The greedy method constructs an optimal solution in stages by making locally optimal choices at each stage without reconsidering past decisions. It selects the choice that appears best at the current time without regard for its long-term consequences. The general greedy algorithm procedure selects the best choice from available inputs at each stage until a complete solution is reached. Examples demonstrate both when the greedy method succeeds in finding an optimal solution and when it fails to do so compared to alternative methods like dynamic programming.
Dynamic programming is used to solve optimization problems by breaking them down into overlapping subproblems. It solves subproblems only once, storing the results in a table to lookup when the same subproblem occurs again, avoiding recomputing solutions. Key steps are characterizing optimal substructures, defining solutions recursively, computing solutions bottom-up, and constructing the overall optimal solution. Examples provided are matrix chain multiplication and longest common subsequence.
The document discusses greedy algorithms and their use for optimization problems. It provides examples of using greedy approaches to solve scheduling and knapsack problems. Specifically:
- A greedy algorithm makes locally optimal choices at each step in the hope of finding a globally optimal solution. However, greedy algorithms do not always yield optimal solutions.
- For scheduling jobs, a greedy approach of scheduling the longest jobs first is not optimal, while scheduling the shortest jobs first is also not optimal. An optimal solution can be found by trying all possible assignments.
- For the knapsack problem, different greedy strategies like selecting items with highest value or lowest weight are discussed. The optimal greedy strategy is to select items in order of highest value to
This document discusses integration, which is the inverse process of differentiation. Integration allows us to find the original function given its derivative. Several integration techniques are explained, including substitution, integration by parts, and finding volumes of revolution. Standard integrals are presented along with examples of calculating areas under curves and volumes obtained by rotating areas about axes. Definite integrals are used to find the area between curves over a specified interval.
Math lecture 10 (Introduction to Integration)Osama Zahid
Integration is a process of adding slices of area to find the total area under a curve. There are three main methods for integration:
1) Slicing the area into thin strips and adding them up as the width approaches zero.
2) Using shortcuts like knowing the integral of 2x is x^2 based on derivatives.
3) Performing u-substitutions to rewrite integrals in a form where the inner function can be integrated.
This document provides an overview of integration and techniques for evaluating integrals. It discusses integration as the inverse operation of differentiation and indefinite integrals. Several methods for evaluating integrals are described, including substitution, integration by parts, partial fractions, and special integrals involving trigonometric, exponential, logarithmic and hyperbolic functions. Examples of standard forms that can be converted into special integrals are given. The document also compares properties of differentiation and integration and provides contact information for the author.
The document discusses algorithms and data structures using divide and conquer and greedy approaches. It covers topics like matrix multiplication, convex hull, binary search, activity selection problem, knapsack problem, and their algorithms and time complexities. Examples are provided for convex hull, binary search, activity selection, and knapsack problem algorithms. The document is intended as teaching material on design and analysis of algorithms.
This document discusses the Galerkin method for solving differential equations. It begins by introducing how engineering problems can be expressed as differential equations with boundary conditions. It then explains that the Galerkin method uses an approximation approach to find the function that satisfies the equations. The key steps of the Galerkin method are to introduce a trial solution as a linear combination of basis functions, choose weight functions, take the inner product of the residual and weight functions to generate a system of equations for the unknown coefficients, and solve this system to obtain the approximate solution. An example of applying the Galerkin method to solve a second order differential equation is also provided.
Dynamic programming is used to solve optimization problems by breaking them down into overlapping subproblems. It solves subproblems only once, storing the results in a table to lookup when the same subproblem occurs again, avoiding recomputing solutions. Key steps are characterizing optimal substructures, defining solutions recursively, computing solutions bottom-up, and constructing the overall optimal solution. Examples provided are matrix chain multiplication and longest common subsequence.
The document discusses greedy algorithms and their use for optimization problems. It provides examples of using greedy approaches to solve scheduling and knapsack problems. Specifically:
- A greedy algorithm makes locally optimal choices at each step in the hope of finding a globally optimal solution. However, greedy algorithms do not always yield optimal solutions.
- For scheduling jobs, a greedy approach of scheduling the longest jobs first is not optimal, while scheduling the shortest jobs first is also not optimal. An optimal solution can be found by trying all possible assignments.
- For the knapsack problem, different greedy strategies like selecting items with highest value or lowest weight are discussed. The optimal greedy strategy is to select items in order of highest value to
This document discusses integration, which is the inverse process of differentiation. Integration allows us to find the original function given its derivative. Several integration techniques are explained, including substitution, integration by parts, and finding volumes of revolution. Standard integrals are presented along with examples of calculating areas under curves and volumes obtained by rotating areas about axes. Definite integrals are used to find the area between curves over a specified interval.
Math lecture 10 (Introduction to Integration)Osama Zahid
Integration is a process of adding slices of area to find the total area under a curve. There are three main methods for integration:
1) Slicing the area into thin strips and adding them up as the width approaches zero.
2) Using shortcuts like knowing the integral of 2x is x^2 based on derivatives.
3) Performing u-substitutions to rewrite integrals in a form where the inner function can be integrated.
This document provides an overview of integration and techniques for evaluating integrals. It discusses integration as the inverse operation of differentiation and indefinite integrals. Several methods for evaluating integrals are described, including substitution, integration by parts, partial fractions, and special integrals involving trigonometric, exponential, logarithmic and hyperbolic functions. Examples of standard forms that can be converted into special integrals are given. The document also compares properties of differentiation and integration and provides contact information for the author.
The document discusses algorithms and data structures using divide and conquer and greedy approaches. It covers topics like matrix multiplication, convex hull, binary search, activity selection problem, knapsack problem, and their algorithms and time complexities. Examples are provided for convex hull, binary search, activity selection, and knapsack problem algorithms. The document is intended as teaching material on design and analysis of algorithms.
This document discusses the Galerkin method for solving differential equations. It begins by introducing how engineering problems can be expressed as differential equations with boundary conditions. It then explains that the Galerkin method uses an approximation approach to find the function that satisfies the equations. The key steps of the Galerkin method are to introduce a trial solution as a linear combination of basis functions, choose weight functions, take the inner product of the residual and weight functions to generate a system of equations for the unknown coefficients, and solve this system to obtain the approximate solution. An example of applying the Galerkin method to solve a second order differential equation is also provided.
Classical optimization theory Unconstrained ProblemSurya Teja
This document discusses classical optimization theory and methods for finding extrema of functions. It defines local and global minima/maxima, and presents necessary and sufficient conditions for identifying extrema. Specifically, it states that a necessary condition is for the gradient to be zero at a point, while sufficient conditions involve the Hessian matrix being positive/negative definite. It then introduces the Newton-Raphson method for numerically solving the necessary condition of setting the gradient to zero. An example demonstrates applying Newton-Raphson to find an extremum of a sample function.
The document summarizes key concepts from Lesson 28 on Lagrange multipliers, including:
1) Restating the method of Lagrange multipliers and providing justifications through elimination, graphical, and symbolic approaches.
2) Discussing second order conditions for constrained optimization problems, noting the importance of compact feasibility sets.
3) Providing the theorem on Lagrange multipliers and examples of its application to problems with more than two variables or multiple constraints.
This document provides an introduction to optimization theory, beginning with an overview of different optimization problem types such as nonlinear equations, nonlinear least squares, constrained and unconstrained optimization. It then presents some key concepts in optimization theory including Taylor's theorem, positive definiteness, convexity, local and global minima, first and second order necessary/sufficient conditions, and uniqueness of minima for convex functions. The document concludes with an overview of the linear least squares problem and its properties.
The document discusses the longest common subsequence (LCS) problem and how to solve it using dynamic programming. It begins by defining LCS as the longest sequence of characters that appear left-to-right in two given strings. It then describes solving LCS using a brute force method with exponential time complexity and using dynamic programming with polynomial time complexity. Finally, it provides an example of finding the LCS of two strings and discusses applications and references.
The document discusses greedy algorithms, which attempt to find optimal solutions to optimization problems by making locally optimal choices at each step that are also globally optimal. It provides examples of problems that greedy algorithms can solve optimally, such as minimum spanning trees and change making, as well as problems they can provide approximations for, like the knapsack problem. Specific greedy algorithms covered include Kruskal's and Prim's for minimum spanning trees.
Fractional programming (A tool for optimization)VARUN KUMAR
The document discusses fractional programming problems (FPP), which involve optimizing an objective function that is the ratio of two other functions. It outlines three common transforms used to convert FPPs into more tractable forms: 1) Charnes-Cooper transform decouples the numerator and denominator, 2) Dinkelbach's transform iteratively updates an auxiliary variable, and 3) quadratic transform ensures the transformed objective function is concave to allow convex optimization techniques to be applied. The document provides detailed mathematical derivations of the quadratic transform to justify its formulation.
Indefinite and Definite Integrals Using the Substitution MethodScott Bailey
The document describes the substitution method for integrating functions. It provides an example of using u-substitution to evaluate the integral from 0 to 4 of (2x + 1) dx by letting u = 2x + 1. This changes the limits of integration from 0 to 1 and 9, allowing the integral to be evaluated as the integral from 1 to 9 of du/2 = 4.5.
The document discusses the 0/1 knapsack problem and the greedy algorithm approach. It describes the knapsack problem as selecting a subset of items with weights and values that fit within a knapsack capacity while maximizing the total value. The greedy algorithm works by selecting the highest value item at each step that fits within remaining capacity. The document provides an example problem of selecting boxes to fill a knapsack of 15kg capacity to maximize profit. It outlines the recurrence relation and time/space complexity of the greedy knapsack algorithm.
- The document outlines a BSc research project on pricing financial derivatives using the Black-Scholes model.
- The project aims to learn established financial models, compare pricing techniques, and see how newer models relate to existing ones.
- It provides background on the student's motivation and experience, and introduces key concepts like options, the Black-Scholes equation, and its derivation and solution.
- The student will present their work on applying and extending the Black-Scholes model to price derivatives.
The document provides instructions for a series of exercises using the Scheme programming language in the DrScheme environment. The exercises cover basic Scheme concepts like defining functions, performing calculations, conditional evaluation, and converting between measurement units. Later exercises involve defining more complex functions to calculate things like interest, taxes, distances, and volumes. The goal is to practice basic Scheme programming skills through a set of tutorial exercises.
This document discusses dynamic programming and its application to solve the knapsack problem.
It begins by defining dynamic programming as a technique for solving problems with overlapping subproblems where each subproblem is solved only once and the results are stored in a table.
It then defines the knapsack problem as selecting a subset of items with weights and values that fit in a knapsack of capacity W to maximize the total value.
The document shows how to solve the knapsack problem using dynamic programming by constructing a table where each entry table[i,j] represents the maximum value for items 1 to i with weight ≤ j. It provides an example problem and walks through filling the table and backtracking to find
This document discusses greedy algorithms and dynamic programming techniques for solving optimization problems. It covers the activity selection problem, which can be solved greedily by always selecting the shortest remaining activity. It also discusses the knapsack problem and how the fractional version can be solved greedily while the 0-1 version requires dynamic programming due to its optimal substructure but non-greedy nature. Dynamic programming builds up solutions by combining optimal solutions to overlapping subproblems.
Minimum mean square error estimation and approximation of the Bayesian updateAlexander Litvinenko
This document discusses methods for approximating the Bayesian update used in parameter identification problems with partial differential equations containing uncertain coefficients. It presents:
1) Deriving the Bayesian update from conditional expectation and proposing polynomial chaos expansions to approximate the full Bayesian update.
2) Describing minimum mean square error estimation to find estimators that minimize the error between the true parameter and its estimate given measurements.
3) Providing an example of applying these methods to identify an uncertain coefficient in a 1D elliptic PDE using measurements at two points.
The document discusses different decrease-and-conquer algorithms. It explains that decrease-and-conquer works by establishing a relationship between solving a problem for an instance of size n and solving a smaller instance of size n-c, where c is some constant. It then decomposes the problem size recursively until reaching a base case. The document provides examples of decrease by a constant, constant factor, and variable amounts. It also discusses insertion sort, binary search, and Euclid's algorithm as examples of decrease-and-conquer approaches.
The newsboy problem in determining optimal quantity inAlexander Decker
This document summarizes a research paper that examines determining the optimal order quantity in a stochastic inventory problem where demand is a continuous random variable. The paper:
1) Describes the classic "newsboy problem" of determining the optimal order quantity when demand is uncertain.
2) Develops a profit function and uses calculus to derive the formula for optimal order quantity as the inverse of the demand distribution function.
3) Applies the model and formula to inventory data for cocoa, calculating the optimal order quantity as 104,000 tons.
Fixed points of contractive and Geraghty contraction mappings under the influ...IJERA Editor
In this paper, we prove the existence of fixed points of contractive and Geraghty contraction maps in complete metric spaces under the influence of altering distances. Our results extend and generalize some of the known results.
Comparison market implied volatilities with implied volatilities computed by ...Yuan Jing
VBA coding application in fx option market, the mission is 1) to compare the implied volatility in market and the implied volatilities computed by Newton method while inputting different times of maturity, strike price, we find that there are big discrepancies when the moneyness is quite great or quite small, which means that there might be some arbitrage. 2) To build the model by using object structure in VBA ---- Class, which allows us to facilitate the coding and we use it to compute the delta when there are different time of maturity and strike price.
This document provides estimates for several number theory functions without assuming the Riemann Hypothesis (RH), including bounds for ψ(x), θ(x), and the kth prime number pk. The following estimates are derived:
1) θ(x) - x < 1/36,260x for x > 0.
2) |θ(x) - x| < ηk x/lnkx for certain values of x, where ηk decreases as k increases.
3) Estimates are obtained for θ(pk), the value of θ at the kth prime number pk, showing θ(pk) is approximately k ln k + ln2 k - 1.
The document discusses greedy algorithms and their use for optimization problems. It provides examples of using greedy approaches to solve scheduling and knapsack problems. Specifically, it describes how a greedy algorithm works by making locally optimal choices at each step in hopes of reaching a globally optimal solution. While greedy algorithms do not always find the true optimal, they often provide good approximations. The document also proves that certain greedy strategies, such as always selecting the item with the highest value to weight ratio for the knapsack problem, will find the true optimal solution.
Classical optimization theory Unconstrained ProblemSurya Teja
This document discusses classical optimization theory and methods for finding extrema of functions. It defines local and global minima/maxima, and presents necessary and sufficient conditions for identifying extrema. Specifically, it states that a necessary condition is for the gradient to be zero at a point, while sufficient conditions involve the Hessian matrix being positive/negative definite. It then introduces the Newton-Raphson method for numerically solving the necessary condition of setting the gradient to zero. An example demonstrates applying Newton-Raphson to find an extremum of a sample function.
The document summarizes key concepts from Lesson 28 on Lagrange multipliers, including:
1) Restating the method of Lagrange multipliers and providing justifications through elimination, graphical, and symbolic approaches.
2) Discussing second order conditions for constrained optimization problems, noting the importance of compact feasibility sets.
3) Providing the theorem on Lagrange multipliers and examples of its application to problems with more than two variables or multiple constraints.
This document provides an introduction to optimization theory, beginning with an overview of different optimization problem types such as nonlinear equations, nonlinear least squares, constrained and unconstrained optimization. It then presents some key concepts in optimization theory including Taylor's theorem, positive definiteness, convexity, local and global minima, first and second order necessary/sufficient conditions, and uniqueness of minima for convex functions. The document concludes with an overview of the linear least squares problem and its properties.
The document discusses the longest common subsequence (LCS) problem and how to solve it using dynamic programming. It begins by defining LCS as the longest sequence of characters that appear left-to-right in two given strings. It then describes solving LCS using a brute force method with exponential time complexity and using dynamic programming with polynomial time complexity. Finally, it provides an example of finding the LCS of two strings and discusses applications and references.
The document discusses greedy algorithms, which attempt to find optimal solutions to optimization problems by making locally optimal choices at each step that are also globally optimal. It provides examples of problems that greedy algorithms can solve optimally, such as minimum spanning trees and change making, as well as problems they can provide approximations for, like the knapsack problem. Specific greedy algorithms covered include Kruskal's and Prim's for minimum spanning trees.
Fractional programming (A tool for optimization)VARUN KUMAR
The document discusses fractional programming problems (FPP), which involve optimizing an objective function that is the ratio of two other functions. It outlines three common transforms used to convert FPPs into more tractable forms: 1) Charnes-Cooper transform decouples the numerator and denominator, 2) Dinkelbach's transform iteratively updates an auxiliary variable, and 3) quadratic transform ensures the transformed objective function is concave to allow convex optimization techniques to be applied. The document provides detailed mathematical derivations of the quadratic transform to justify its formulation.
Indefinite and Definite Integrals Using the Substitution MethodScott Bailey
The document describes the substitution method for integrating functions. It provides an example of using u-substitution to evaluate the integral from 0 to 4 of (2x + 1) dx by letting u = 2x + 1. This changes the limits of integration from 0 to 1 and 9, allowing the integral to be evaluated as the integral from 1 to 9 of du/2 = 4.5.
The document discusses the 0/1 knapsack problem and the greedy algorithm approach. It describes the knapsack problem as selecting a subset of items with weights and values that fit within a knapsack capacity while maximizing the total value. The greedy algorithm works by selecting the highest value item at each step that fits within remaining capacity. The document provides an example problem of selecting boxes to fill a knapsack of 15kg capacity to maximize profit. It outlines the recurrence relation and time/space complexity of the greedy knapsack algorithm.
- The document outlines a BSc research project on pricing financial derivatives using the Black-Scholes model.
- The project aims to learn established financial models, compare pricing techniques, and see how newer models relate to existing ones.
- It provides background on the student's motivation and experience, and introduces key concepts like options, the Black-Scholes equation, and its derivation and solution.
- The student will present their work on applying and extending the Black-Scholes model to price derivatives.
The document provides instructions for a series of exercises using the Scheme programming language in the DrScheme environment. The exercises cover basic Scheme concepts like defining functions, performing calculations, conditional evaluation, and converting between measurement units. Later exercises involve defining more complex functions to calculate things like interest, taxes, distances, and volumes. The goal is to practice basic Scheme programming skills through a set of tutorial exercises.
This document discusses dynamic programming and its application to solve the knapsack problem.
It begins by defining dynamic programming as a technique for solving problems with overlapping subproblems where each subproblem is solved only once and the results are stored in a table.
It then defines the knapsack problem as selecting a subset of items with weights and values that fit in a knapsack of capacity W to maximize the total value.
The document shows how to solve the knapsack problem using dynamic programming by constructing a table where each entry table[i,j] represents the maximum value for items 1 to i with weight ≤ j. It provides an example problem and walks through filling the table and backtracking to find
This document discusses greedy algorithms and dynamic programming techniques for solving optimization problems. It covers the activity selection problem, which can be solved greedily by always selecting the shortest remaining activity. It also discusses the knapsack problem and how the fractional version can be solved greedily while the 0-1 version requires dynamic programming due to its optimal substructure but non-greedy nature. Dynamic programming builds up solutions by combining optimal solutions to overlapping subproblems.
Minimum mean square error estimation and approximation of the Bayesian updateAlexander Litvinenko
This document discusses methods for approximating the Bayesian update used in parameter identification problems with partial differential equations containing uncertain coefficients. It presents:
1) Deriving the Bayesian update from conditional expectation and proposing polynomial chaos expansions to approximate the full Bayesian update.
2) Describing minimum mean square error estimation to find estimators that minimize the error between the true parameter and its estimate given measurements.
3) Providing an example of applying these methods to identify an uncertain coefficient in a 1D elliptic PDE using measurements at two points.
The document discusses different decrease-and-conquer algorithms. It explains that decrease-and-conquer works by establishing a relationship between solving a problem for an instance of size n and solving a smaller instance of size n-c, where c is some constant. It then decomposes the problem size recursively until reaching a base case. The document provides examples of decrease by a constant, constant factor, and variable amounts. It also discusses insertion sort, binary search, and Euclid's algorithm as examples of decrease-and-conquer approaches.
The newsboy problem in determining optimal quantity inAlexander Decker
This document summarizes a research paper that examines determining the optimal order quantity in a stochastic inventory problem where demand is a continuous random variable. The paper:
1) Describes the classic "newsboy problem" of determining the optimal order quantity when demand is uncertain.
2) Develops a profit function and uses calculus to derive the formula for optimal order quantity as the inverse of the demand distribution function.
3) Applies the model and formula to inventory data for cocoa, calculating the optimal order quantity as 104,000 tons.
Fixed points of contractive and Geraghty contraction mappings under the influ...IJERA Editor
In this paper, we prove the existence of fixed points of contractive and Geraghty contraction maps in complete metric spaces under the influence of altering distances. Our results extend and generalize some of the known results.
Comparison market implied volatilities with implied volatilities computed by ...Yuan Jing
VBA coding application in fx option market, the mission is 1) to compare the implied volatility in market and the implied volatilities computed by Newton method while inputting different times of maturity, strike price, we find that there are big discrepancies when the moneyness is quite great or quite small, which means that there might be some arbitrage. 2) To build the model by using object structure in VBA ---- Class, which allows us to facilitate the coding and we use it to compute the delta when there are different time of maturity and strike price.
This document provides estimates for several number theory functions without assuming the Riemann Hypothesis (RH), including bounds for ψ(x), θ(x), and the kth prime number pk. The following estimates are derived:
1) θ(x) - x < 1/36,260x for x > 0.
2) |θ(x) - x| < ηk x/lnkx for certain values of x, where ηk decreases as k increases.
3) Estimates are obtained for θ(pk), the value of θ at the kth prime number pk, showing θ(pk) is approximately k ln k + ln2 k - 1.
The document discusses greedy algorithms and their use for optimization problems. It provides examples of using greedy approaches to solve scheduling and knapsack problems. Specifically, it describes how a greedy algorithm works by making locally optimal choices at each step in hopes of reaching a globally optimal solution. While greedy algorithms do not always find the true optimal, they often provide good approximations. The document also proves that certain greedy strategies, such as always selecting the item with the highest value to weight ratio for the knapsack problem, will find the true optimal solution.
The document discusses various greedy algorithms including knapsack problems, minimum spanning trees, shortest path algorithms, and job sequencing. It provides descriptions of greedy algorithms, examples to illustrate how they work, and pseudocode for algorithms like fractional knapsack, Prim's, Kruskal's, Dijkstra's, and job sequencing. Key aspects covered include choosing the best option at each step and building up an optimal solution incrementally using greedy choices.
The document discusses greedy algorithms and provides examples of how they can be applied to solve optimization problems like the knapsack problem. It defines greedy techniques as making locally optimal choices at each step to arrive at a global solution. Examples where greedy algorithms are used include finding the shortest path, minimum spanning tree (using Prim's and Kruskal's algorithms), job sequencing with deadlines, and the fractional knapsack problem. Pseudocode and examples are provided to demonstrate how greedy algorithms work for the knapsack problem and job sequencing problem.
Dynamic programming is used to solve optimization problems by breaking them down into subproblems. It solves each subproblem only once, storing the results in a table to lookup when the subproblem recurs. This avoids recomputing solutions and reduces computation. The key is determining the optimal substructure of problems. It involves characterizing optimal solutions recursively, computing values in a bottom-up table, and tracing back the optimal solution. An example is the 0/1 knapsack problem to maximize profit fitting items in a knapsack of limited capacity.
The document discusses approximation algorithms for NP-complete problems. It introduces the concept of approximation ratios, which measure how close an approximate solution from a polynomial-time algorithm is to the optimal solution. The document then provides examples of approximation algorithms with a ratio of 2 for the vertex cover and traveling salesman problems. It also discusses using backtracking to find all possible solutions to the subset sum problem.
This document discusses dynamic programming and algorithms for solving all-pair shortest path problems. It begins by defining dynamic programming as avoiding recalculating solutions by storing results in a table. It then describes Floyd's algorithm for finding shortest paths between all pairs of nodes in a graph. The algorithm iterates through nodes, calculating shortest paths that pass through each intermediate node. It takes O(n3) time for a graph with n nodes. Finally, it discusses the multistage graph problem and provides forward and backward algorithms to find the minimum cost path from source to destination in a multistage graph in O(V+E) time, where V and E are the numbers of vertices and edges.
Mastering Greedy Algorithms: Optimizing Solutions for Efficiency"22bcs058
Greedy algorithms are fundamental techniques used in computer science and optimization problems. They belong to a class of algorithms that make decisions based on the current best option without considering the overall future consequences. Despite their simplicity and intuitive appeal, greedy algorithms can provide efficient solutions to a wide range of problems across various domains.
At the core of greedy algorithms lies a simple principle: at each step, choose the locally optimal solution that seems best at the moment, with the hope that it will lead to a globally optimal solution. This principle makes greedy algorithms easy to understand and implement, as they typically involve iterating through a set of choices and making decisions based on some criteria.
One of the key characteristics of greedy algorithms is their greedy choice property, which states that at each step, the locally optimal choice leads to an optimal solution overall. This property allows greedy algorithms to make decisions without needing to backtrack or reconsider previous choices, resulting in efficient solutions for many problems.
Greedy algorithms are commonly used in problems involving optimization, scheduling, and combinatorial optimization. Examples include finding the minimum spanning tree in a graph (Prim's and Kruskal's algorithms), finding the shortest path in a weighted graph (Dijkstra's algorithm), and scheduling tasks to minimize completion time (interval scheduling).
Despite their effectiveness in many situations, greedy algorithms may not always produce the optimal solution for a given problem. In some cases, a greedy approach can lead to suboptimal solutions that are not globally optimal. This occurs when the greedy choice property does not guarantee an optimal solution at each step, or when there are conflicting objectives that cannot be resolved by a greedy strategy alone.
To mitigate these limitations, it is essential to carefully analyze the problem at hand and determine whether a greedy approach is appropriate. In some cases, greedy algorithms can be augmented with additional techniques or heuristics to improve their performance or guarantee optimality. Alternatively, other algorithmic paradigms such as dynamic programming or divide and conquer may be better suited for certain problems.
Overall, greedy algorithms offer a powerful and versatile tool for solving optimization problems efficiently. By understanding their principles and characteristics, programmers and researchers can leverage greedy algorithms to tackle a wide range of computational challenges and design elegant solutions that balance simplicity and effectiveness.
The document discusses the greedy method algorithm design paradigm. It can be used to solve optimization problems with the greedy-choice property, where choosing locally optimal decisions at each step leads to a globally optimal solution. Examples discussed include fractional knapsack problem, task scheduling, and making change problem. The greedy algorithm works by always making the choice that looks best at the moment, without considering future implications of that choice.
The document discusses the greedy method algorithmic approach. It provides an overview of greedy algorithms including that they make locally optimal choices at each step to find a global optimal solution. The document also provides examples of problems that can be solved using greedy methods like job sequencing, the knapsack problem, finding minimum spanning trees, and single source shortest paths. It summarizes control flow and applications of greedy algorithms.
Dynamic programming (DP) is a powerful technique for solving optimization problems by breaking them down into overlapping subproblems and storing the results of already solved subproblems. The document provides examples of how DP can be applied to problems like rod cutting, matrix chain multiplication, and longest common subsequence. It explains the key elements of DP, including optimal substructure (subproblems can be solved independently and combined to solve the overall problem) and overlapping subproblems (subproblems are solved repeatedly).
The document discusses assignment problems and provides examples to illustrate how to solve them. Assignment problems involve allocating jobs to people or machines in a way that minimizes costs or maximizes profits. The key steps to solve assignment problems are: (1) construct a cost matrix, (2) perform row and column reductions to obtain zeros, (3) draw lines to cover zeros and determine optimal assignments. Traveling salesman problems, which involve finding the lowest cost route to visit all cities once, can also be formulated as assignment problems.
sublabel accurate convex relaxation of vectorial multilabel energiesFujimoto Keisuke
This document summarizes a presentation on the paper "Sublabel-Accurate Convex Relaxation of Vectorial Multilabel Energies". It discusses how the paper proposes a method to efficiently solve high-dimensional, nonlinear vectorial labeling problems by approximating them as convex problems. Specifically, it divides the problem domain into subregions and approximates each subregion with a convex function, yielding an overall approximation that is still non-convex but with higher accuracy. This lifting technique transforms the variables into a higher-dimensional space to formulate the data and regularization terms in a way that allows solving the problem as a convex optimization.
The document discusses implementing the Gauss-Jacobi iterative method to solve systems of linear equations. It begins by providing an overview of the Gauss-Jacobi method and its application to solve a sample system of 3 equations with 3 unknowns. It then compares the Gauss-Jacobi method to the Gauss-Seidel method, noting that Gauss-Seidel uses updated values in the current iteration while Gauss-Jacobi uses values from the previous iteration. The document concludes by providing C code to implement the Gauss-Jacobi method and listing its main advantages as being iterative, and its disadvantages as being inflexible and requiring large set-up time.
Backtracking is used to systematically search for solutions to problems by trying possibilities and abandoning ("backtracking") partial candidate solutions that fail to satisfy constraints. Examples discussed include the rat in a maze problem, the traveling salesperson problem, container loading, finding the maximum clique in a graph, and board permutation problems. Backtracking algorithms traverse the search space depth-first and use pruning techniques like discarding partial solutions that cannot lead to better complete solutions than those already found.
The MODI method is used to find the optimal solution to a transportation problem in 3 steps:
1) Obtain an initial basic feasible solution using the Matrix Minimum method
2) Evaluate unoccupied cells to find their opportunity costs by calculating implicit costs as the sum of dual variables for each row and column
3) Find the most negative opportunity cost and draw a closed path, then adjust quantities along the path to make an unoccupied cell occupied and recalculate, repeating until all costs are non-negative
1) The document discusses the Longest Increasing Subsequence (LIS) problem to find the longest subsequence of a given sequence where elements are in increasing order. It provides an example LIS of length 6 for a sample input array. A dynamic programming table is used to store the LIS value for each array element.
2) The problem of counting the number of ways to make change for an amount N using coins of values in S is discussed. A 2D dynamic programming table is used where one dimension tracks coins and the other tracks the change value.
3) The 0-1 Knapsack problem is described, to find the maximum value subset of items fitting in a knapsack of capacity
Or: Beyond linear.
Abstract: Equivariant neural networks are neural networks that incorporate symmetries. The nonlinear activation functions in these networks result in interesting nonlinear equivariant maps between simple representations, and motivate the key player of this talk: piecewise linear representation theory.
Disclaimer: No one is perfect, so please mind that there might be mistakes and typos.
dtubbenhauer@gmail.com
Corrected slides: dtubbenhauer.com/talks.html
When I was asked to give a companion lecture in support of ‘The Philosophy of Science’ (https://shorturl.at/4pUXz) I decided not to walk through the detail of the many methodologies in order of use. Instead, I chose to employ a long standing, and ongoing, scientific development as an exemplar. And so, I chose the ever evolving story of Thermodynamics as a scientific investigation at its best.
Conducted over a period of >200 years, Thermodynamics R&D, and application, benefitted from the highest levels of professionalism, collaboration, and technical thoroughness. New layers of application, methodology, and practice were made possible by the progressive advance of technology. In turn, this has seen measurement and modelling accuracy continually improved at a micro and macro level.
Perhaps most importantly, Thermodynamics rapidly became a primary tool in the advance of applied science/engineering/technology, spanning micro-tech, to aerospace and cosmology. I can think of no better a story to illustrate the breadth of scientific methodologies and applications at their best.
Mending Clothing to Support Sustainable Fashion_CIMaR 2024.pdfSelcen Ozturkcan
Ozturkcan, S., Berndt, A., & Angelakis, A. (2024). Mending clothing to support sustainable fashion. Presented at the 31st Annual Conference by the Consortium for International Marketing Research (CIMaR), 10-13 Jun 2024, University of Gävle, Sweden.
The binding of cosmological structures by massless topological defectsSérgio Sacani
Assuming spherical symmetry and weak field, it is shown that if one solves the Poisson equation or the Einstein field
equations sourced by a topological defect, i.e. a singularity of a very specific form, the result is a localized gravitational
field capable of driving flat rotation (i.e. Keplerian circular orbits at a constant speed for all radii) of test masses on a thin
spherical shell without any underlying mass. Moreover, a large-scale structure which exploits this solution by assembling
concentrically a number of such topological defects can establish a flat stellar or galactic rotation curve, and can also deflect
light in the same manner as an equipotential (isothermal) sphere. Thus, the need for dark matter or modified gravity theory is
mitigated, at least in part.
(June 12, 2024) Webinar: Development of PET theranostics targeting the molecu...Scintica Instrumentation
Targeting Hsp90 and its pathogen Orthologs with Tethered Inhibitors as a Diagnostic and Therapeutic Strategy for cancer and infectious diseases with Dr. Timothy Haystead.
hematic appreciation test is a psychological assessment tool used to measure an individual's appreciation and understanding of specific themes or topics. This test helps to evaluate an individual's ability to connect different ideas and concepts within a given theme, as well as their overall comprehension and interpretation skills. The results of the test can provide valuable insights into an individual's cognitive abilities, creativity, and critical thinking skills
PPT on Direct Seeded Rice presented at the three-day 'Training and Validation Workshop on Modules of Climate Smart Agriculture (CSA) Technologies in South Asia' workshop on April 22, 2024.
ESA/ACT Science Coffee: Diego Blas - Gravitational wave detection with orbita...Advanced-Concepts-Team
Presentation in the Science Coffee of the Advanced Concepts Team of the European Space Agency on the 07.06.2024.
Speaker: Diego Blas (IFAE/ICREA)
Title: Gravitational wave detection with orbital motion of Moon and artificial
Abstract:
In this talk I will describe some recent ideas to find gravitational waves from supermassive black holes or of primordial origin by studying their secular effect on the orbital motion of the Moon or satellites that are laser ranged.
EWOCS-I: The catalog of X-ray sources in Westerlund 1 from the Extended Weste...Sérgio Sacani
Context. With a mass exceeding several 104 M⊙ and a rich and dense population of massive stars, supermassive young star clusters
represent the most massive star-forming environment that is dominated by the feedback from massive stars and gravitational interactions
among stars.
Aims. In this paper we present the Extended Westerlund 1 and 2 Open Clusters Survey (EWOCS) project, which aims to investigate
the influence of the starburst environment on the formation of stars and planets, and on the evolution of both low and high mass stars.
The primary targets of this project are Westerlund 1 and 2, the closest supermassive star clusters to the Sun.
Methods. The project is based primarily on recent observations conducted with the Chandra and JWST observatories. Specifically,
the Chandra survey of Westerlund 1 consists of 36 new ACIS-I observations, nearly co-pointed, for a total exposure time of 1 Msec.
Additionally, we included 8 archival Chandra/ACIS-S observations. This paper presents the resulting catalog of X-ray sources within
and around Westerlund 1. Sources were detected by combining various existing methods, and photon extraction and source validation
were carried out using the ACIS-Extract software.
Results. The EWOCS X-ray catalog comprises 5963 validated sources out of the 9420 initially provided to ACIS-Extract, reaching a
photon flux threshold of approximately 2 × 10−8 photons cm−2
s
−1
. The X-ray sources exhibit a highly concentrated spatial distribution,
with 1075 sources located within the central 1 arcmin. We have successfully detected X-ray emissions from 126 out of the 166 known
massive stars of the cluster, and we have collected over 71 000 photons from the magnetar CXO J164710.20-455217.
EWOCS-I: The catalog of X-ray sources in Westerlund 1 from the Extended Weste...
Data structure notes
1. The greedy method
In the greedy method we attempt to construct an optimal solution in stages. At each
stage we make a decision that appears to be the best at that time ( optimal one ). A
decision made in one stage is not changed later and should assure feasibility ( i.e it must
satisfy all constraints ). It is called ‘greedy’ because it chooses the best one at that stage
without considering whether this will prove to be a sound decision in the long run.
General Method
Procedure GREEDY(A,n)
Solutionnull
for I 1 to n do
x SELECT(A)
if FEASIBLE(Solution,x)
then Solution UNION(Solution,x)
end if
repeat
return(Solution)
end GREEDY
A is a set of all possible inputs
SELECT function selects a input from A
FEASIBLE determines whether x can be included into the solution vector
UNION combines x with solution and updates the objective function
Example:
Suppose we are living in a place having coins of 1, 4, and 6 units we want to
make change of 8 units, we greedily select one 6 units and 2 one units instead of
selecting two 4 units which is a better solution.
Example:
Assume that we have a knapsack of capacity m and n objects having weight wi
and profit pi. Now we have to fill the knapsack with the object or fraction of it
in such a way that it brings maximum profit.
i.e maximize pi xi
xi fraction of object
pi profit for object i
subject to xi wi < = m
m total weight
eg:
Let m = 20, w1, w2, w3 = 18, 15, 10 and p1,p2,p3 =25,24,15
xi xi xi
wi xi pi xi
Comments
1 2/15 0 20 28.2 Increasing order of profit
0 2/3 1 20 31 Decreasing order of weight
0 1 1/2 20 31.5 Decreasing order of p/w ratio
2. From the above example we see that we can get the maximum profit by
arranging the elements in the decreasing order of p/w ratio.
Procedure GREEDY_KNAPSACK
(Assume the objects are arranged in the decreasing order of p/w ratio)
x 0
remweight m
for i 1 to n do
if w(i) > remweight then exit
end if
x(i) 1
remweight remweight – w(i)
repeat
if i < = n then x(i) remweight/w(i)
end if
end GREEDY_KNAPSACK
Efficiency in terms of time without considering time to sort is O(n)
x is an array representing solutions
remweight denotes remaining weight initialized to m first
Example 2:
Minimum spanning tree:
Given an n – vertex undirected network G having n-1 edges, our problem is to
select n-1 edges in such a way that selected edges form a least cost spanning
tree.
There are two different greedy techniques to solve this problem:
(1) Kruskals
(2) Prims
Kruskals Algorithmn:
General method:
From the remaining edges we select a least cost edge that does not result in a
cycle when added to the set of already selected edges.
Consider:
4
3
2
1
6
7
5
24
0
28
28
0
25
0
22
0
12
0
16
0
14
0
10
18
0
3. First arrange the edges in the ascending order of cost.
{1,6},{3,4},{2,6},{2,3},{7,4},{5,4},{5,7},{6,5},{1,2}
We will now strart constructing the shortest path according to Kruskal’s
algorithmn.
1. Add {1,6}
2. Add {3,4}
3. Add {2,7}
4
3
2
1
6
7
5
10
4
3
2
1
6
7
5 12
0
10
4
3
2
1
6
7
5 12
0
14
0
10
5. Function Kruskal
Sort all the edges by increasing order of weights
N number of weights
T null
Repeat
e {u,v} // Shortest edge not yet considered
ucomp find(u) //find(u) tells us to which component does u belong to
vcomp find(v)
if ucomp <> vcomp then //doesn’t form a cycle
merge(ucomp,vcomp)
T T U {e}
Until T contains n-1 edges
Return T
Time taken for this algorithm is O(alogn)
A number of edges
N number of nodes
Example:
From the remaining edges select a least cost edge whose addition to the set of
selected edges forms a tree.
Consider the following example as that of kruskals:
4
3
2
1
6
7
5
10
4
3
2
1
6
7
5
25
0
10
4
3
2
1
6
7
5
25
0
22
0
10
4
3
2
1
6
7
5
25
0
22
0
12
0
10
6. Efficiency of the algorithm is O(n2)
Algorithm PRIMS(E,COST,n,T,mincost)
(k,l) edge with minimum cost;
mincost COST(k,l)
T[1,1] k;
T[1,2] l;
For i 1 to n do
If COST(i,l) < COST(I,k) then
NEAR(i) l
Else
NEAR(i) k
End if
Repeat
NEAR(l) NEAR(l) 0
For i 2 to n-1 do
If NEAR(j) 0 and COST(j,NEAR(j)) is minimum
T[i,1] j;
T[I,2] NEAR(j);
mincost mincost + COST(j,NEAR(j))
NEAR(j) 0
For k1 to n do
If NEAR(k) 0 and COST(k,NEAR(k)) > COST(k,j)
Then NEAR(k) j
End if
Repeat
Repeat
If mincost then print (‘no spanning tree’) end if
End PRIM
4
3
2
1
6
7
5
25
0
22
0
12
0
16
0
10
4
3
2
1
6
7
5
25
0
22
0
12
0
16
0
14
0
10
7. Dynamic Programming method
It is a bottom up approach in which we avoid calculating the same thing twice by keeping
a table of known results that fills up as sub instances are solved. In greedy method we
make irrevocable decisions one at a time using greedy criteria but there here we examine
the decision sequence to see whether the optimal decision sequence contains optimal
decision subsequences. These optimal sequences of decisions are arrived by making use
of the PRINCIPLE OF OPTIMALITY.
This principle states that an optimal sequence of decisions has the property that whatever
the initial state and decisions are the remaining decisions must constitute an optimal
decision sequence with regard to the state resulting from the first decision.
Example:
Making changes:
Suppose we live in an area where there are coins for 1, 4, 6 units . If we have to
make change for 8 units the greedy algorithm will propose one 6 unit and two 1
unit coins making a total of three coins. The better solution is giving two 4 unit
coins.
To solve this problem by dynamic programming we assume a table [1..n,0..N]
One row for each denomination [1---n]
One column for each amount [1---N]
C[i,j] will be the minimum number of coins required to pay an amount
of j units using only denominations from 1 to i.
To pay an amount j using coins of denominations 1 to I we have 2
choices
1. First we may choose not to use any coins of that denominations i
that is C [i , j]=C[i-1,j]
2. we may choose to use atleast one coin of this denomination
hence
C[i,j]=one coin from this denomination+least number of
coins that make up the rest of the amount i.e j-di
= 1 + C[i,j-di]
In general C[i,j] = min( C[i-1,j],1+C[i,j-di])
i – denomination row
j – amount requiring change
8. AMOUNT/
DENOMINATION 0 1 2 3 4 5 6 7 8
d1=1 0 1 2 3 4 5 6 7 8
d2=4 0 1 2 3 1 2 3 4 2
d3=6 0 1 2 3 1 2 1 2 2
The above table is the cost matrix C[3,9]
C[i,j] – The number of coins needed to make change for j unit using
denominations less than or equal to dj.
From this table if we want to find out the number of minimum coins to make an
amount of 7 units with only 2 denominations then we look up at C[2,7] we get 4
coins one 4 unit coins and 3 one unit coins
C[1,0]=0
C[1,1]=C[1,0]+1=1 since C[0,0] is not possible we cannot consider min(C[0,0],1+C[1,0])
C[2,0]=C[1,0] = 0
C[2,6]=min(C[2,6-4]+1,C[1,6])=min(C[2,2]+1,C[1,6])=min(2+1,6)=3
C[3,8]=min(C[3,8-6]+1,C[2,8])=min(C[3,2]+1,C[2,8])=min(2+1,2)=2
Algorithm:
Function Coins(N)
(gives minimum number of coins to make change for N units)
for i 1 to n do C[i,0] 0 //for the amount 0 the change required
for i 1 to n do
for j 1 to N do
C [I,j] (if i =1 and j<d[i] then +
Else if i=1 then 1+C [1,j-d [1]]
Else if j<d [i] then C [i-1,j]
Else min(C [i-1,j],1+C[i,j-d[i]])
Return C[n,N]
The total time required for this algorithm is O(n, C[n,N])
n Number of denominations
N amount
Example:
0/1 Knapsack problem:
Assume that we have a knapsack of capacity m and n objects having weight wi
and profit pi. Now we have to fill the knapsack with the object or we discard it,
(We allow no fractions)t in such a way, that it brings maximum profit.
i.e maximize pi xi
xi 0 or 1
pi profit for object i
subject to xi wi < = m
m total weight
Similar to the previous problem we create a table V[1…n,0…w] with one row
for each available object and one column for each weight from 0 to w
9. The criterion for filling the table depends upon two choices
1. adding the object .
2. Neglecting the object.
Let us assume that we have five objects of weights 1, 2, 5, 6, and 7 units
having a profit of 1, 6, 18, 22, and 28 respectively. We have to fill a knapsack
in such a manner the total weight of the objects does not exceed the maximum
capacity of the knapsack i.e. 11 units.
CAPACITY
/WEIGHT 0 1 2 3 4 5 6 7 8 9 10 11
w1=1,p1=1 0 1 1 1 1 1 1 1 1 1 1 1
w2=2,p2=6 0 1 6 7 7 7 7 7 7 7 7 7
w3=5,p3=18 0 1 6 7 7 18 19 24 25 25 25 25
w4=6,p4=22 0 1 6 7 7 18 22 24 28 29 29 40
w5=7,p5=28 0 1 6 7 7 18 22 28 29 34 35 40
In the above table construction we have assumed the following formula
v[i, j] = max ( v[i-1, j], v[i-1, j-wi] + pi )
each v[i, j] = pi of selected objects
v[4, 7] = max ( v[3,7], v[3, 7-6]+22)=max ( 24,1+22)=24
In the above example it was more profitable to put one object of w3=5 and
one object of w2=2 to get a profit of p2+p3=18+6, rather than just putting in
one object of w4=6 and one object of w1 = 1 to get a profit of p4+p1=22+1.
Example:
Traveling salesman problem:
This problem deals with finding a tour of minimum cost covering all the nodes in
a weighted undirected graph, starting and ending at a particular node.
Similar to the previous examples, here also we make use of one function to deal
with the principle of optimality i.e.
g(i,S) = min { Cij+ g(j, S – {j})}
g( I, s) minimum cost of traveling from node i and covering all the nodes in
S and reaching back to the node i.
Cij denotes cost of going from node i to node j
S the set of all nodes except node i
S – {j} set of all nodes except nodes i and j
Suppose we want to find a minimum path from node 1 covering all the nodes
in set V and coming back to node 1, we denote it by
g(1,V-{1}) = min(cik+g(k,V-{1,k})} 2 k n
n number of nodes
10. Consider the cost Matrix of the salesman
Now let us try to find the minimum path starting at node 1 and ending at the same
node and covering all the other nodes.
g(1,{2, ,3, 4}) = min {C12+ g(2,{3, 4}),C13+g(3,{2,4}),C14+g(4,{2,3})}
= min (10+25,15+25,20+23)
= 35
g(2,{3, 4}) = min {C23+ g(3,{4}),C24+g(4,{3})}=min(9+20,10+15) =25
g(3,{2, 4}) = min {C32+ g(2,{4}),C34+g(4,{2})}=min(13+18,12+13)=25
g(4,{2, 3}) = min {C42+ g(2,{3}),C43+g(3,{2})}=min(8+15,9+18) =23
g(3,{4}) = C34+g{4,}=12+8 =20
g(4,{3}) = C43+g{3,}=9+6 =15
g(2,{4}) = C24+g{4,}=10+8 =18
g(4,{2}) = C42+g{2,}=8+5 =13
g(2,{3}) = C23+g{3,}=9+6 =15
g(3,{2}) = C32+g{2,}=13+5 =18
g{2,}=C21=5; g{3,}=C31=6; g{4,}=C41=8;
The required cost is g(1,{2,3,4}) = 35
Path is 1 2 4 3 1
Efficiency is in terms of time is O(n22n)
0 10 15 20
5 0 9 10
6 13 0 12
8 8 9 0
1
2
3
4
1 2 3 4
11. Backtracking
This method is based on the systematic examination of the possible solutions. We have
a procedure that looks thorough the set of possible solutions and the possible solutions
are rejected even before they are completely examined hence the number of possible
solutions are getting smaller. We reject solutions on the basis that certain solutions do not
fulfill certain requirements set beforehand. The name backtracking was coined by
Dr.Lehmer.
We have a finite set of solution space Si each element in the solution space is given as a
n - tuple (x1,x2,….,xn) and a certain set of constraints to be satisfied by any of the
solutions in the solution space.
Constraints are categorized into implicit and explicit constraints. Explicit constraints
are rules that restrict each xi to take on values from a given set. Eg: each xi > 0
Implicit constraints are rules that determine which tuple satisfy the criterion function.
Example:
N Queens problem:
N queens are to be placed on a n x n chessboard so that no two attack. Generally
we have n! Permutation possibilities.
Let us consider for n=4 we have 24 possibilities. We will construct a tree
structure to represent all the possibilities. In the following tree the 1st level is for
the first row in the chessboard and the 2nd level for the 2nd row and so on.
12. Q
Q
Q
Dead end Hence Backtrack Dead end hence Backtrack
Q
Q
Q
Q
Solution
Algorithm Nqueens(k,n)
{it gives all possible placements of n queens on an n x n Matrix)
{
for(I=1 to n do
{
if Place(k,I) then
{
x[k] := I
if(k==n) then Write(x[i:n])//Solution found all queens placed
else Nqueens(k+1,n)
}
}
}
Algorithm Place(k,I)
{returns true if a queen can be placed in k th row and i th column)
{
for j=1 to k-1 do
if (x[j] = I or (Abs(x[j] –1) = Abs(j-k)) then
return false
else return true;
}
Q
Q
13. The algorithm Nqueens can be invoked by calling NQueens(1,n)
To place a queen on the chessboard we have to check for three conditions
1. Must not be on the same row
2. Must not be in the same column
3. Must not remain in the same diagonal
Suppose 2 queens are placed at the positions (i,j) and (k,l) then
i-j=k-l
These computations are verified in the algorithm.
The computing time of the algorithm Place is O(k-1).
Example
Graph coloring:
Let G be a graph and m be a given positive integer.We have to color the nosdes of G in
such a way so that no two adjacent nodes have the same color, yet only m colors are
being used. Chromatic number indicates smallest integer m for which the graph can be
colored. Here we use backtracking technique to color a given graph using atmost m
colors.
Assume that our graph is represented by an adjacency matrix GRAPH(1…n,1..n) where
GRAPH(i,j)= true or 1 if thee exists an edge between node i and node j otherwise it is 0
or false. The colors are represented by integers 1….m .Solution is given by an array x[]
where x[I] gives the color of the node i.
Consider:
1 2
4
3
5
1
2
1
3
3
0 1 1 0 1
1 0 1 0 1
1 1 0 1 0
0 0 1 0 1
1 1 0 1 0
1
2
3
4
5
1 2 3 4 5
14. The Graph in the figure can be colored by 3 colos as indicated ihn the figue.
Solution is:
x[1]=1, x[2]=2, x[3]=3, x[4]=1, x[5]=3
Algorithm mcoloing(k)
(k ios the index of the next vector to be colored)
{
}