawing ObjectWaqas Tariq
Several researches had worked on the development of sketch interpreters. However, very few of them gave a complete cycle of the sketch interpreter which can be used to transform an engineering sketch to a valid solid object. In this paper, a framework of the complete cycle of the sketch interpreter is presented. The discussion in this paper will stress on the usage of line labeling and graph coloring application in the validation of two dimensional (2D) line drawing phase. Both applications are needed to determine whether the given 2D line drawing represent possible or impossible object. In 2008, previous work by Matondang et al., has used line labeling algorithm to validate several 2D line drawings. However, the result shows that line labeling algorithm is not sufficient, as the algorithm does not have a validation technique for the result. Therefore, in this research study, it is going to be shown that if a 2D line drawing is valid as a possible object by using the line labeling algorithm, then it can be colored using graph coloring concept with a determine-able minimum numbers of color needed. This is equal in vice versa. The expected output from this phase is a valid-labeled of 2D line drawing with different colors at each edge and ready for the reconstruction phase. As a preliminary result, a high programming language MATLAB R2009a and several primitive 2D line drawings has been used and presented in this paper to test the graph coloring concept in labeled 2D line drawing.
The document discusses using the branch and bound technique to solve optimization problems like the traveling salesman problem and water jug problem. It explains that branch and bound systematically breaks problems into smaller subsets, calculates bounds on objective functions, and discards subsets that cannot produce better solutions than what is currently known. For the water jug problem, it provides an example path and explains how branch and bound finds the optimal 6-state solution. For traveling salesman, it notes the problem is to minimize travel distance by visiting all cities once and returning home.
The document discusses the dynamic programming approach to solving the matrix chain multiplication problem. It explains that dynamic programming breaks problems down into overlapping subproblems, solves each subproblem once, and stores the solutions in a table to avoid recomputing them. It then presents the algorithm MATRIX-CHAIN-ORDER that uses dynamic programming to solve the matrix chain multiplication problem in O(n^3) time, as opposed to a brute force approach that would take exponential time.
In many decision situations, decision-makers face a kind of complex problems. In these decision-making problems, different types of fuzzy numbers are defined and, have multiple types of membership functions. So, we need a standard form to formulate uncertain numbers in the problem. Shadowed fuzzy numbers are considered granule numbers which approximate different types and different forms of fuzzy numbers. In this paper, a new ranking approach for shadowed fuzzy numbers is developed using value, ambiguity and fuzziness for shadowed fuzzy numbers. The new ranking method has been compared with other existing approaches through numerical examples. Also, the new method is applied to a hybrid multi-attribute decision making problem in which the evaluations of alternatives are expressed with different types of uncertain numbers. The comparative study for the results of different examples illustrates the reliability of the new method.
A study-of-vertex-edge-coloring-techniques-with-applicationIjcem Journal
This document discusses applications of graph coloring techniques. It begins with an overview of vertex coloring and edge coloring of graphs, including definitions of key terms like chromatic number and edge chromatic number. It then provides several examples of applications of graph coloring, including scheduling problems like job scheduling, aircraft scheduling, and timetabling problems. It describes how these problems can be modeled as graph coloring problems and solved using techniques like precoloring, list coloring, and minimum sum coloring. The document concludes with additional examples of applying graph coloring, such as map coloring for cellular networks.
Numerical Solution to Ninth order Non- Linear Differential Equation Using the...rahulmonikasharma
This document presents a numerical solution to a ninth order non-linear differential equation using a ninth degree B-spline collocation method. The method develops a recursive form of ninth degree B-spline functions as a basis for the collocation method. The method is tested on a sample ninth order non-linear boundary value problem. The numerical results generated by the method closely match the exact solution, demonstrating the effectiveness of using ninth degree B-splines in the collocation method to solve ninth order differential equations.
The document provides a table of contents and questions on various technical aptitude topics such as data structures, C, C++, quantitative aptitude, UNIX concepts, RDBMS concepts, SQL, computer networks, and operating systems. It contains 35 questions related to data structures concepts like linked lists, trees, graphs, hashing etc. and their applications.
This document proposes representing complex numbers using colors (chromatic numbers) and developing ternary algebra based on this. It defines chromatic numbers and shows how to perform calculations on them. It introduces the 3-RSS NP-complete problem and shows how to formulate it and 3-colorability using ternary algebra and ternary variables. It defines ternary resolution rules to operate on ternary-CNF formulas and proves soundness and completeness of the ternary resolution system.
An analysis between different algorithms for the graph vertex coloring problem IJECEIAES
The document analyzes and compares different algorithms for solving the graph vertex coloring problem. It presents an exact backtracking algorithm that finds optimal solutions and two heuristic algorithms (greedy coloring and Welsh-Powell) that provide approximate solutions. Experiments were conducted on 27 graphs with varying numbers of vertices and edges to analyze the performance of the algorithms. The results show that the exact algorithm can efficiently solve graphs with up to around 35 vertices, finding optimal solutions for half the test graphs. The approximate algorithms provided identical solutions in terms of number of colors used but were faster for larger graphs.
This document contains 71 multiple choice questions about regression analysis from the textbook "Quantitative Analysis for Management, 11e". The questions cover topics such as simple and multiple linear regression, assumptions of regression models, measuring model fit, and testing models for significance. Correct answers are provided along with a difficulty rating and topic for each question.
This document contains 26 multiple choice questions and their answers related to graph theory. It begins by defining key graph theory terms like graphs, vertices, edges, simple graphs, and applications of graph theory. It then discusses incidence, adjacency, degrees, finite and infinite graphs, isolated and pendant vertices, null graphs, and multigraphs. The document also defines complete graphs, regular graphs, cycles, isomorphism, subgraphs, walks, paths, circuits, connectivity, components, Euler graphs, Hamiltonian circuits/paths, trees, properties of trees, distance in trees, eccentricity, center, distance metric, radius, diameter, rooted trees, and binary rooted trees.
The document presents a method for solving fuzzy assignment problems using triangular and trapezoidal fuzzy numbers. It formulates the fuzzy assignment problem into a crisp linear programming problem that can be solved using the Hungarian method. The paper also uses Robust's ranking method to transform fuzzy costs into crisp values, allowing conventional solution methods to be applied. It aims to provide a more realistic approach to assignment problems by considering costs as fuzzy numbers rather than deterministic values.
International Refereed Journal of Engineering and Science (IRJES)irjes
International Refereed Journal of Engineering and Science (IRJES) is a leading international journal for publication of new ideas, the state of the art research results and fundamental advances in all aspects of Engineering and Science. IRJES is a open access, peer reviewed international journal with a primary objective to provide the academic community and industry for the submission of half of original research and applications
Graph theory is a branch of mathematics that studies networks and their structures. It began in 1736 when Euler solved the Konigsberg bridges problem. Graphs represent objects called nodes connected by relationships called edges. They are used to model many real-world applications. Similarity searching uses graph representations of molecules to identify structurally or pharmacologically similar compounds. Various molecular descriptors and similarity coefficients are used with substructure, reduced graph, and 3D methods. Data fusion combines results from multiple search methods to improve accuracy. Graph theory continues to be important for modeling complex systems and networks in science.
This document contains exercises, hints, and solutions for Chapter 1 of the book "Introduction to the Design and Analysis of Algorithms." It includes 11 exercises related to algorithms for computing greatest common divisors, square roots, binary representations, and other topics. The document also provides hints for each exercise to help students solve them and includes the solutions.
SMU BSC IT FALL / SUMMER 2013 SOLVED ASSIGNMENTSsmumbahelp
This document provides sample questions for various subjects for different semesters. It begins by providing contact information for getting fully solved assignments. It then lists sample questions in 3 sentences or less for subjects like Computer Organization and Architecture, Discrete Mathematics, Operating Systems, Technical Communication, Computer Networks, Mobile Computing, WML and WAP Programming, Cryptography and Network Security, C# Programming, Data Mining, Software Project Management, Computer Graphics, Artificial Intelligence, Fundamentals of IT, Mathematics for IT, Logic Design, C Programming and Data Structures, Database Management Systems, Fundamentals of Algorithms, Software Engineering, and Visual Basic. Students can check answers by contacting provided details or checking blog archives.
Different Solutions to a Mathematical Problem: A Case Study of Calculus 12theijes
An important duty of the mathematics teacher is to train and to develop thinking for students. To accomplish this duty, teachers can organize creative activities for students through activities of solving problems. In particular, there is an effective way to train students to think is that teachers can organize activities of solving problems in many different ways. Based on this idea, we implement an experiment for students in grade 12 to calculate integrals in various ways. The results of the study showed that students were active to find out different solutions to the given problem.
The document provides a sample paper for the Common Admission Test (CAT) with 20 multiple-choice questions covering quantitative ability, logical reasoning, and data interpretation. It also provides solutions and explanations for the questions. The sample paper is intended to help CAT exam preparation by providing practice questions.
IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
Role Model of Graph Coloring Application in Labeled 2D Line Drawing ObjectWaqas Tariq
Several researches had worked on the development of sketch interpreters. However, very few of them gave a complete cycle of the sketch interpreter which can be used to transform an engineering sketch to a valid solid object. In this paper, a framework of the complete cycle of the sketch interpreter is presented. The discussion in this paper will stress on the usage of line labeling and graph coloring application in the validation of two dimensional (2D) line drawing phase. Both applications are needed to determine whether the given 2D line drawing represent possible or impossible object. In 2008, previous work by Matondang et al., has used line labeling algorithm to validate several 2D line drawings. However, the result shows that line labeling algorithm is not sufficient, as the algorithm does not have a validation technique for the result. Therefore, in this research study, it is going to be shown that if a 2D line drawing is valid as a possible object by using the line labeling algorithm, then it can be colored using graph coloring concept with a determine-able minimum numbers of color needed. This is equal in vice versa. The expected output from this phase is a valid-labeled of 2D line drawing with different colors at each edge and ready for the reconstruction phase. As a preliminary result, a high programming language MATLAB R2009a and several primitive 2D line drawings has been used and presented in this paper to test the graph coloring concept in labeled 2D line drawing.
The document discusses using the branch and bound technique to solve optimization problems like the traveling salesman problem and water jug problem. It explains that branch and bound systematically breaks problems into smaller subsets, calculates bounds on objective functions, and discards subsets that cannot produce better solutions than what is currently known. For the water jug problem, it provides an example path and explains how branch and bound finds the optimal 6-state solution. For traveling salesman, it notes the problem is to minimize travel distance by visiting all cities once and returning home.
The document discusses the dynamic programming approach to solving the matrix chain multiplication problem. It explains that dynamic programming breaks problems down into overlapping subproblems, solves each subproblem once, and stores the solutions in a table to avoid recomputing them. It then presents the algorithm MATRIX-CHAIN-ORDER that uses dynamic programming to solve the matrix chain multiplication problem in O(n^3) time, as opposed to a brute force approach that would take exponential time.
In many decision situations, decision-makers face a kind of complex problems. In these decision-making problems, different types of fuzzy numbers are defined and, have multiple types of membership functions. So, we need a standard form to formulate uncertain numbers in the problem. Shadowed fuzzy numbers are considered granule numbers which approximate different types and different forms of fuzzy numbers. In this paper, a new ranking approach for shadowed fuzzy numbers is developed using value, ambiguity and fuzziness for shadowed fuzzy numbers. The new ranking method has been compared with other existing approaches through numerical examples. Also, the new method is applied to a hybrid multi-attribute decision making problem in which the evaluations of alternatives are expressed with different types of uncertain numbers. The comparative study for the results of different examples illustrates the reliability of the new method.
A study-of-vertex-edge-coloring-techniques-with-applicationIjcem Journal
This document discusses applications of graph coloring techniques. It begins with an overview of vertex coloring and edge coloring of graphs, including definitions of key terms like chromatic number and edge chromatic number. It then provides several examples of applications of graph coloring, including scheduling problems like job scheduling, aircraft scheduling, and timetabling problems. It describes how these problems can be modeled as graph coloring problems and solved using techniques like precoloring, list coloring, and minimum sum coloring. The document concludes with additional examples of applying graph coloring, such as map coloring for cellular networks.
Numerical Solution to Ninth order Non- Linear Differential Equation Using the...rahulmonikasharma
This document presents a numerical solution to a ninth order non-linear differential equation using a ninth degree B-spline collocation method. The method develops a recursive form of ninth degree B-spline functions as a basis for the collocation method. The method is tested on a sample ninth order non-linear boundary value problem. The numerical results generated by the method closely match the exact solution, demonstrating the effectiveness of using ninth degree B-splines in the collocation method to solve ninth order differential equations.
The document provides a table of contents and questions on various technical aptitude topics such as data structures, C, C++, quantitative aptitude, UNIX concepts, RDBMS concepts, SQL, computer networks, and operating systems. It contains 35 questions related to data structures concepts like linked lists, trees, graphs, hashing etc. and their applications.
This document proposes representing complex numbers using colors (chromatic numbers) and developing ternary algebra based on this. It defines chromatic numbers and shows how to perform calculations on them. It introduces the 3-RSS NP-complete problem and shows how to formulate it and 3-colorability using ternary algebra and ternary variables. It defines ternary resolution rules to operate on ternary-CNF formulas and proves soundness and completeness of the ternary resolution system.
An analysis between different algorithms for the graph vertex coloring problem IJECEIAES
The document analyzes and compares different algorithms for solving the graph vertex coloring problem. It presents an exact backtracking algorithm that finds optimal solutions and two heuristic algorithms (greedy coloring and Welsh-Powell) that provide approximate solutions. Experiments were conducted on 27 graphs with varying numbers of vertices and edges to analyze the performance of the algorithms. The results show that the exact algorithm can efficiently solve graphs with up to around 35 vertices, finding optimal solutions for half the test graphs. The approximate algorithms provided identical solutions in terms of number of colors used but were faster for larger graphs.
A Modified Dna Computing Approach To Tackle The Exponential Solution Space Of...ijfcstjournal
This document summarizes a research paper that proposes a modified DNA computing approach to solve the graph coloring problem. The approach colors the vertices of a graph one by one, expanding DNA strands encoding promising solutions at each step and discarding infeasible solutions. This avoids an exponential number of DNA strands by not expanding the entire solution space at once. The approach is simulated on a sample 12-vertex graph, requiring fewer DNA strands than typical approaches. The modified approach provides an improvement over previous DNA computing methods for graph coloring by reducing the solution space in a step-wise manner.
Graph coloring is an important concept in graph theory. It is a special kind of problem in which we have assign colors to certain elements of the graph along with certain constraints. Suppose we are given K colors, we have to color the vertices in such a way that no two adjacent vertices of the graph have the same color, this is known as vertex coloring, similarly we have edge coloring and face coloring. The coloring problem has a huge number of applications in modern computer science such as making schedule of time table , Sudoku, Bipartite graphs , Map coloring, data mining, networking. In this paper we are going to focus on certain applications like Final exam timetabling, Aircraft Scheduling, guarding an art gallery.
This document introduces Matlab and how it can be used for computer vision and image processing tasks. Matlab is a useful language for these applications because images can be represented as matrices and many vision algorithms are naturally implemented using matrix operations in Matlab. The document provides examples of reading image files in common formats like PGM and PPM, and demonstrates basic Matlab operations for creating, accessing, and manipulating image matrices.
Greedy Edge Colouring for Lower Bound of an Achromatic Index of Simple Graphsinventionjournals
International Journal of Engineering and Science Invention (IJESI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJESI publishes research articles and reviews within the whole field Engineering Science and Technology, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
Distributed coloring with O(sqrt. log n) bitsSubhajit Sahu
Distributed Coloring with O˜(√log n) Bits
K Kothapalli, M Onus, C Scheideler, C Schindelhauer
Proc. of IEEE International Parallel and Distributed Processing Symposium …
We consider the well-known vertex coloring problem: given a graph G, find a coloring of its vertices so that no two neighbors in G have the same color. It is trivial to see that every graph of maximum degree∆ can be colored with∆+ 1 colors, and distributed algorithms that find a (∆+ 1)-coloring in a logarithmic number of communication rounds, with high probability, are known since more than a decade. This is in general the best possible if only a constant number of bits can be sent along every edge in each round. In fact, we show that for the n-node cycle the bit complexity of the coloring problem is
Ω (log n). More precisely, if only one bit can be sent along each edge in a round, then every distributed coloring algorithm (ie, algorithms in which every node has the same initial state and initially only knows its own edges) needs at least Ω (log n) rounds, with high probability, to color the n–node cycle, for any finite number of colors. But what if the edges have orientations, ie, the endpoints of an edge agree on its orientation (while bits may still flow in both directions)? Edge orientations naturally occur in dynamic networks where new nodes establish connections to old nodes. Does this allow one to provide faster coloring algorithms?
This document contains the instructions and questions for the final exam in the ICS 161 Algorithms course. It instructs students to answer 8 questions on weighted graphs, recurrences, longest common subsequence, convex hulls, vertex cover problems and their approximability. Students are provided 3 answer sheets to write their responses and are informed that answers written elsewhere will not be scored. They must also write their name and student ID on each sheet.
The document discusses graph coloring and its applications. It defines graph coloring as assigning colors to vertices of a graph such that no two adjacent vertices have the same color. Applications of graph coloring include frequency assignment in mobile networks, mapping coloration, scheduling problems, Sudoku puzzles, and register allocation. The minimum number of colors needed for a graph is called its chromatic number.
The document provides instructions for a GATE exam. It states that the exam is 3 hours long and contains questions worth 1 or 2 marks. It provides details on how answers should be marked on the answer sheet, notes on calculators and rough work, and information that negative marks will be given for incorrect answers. The instructions emphasize accurately filling out identification details on the answer sheet and carefully reading the entire paper.
This document discusses graph coloring and its applications. It begins by defining graph coloring as assigning labels or colors to elements of a graph such that no two adjacent elements have the same color. It then provides examples of vertex coloring, edge coloring, and face coloring. The document also discusses the chromatic number and chromatic polynomial. It describes several real-world applications of graph coloring, including frequency assignment in cellular networks.
This document discusses methods for summarizing Lego-like sphere and torus maps. It begins by introducing the concept of ({a,b},k)-maps, which are k-valent maps with faces of size a or b. It then discusses several challenges in enumerating and drawing such maps, including enumerating all possible Lego decompositions. Specific enumeration methods are described, such as using exact covering problems or satisfiability problems. The document also discusses challenges in graph drawing representations, and suggests using primal-dual circle packings as a promising approach.
Graph coloring is the assignment of colors to the graph vertices and edges in the graph theory. We can
divide the graph coloring in two types. The first is vertex coloring and the second is edge coloring. The
condition which we follow in graph coloring is that the incident vertices/edges have not the same color.
There are some algorithms which solve the problem of graph coloring. Some are offline algorithm and
others are online algorithm. Where offline means the graph is known in advance and the online means that
the edges of the graph are arrive one by one as an input, and We need to color each edge as soon as it is
added to the graph and the main issue is that we want to minimize the number of colors. We cannot change
the color of an edge after colored in an online algorithm. In this paper, we improve the online algorithm
for edge coloring. There is also a theorem which proves that if the maximum degree of a graph is Δ, then it
is possible to color its edges, in polynomial time, using at most Δ+ 1 color. The algorithm provided by
Vizing is offline, i.e., it assumes the whole graph is known in advance. In online algorithm edges arrive one
by one in a random permutation. This online algorithm is inspired by a distributed offline algorithm of
Panconesi and Srinivasan, referred as PS algorithm, works on 2-rounds which we extend by reusing colors
online in multiple rounds.
Graph Dynamical System on Graph ColouringClyde Shen
This document discusses using a graph dynamical system to model the vertex coloring problem. It proposes using a Graph-cellular Automaton (GA), which is an extension of cellular automata, to distribute the coloring process. In the GA model, each vertex independently chooses its optimal color based on the colors of its adjacent vertices. The document outlines testing the GA approach on different types of graphs and analyzing whether it can find colorings for them. It provides background on graph coloring, planar graphs, and introduces the graph dynamical system and GA models.
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The document describes a method for reducing generic software routines that accept alphanumeric inputs and outputs into mathematical functions. It presents a simple example of a routine that reverses a string input. To model this as a function, it introduces a method to assign unique numbers to strings using a dictionary of allowed characters and representing strings as numeric expressions in a given base. It then proves this method mathematically. The goal is to enable testing software routines using standard mathematical testing techniques by mapping routines to well-defined functions.
Cs6402 design and analysis of algorithms may june 2016 answer keyappasami
The document discusses algorithms and complexity analysis. It provides Euclid's algorithm for computing greatest common divisor, compares the orders of growth of n(n-1)/2 and n^2, and describes the general strategy of divide and conquer methods. It also defines problems like the closest pair problem, single source shortest path problem, and assignment problem. Finally, it discusses topics like state space trees, the extreme point theorem, and lower bounds.
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The progressive development of Synthetic Aperture Radar (SAR) systems diversify the exploitation of the generated images by these systems in different applications of geoscience. Detection and monitoring surface deformations, procreated by various phenomena had benefited from this evolution and had been realized by interferometry (InSAR) and differential interferometry (DInSAR) techniques. Nevertheless, spatial and temporal decorrelations of the interferometric couples used, limit strongly the precision of analysis results by these techniques. In this context, we propose, in this work, a methodological approach of surface deformation detection and analysis by differential interferograms to show the limits of this technique according to noise quality and level. The detectability model is generated from the deformation signatures, by simulating a linear fault merged to the images couples of ERS1 / ERS2 sensors acquired in a region of the Algerian south.
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Denis is a dynamic and results-driven Chief Information Officer (CIO) with a distinguished career spanning information systems analysis and technical project management. With a proven track record of spearheading the design and delivery of cutting-edge Information Management solutions, he has consistently elevated business operations, streamlined reporting functions, and maximized process efficiency.
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Date: May 29, 2024
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How to Make a Field Mandatory in Odoo 17Celine George
In Odoo, making a field required can be done through both Python code and XML views. When you set the required attribute to True in Python code, it makes the field required across all views where it's used. Conversely, when you set the required attribute in XML views, it makes the field required only in the context of that particular view.
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LAND USE LAND COVER AND NDVI OF MIRZAPUR DISTRICT, UPRAHUL
This Dissertation explores the particular circumstances of Mirzapur, a region located in the
core of India. Mirzapur, with its varied terrains and abundant biodiversity, offers an optimal
environment for investigating the changes in vegetation cover dynamics. Our study utilizes
advanced technologies such as GIS (Geographic Information Systems) and Remote sensing to
analyze the transformations that have taken place over the course of a decade.
The complex relationship between human activities and the environment has been the focus
of extensive research and worry. As the global community grapples with swift urbanization,
population expansion, and economic progress, the effects on natural ecosystems are becoming
more evident. A crucial element of this impact is the alteration of vegetation cover, which plays a
significant role in maintaining the ecological equilibrium of our planet.Land serves as the foundation for all human activities and provides the necessary materials for
these activities. As the most crucial natural resource, its utilization by humans results in different
'Land uses,' which are determined by both human activities and the physical characteristics of the
land.
The utilization of land is impacted by human needs and environmental factors. In countries
like India, rapid population growth and the emphasis on extensive resource exploitation can lead
to significant land degradation, adversely affecting the region's land cover.
Therefore, human intervention has significantly influenced land use patterns over many
centuries, evolving its structure over time and space. In the present era, these changes have
accelerated due to factors such as agriculture and urbanization. Information regarding land use and
cover is essential for various planning and management tasks related to the Earth's surface,
providing crucial environmental data for scientific, resource management, policy purposes, and
diverse human activities.
Accurate understanding of land use and cover is imperative for the development planning
of any area. Consequently, a wide range of professionals, including earth system scientists, land
and water managers, and urban planners, are interested in obtaining data on land use and cover
changes, conversion trends, and other related patterns. The spatial dimensions of land use and
cover support policymakers and scientists in making well-informed decisions, as alterations in
these patterns indicate shifts in economic and social conditions. Monitoring such changes with the
help of Advanced technologies like Remote Sensing and Geographic Information Systems is
crucial for coordinated efforts across different administrative levels. Advanced technologies like
Remote Sensing and Geographic Information Systems
9
Changes in vegetation cover refer to variations in the distribution, composition, and overall
structure of plant communities across different temporal and spatial scales. These changes can
occur natural.
2. 136 Computer Science & Information Technology (CS & IT)
The work reported in this report is organized in the remaining chapters as follows. In Chapter 2
we firstexplain graph colouring, then show how a graphcolouringproblem can be formulated and
solved in SAT using a deterministic approach. We also list the program code andthe output.In
Chapter 3we describe how the graphcolouringproblem can be formulated in the GSAT
framework, and show how it can be solved using GSAT with a tabulist. Chapter 4lists the GSAT
Experimental Results. Chapter 5 summarizes the report
2. GRAPH COLOURING
Here only undirected graphs are considered. To colour a graph G means to assigns a colour to
each vertex of G with the restriction that two adjacent vertices are not the same colour.
2.1 Graph Colouring Using SAT
This method is based on propositional logic. Let A = {a1, a2,..., an} be a set of n > 1 Boolean
variables. If a is a variable in A then a and ~a are called literals over A. A truth assignment for A
is a total function t: A → {true, false}. Under any truth assignment, aistrue if and only if ~a is
false, and a and ~~a have identical truth-values. A clause C over A is a set of literals over A. It
represents the disjunction of the literals, and is valid (or satisfied) under a truth assignment if and
only if at least one of its literals is true under the assignment. It is invalid (or unsatisfied) under
the truth assignment if every literal in it is false under the assignment. A set C of clauses over A is
satisfiable if there exist a truth assignment for A such that every clause in C is valid under the
assignment. In the Satisfiability Problem (SAT) we are required to determine whether a given set C
of clauses is satisfiable. In SAT the clauses represent the constraints to be satisfied when assigning
truth-values to the Boolean variables. In SAT, the number of literals in a clause can vary.
2.1.1 SAT formulation
In the SAT formulation there are three types of clauses. A Type 1 clause states that two adjacent
nodes cannot have the same colour. A Type 2 clause states that each node must be assigned at
least one of the available s colours. There is an Extended Formulation that has Type 3 clauses in
addition to Type 1 and Type 2 clauses. A Type 3 clause states that a node can be assigned at most
one of the colours.
3. Computer Science & Information Technology (CS & IT)
2.1.2 Example
[
Figure 1:Example of an Undirected
Figure 2: Result of the possib
Figure 3: Screenshot of calculating minimum colour needed to
2.2 GraphColouring Problem Using
Graph Coloring Problem can be formulated in the SAT framework
follows:
Step 1. INPUT: a) Number of nodes.
b)Number of edges.
c)Number of colours.
Step 2: Identification of variables
Step 3: Construction of the clauses
Step 4: Assignment of truth value
Step 5:Checking that each clauses
Step 6: Steps 4 and 5 are repeated
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Table 1: Adjacency matrix
Figure 1:Example of an Undirected Graph
Result of the possible colouring of the above graph.
t of calculating minimum colour needed to colour the above graph
roblem Using SAT framework
Problem can be formulated in the SAT framework. The solution steps are as
INPUT: a) Number of nodes.
Identification of variables
clauses
Assignment of truth values to literals
clauses is valid.
ed until all the clauses are true.
0 1 1 0 1 1 0
1 0 0 1 0 1 1
1 0 0 1 1 1 0
0 1 1 0 0 1 1
1 0 1 0 0 0 1
1 1 1 1 0 0 0
0 1 0 1 1 0 0
137
colour the above graph.
The solution steps are as
4. 138 Computer Science & Information Technology (CS & IT)
2.2.1 Example
This example will show how a graph coloring problem can be solved using SAT.
Figure 4: Example of an undirected graph with colour
Figure 4 shows a simple graph consisting of 4 nodes and 4 edges. Suppose we have to determine
whether it can be colored with 3 colors named 1, 2 and 3.a1 is true when a is coloured with colour
1 and is false otherwise .We have seen that constraints here are represented by clauses. The
constraint that two adjacent clauses cannot be colored with the same color can be expressed by
the following three
Type 1 clauses:~a1 / ~c1 ~a2 / ~c2 ~a3 / ~c3
As there are 4 edges in total, and there are three Type 1 clauses corresponding to each edge, the
total number of such clauses will be 12.
The constraint that a node must not be left uncolored can be expressed in case of node 1 by the
following
Type 2 clause: a1 / a2 / a3
For four nodes in the graph the total number of Type 2 clauses will be 4.
Type 3 clauses force a node such as node to be colored by a single color. The clauses are as
follows:
Type 3 clauses:~a1 / ~a2 ~a1 / ~a3 ~a3/ ~a2
The number of Type 3 clauses for the graph of Figure 1 will be 12, as there are four nodes and three
colours. Thus the total number of clauses in the graph will be 28 in this case.
2.2.2 Example
We illustrate below how SAT solves a typical satisfiability problem.
8. 142 Computer Science & Information Technology (CS & IT)
Figure 8:Screenshot of result for the above example
3. RANDOMIZED SEARCH
Deterministic search methods frequently run slowly on hard CSP instances. This has led to the
formulation and development of randomized search algorithms
based on local search methods and are therefore often called
find near-optimal solutions. A local search method generally initiates the search process in some
randomly chosen point (or set of points) in the
Such a point is often called a candidate
complete set of assignments of values to the search variables. The set of points reachable from a
point by making all possible moves is called the
has to evaluate the neighborhood
made. The evaluation is often accomplished with the aid of an objective function. This function is
generally problem dependent and rates the likelihood
For example, the objective function for the algorithm GSAT for any problem instance formulated as
a Satisfiability Problem is typically taken as the total number of unsatisfied clauses. Auxiliary data
structures such as tab lists can help to improve the performance of the algorithm. The initial solution
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Figure 8:Screenshot of result for the above example
EARCH ALGORITHMS
Deterministic search methods frequently run slowly on hard CSP instances. This has led to the
of randomized search algorithms. Most randomized algorithms are
based on local search methods and are therefore often called Local Search Algorithms
optimal solutions. A local search method generally initiates the search process in some
randomly chosen point (or set of points) in the search space (sometimes called the solution space
candidatesolution. A feasible (or complete) solution consists of a
complete set of assignments of values to the search variables. The set of points reachable from a
point by making all possible moves is called the neighborhood of the point. A local searc
neighborhood points and determine to which neighbor the next move should be
made. The evaluation is often accomplished with the aid of an objective function. This function is
generally problem dependent and rates the likelihood that the point leads closer to a feasible solution.
For example, the objective function for the algorithm GSAT for any problem instance formulated as
a Satisfiability Problem is typically taken as the total number of unsatisfied clauses. Auxiliary data
can help to improve the performance of the algorithm. The initial solution
Deterministic search methods frequently run slowly on hard CSP instances. This has led to the
. Most randomized algorithms are
Search Algorithms. These can
optimal solutions. A local search method generally initiates the search process in some
solution space).
(or complete) solution consists of a
complete set of assignments of values to the search variables. The set of points reachable from a
of the point. A local search method
the next move should be
made. The evaluation is often accomplished with the aid of an objective function. This function is
that the point leads closer to a feasible solution.
For example, the objective function for the algorithm GSAT for any problem instance formulated as
a Satisfiability Problem is typically taken as the total number of unsatisfied clauses. Auxiliary data
can help to improve the performance of the algorithm. The initial solution
9. Computer Science & Information Technology (CS & IT) 143
can be chosen either randomly or by the use of a heuristic. The search then proceeds to a
neighboringpoint and from that point to another neighboring point, and so on in an iterative way.
Although moves are made in a systematic manner, local search algorithms are typically incomplete.
Local search algorithms often make use of stochastic mechanisms.
3.1 Local Search Algorithms
In this section, however, we are concerned only with general local search methods such as GSAT
and Tabu Search.
3.2.1GSAT
The greedy local search procedure GSAT assigns truth values to variables in an effort to satisfy all
the clauses in a given set. At start, the procedure randomly assigns truth-values to variables. This
amounts to picking a solution randomly from the solution space. Typically this truth assignment
fails to satisfy all the clauses. If the truth-value of a variable is flipped (i.e., complemented), the total
number of satisfied clauses will change. Suppose a variable v is flipped. Then the unsatisfied
clauses where v occurs get satisfied. The number of such clauses is represented by make(v). At the
same time, before flipping v, there were some clauses satisfied only by the variable v. As v is
flipped these clauses get unsatisfied. The number of such clauses is given by break(v). The
difference make(v) - break(v) is the net increase in the number of satisfied clauses and is referred to
as gain(v). Flipping a variable means moving to a neighboring point in the solution space. We have
mentioned above that an objective function can help to guide us where to move next in the
neighborhood. In GSAT the value of the function is the number of unsatisfied clauses. GSAT looks
for the variable with the property that its truth-value when flipped causes the largest net decrease in
the number of unsatisfied clauses. It flips the truth-value of this variable, and again looks for such a
variable. Ties are resolved arbitrarily. This is repeated until a satisfying assignment is found. If no
satisfying assignment is found within a specified number of flips (maxflips), the procedure is
restarted with a new initial truth assignment. This cycle is repeated a specified number of times
(maxtries). The algorithm always tries to select the most promising neighbour at each step. It is
therefore called a greedy local search algorithm.
3.3Graph Colouring Problem Using GSAT framework
The solution steps are therefore as follows:
Step1:INPUTa) Size of array
b) Percentage of edges
c)Initial number of colours
d) The number of runs
e)Size of tabu list
Step 2: Creation of list of literals.
Step 3:Construct the clauses
Step 4:Assignrandom truth assignment value to literals
Step 5:Check whether that each clauses is valid.
Step 6:If this truth assignment does not satisfy all the clauses
Step 7:Determine the effect of flipping the truth-value of each variable
Step 8:Count the satisfied clauses after the bit is changed
Step 9:Determine the variables flipping which cause the maximumdecrease in the number of
Unsatisfied clauses;
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Select a variable from these set resolving ties arbitrarily;
Flip the selected variable.
Step 10: Repeat Step 4 and Step 5until all the clauses are true.
3.4Procedure GSAT
Anillustrationis done below how GSAT solves a typical satisfiability problem. Consider the
following set of six clauses:
C1: a3 U a4
C2: a5 U ~a4
C3: a1 U a2 U ~a3
C4: ~a1 U ~a3 U ~a5
C5: ~a2 U ~a3
C6: a3 U ~a4
There are five variables a1, a2, a3, a4 and a5. Suppose the initial random truth assignment is as
follows:
a1 = true,a2 = false,a3 = true, a4 = false, a5 = true.
This truth assignment does not satisfy all the clauses; in particular, clause C4 remains unsatisfied.
We must now determine the effect of flipping the truth-value of each variable. Suppose that when a
variable is flipped, make clauses that are currently unsatisfied get satisfied, and breakclauses that
are currently satisfied become unsatisfied. The overall decrease in the number of unsatisfied clauses
is then given by the expression gain = make – break. We have to find the variable with the largest
gain. Here
gain(x1) = 1 – 1 =0,
gain(x2) = 0 – 1 = -1,
gain(x3) = 1 – 1 = 0,
gain(x4) = 0 – 0 = 0,
gain(x5) = 1 – 0 = 1.
We change the truth value of x5 to false since this results in the largest gain, and find as a result that
all the clauses are satisfied.
3.5 Tabu Search Strategy
GSAT can significantly improved by tabu search strategy to ensure that the same variable is not
flipped again and again.Tabulist is initially empty and is implemented as a FIFO queue.The variable
that has just been flipped is inserted into the list.The variable to be flipped next is selected randomly
from among those variables not in the tabulist.Thus some variables are prevented from being
flipped for a limited period of time,determine by the length of the tabulist. Tabu list in GSAT not
only helped to reduce the running time,it helped us to solve some problems that could not be solved
at all without using tabulist.
This chapter describes how randomized algorithms perform local searches. In spite of being
incomplete in nature, a local search method is often preferred over a deterministic search method
when a near-optimal solution has to be found in real time.GSAT algorithms based on local search
have been described in this chapter. These include Tabu Search. The Satisfiability Problem (SAT)
is explained. The most interesting property of the Satisfiability such as Graph Colouring can be
encoded into SAT has been illustrated. Once encoded into SAT, the problem can be solved by the
11. Computer Science & Information Technology (CS & IT)
use of SAT algorithms and Graph Colouring can b
illustrated with experimental result.
4. EXPERIMENTAL RESULTS
Figure 9:Screenshot of result of the Graph Colouring Problem using GSAT with low time complexity
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and Graph Colouring can be encoded into GSAT and Tabu search has been
experimental result.
ESULTS
Screenshot of result of the Graph Colouring Problem using GSAT with low time complexity
145
and Tabu search has been
Screenshot of result of the Graph Colouring Problem using GSAT with low time complexity
12. 146 Computer Science & Information Technology (CS & IT)
Table 6: Colouring Random Graph with Tabulist
5. CONCLUSION
In this paper both theoretical and experimental studies are done successfully to calculate the
performance of SAT for a Graph coloring problem. The role played by SAT as an intermediate
domain for solving problems. SAT is not sufficient to calculate the minimum number of color
needed to color a graph.SAT technique only extracted that input number of color can sufficient to
color the graph or not. Hera also shows how this problem can be encoded in SAT. Finally, GSAT
method is introduced to use for solving graph coloring problems. GSAT method is enabled to
calculate the number of color need to color a graph. This paper also established a new algorithm
using GSAT which can satisfy the problem with low time complexity.
REFERENCES
[1] Acharyya S, TheSatisfiability Problem: A Constraint Satisfaction Approach, Ph D Thesis, Computer
Science & Engineering, University of Calcutta, 2001
[2] Acharyya S, SAT Algorithms for Colouring Some Special Classes of Graphs: Some Theoretical and
Experimental Results, Journal on Satisfiability, Boolean Modeling and Computation(JSAT),vol 4, no
1, pp 33-55, 2007
[3] Acharyya S &Bagchi A, Local search methods for coloring graphs: Role of tabu list, Artificial
Intelligence: Emerging Trends & Applications (Proc KBCS-2004), International Conf on Knowledge-
Based Computer Systems, Hyderabad, India, Dec 2004, pp 462-472
13. Computer Science & Information Technology (CS & IT) 147
[4] Selman B,Levesque H J & Mitchell D J, A New Method for Solving Hard SatisfiabilityProblems,
Proc AAAI 92, American Association for Artificial Intelligence, 1992, pp 440 446, 1992
AUTHORS
MunmunDey is an Assistant Professorat Sanaka Educational Trust's Group of
Institutions, Durgapur, West Bengal, India since 2012. She received his M.E degree from
Heritage Institute of Technology in year 2012 and B-Tech degree from Meghnad Saha
Institute of Technology in 2009. Her research interests include the field of Optimization
techniques.
Prof. (Dr.) Amitava Bagchi is a Professor at Heritage Institute of Technology, West Bengal, India. He
received his DSc. degree from MIT, USA. Prof Bagchi has 43 years teaching and Industry experiences. His
research interests include the field of Optimization techniques. He has about 55 referred national and
international publications to her credit.