The document provides information on solving the sum of subsets problem using backtracking. It discusses two formulations - one where solutions are represented by tuples indicating which numbers are included, and another where each position indicates if the corresponding number is included or not. It shows the state space tree that represents all possible solutions for each formulation. The tree is traversed depth-first to find all solutions where the sum of the included numbers equals the target sum. Pruning techniques are used to avoid exploring non-promising paths.
This document summarizes the n-queen problem, which involves placing N queens on an N x N chessboard so that no queen can attack any other. It describes the problem's inputs and tasks, provides examples of solutions for different board sizes, and outlines the backtracking algorithm commonly used to solve this problem. The backtracking approach guarantees a solution but can be slow, with complexity rising exponentially with problem size. It is a good benchmark for testing parallel computing systems due to its iterative nature.
This document discusses various problems that can be solved using backtracking, including graph coloring, the Hamiltonian cycle problem, the subset sum problem, the n-queen problem, and map coloring. It provides examples of how backtracking works by constructing partial solutions and evaluating them to find valid solutions or determine dead ends. Key terms like state-space trees and promising vs non-promising states are introduced. Specific examples are given for problems like placing 4 queens on a chessboard and coloring a map of Australia.
The document discusses Strassen's algorithm for matrix multiplication. It begins by explaining traditional matrix multiplication that has a time complexity of O(n3). It then explains how the divide and conquer strategy can be applied by dividing the matrices into smaller square sub-matrices. Strassen improved upon this by reducing the number of multiplications from 8 to 7 terms, obtaining a time complexity of O(n2.81). His key insight was applying different equations on the sub-matrix multiplication formulas to minimize operations.
The document discusses various backtracking techniques including bounding functions, promising functions, and pruning to avoid exploring unnecessary paths. It provides examples of problems that can be solved using backtracking including n-queens, graph coloring, Hamiltonian circuits, sum-of-subsets, 0-1 knapsack. Search techniques for backtracking problems include depth-first search (DFS), breadth-first search (BFS), and best-first search combined with branch-and-bound pruning.
The document discusses the 0/1 knapsack problem and dynamic programming algorithm to solve it. The 0/1 knapsack problem involves selecting a subset of items to pack in a knapsack that maximizes the total value without exceeding the knapsack's weight capacity. The dynamic programming algorithm solves this by building up a table where each entry represents the maximum value for a given weight. It iterates through items, checking if including each item increases the maximum value for that weight.
This presentation discusses the knapsack problem and its two main versions: 0/1 and fractional. The 0/1 knapsack problem involves indivisible items that are either fully included or not included, and is solved using dynamic programming. The fractional knapsack problem allows items to be partially included, and is solved using a greedy algorithm. Examples are provided of solving each version using their respective algorithms. The time complexity of these algorithms is also presented. Real-world applications of the knapsack problem include cutting raw materials and selecting investments.
PPT on Analysis Of Algorithms.
The ppt includes Algorithms,notations,analysis,analysis of algorithms,theta notation, big oh notation, omega notation, notation graphs
This document summarizes the n-queen problem, which involves placing N queens on an N x N chessboard so that no queen can attack any other. It describes the problem's inputs and tasks, provides examples of solutions for different board sizes, and outlines the backtracking algorithm commonly used to solve this problem. The backtracking approach guarantees a solution but can be slow, with complexity rising exponentially with problem size. It is a good benchmark for testing parallel computing systems due to its iterative nature.
This document discusses various problems that can be solved using backtracking, including graph coloring, the Hamiltonian cycle problem, the subset sum problem, the n-queen problem, and map coloring. It provides examples of how backtracking works by constructing partial solutions and evaluating them to find valid solutions or determine dead ends. Key terms like state-space trees and promising vs non-promising states are introduced. Specific examples are given for problems like placing 4 queens on a chessboard and coloring a map of Australia.
The document discusses Strassen's algorithm for matrix multiplication. It begins by explaining traditional matrix multiplication that has a time complexity of O(n3). It then explains how the divide and conquer strategy can be applied by dividing the matrices into smaller square sub-matrices. Strassen improved upon this by reducing the number of multiplications from 8 to 7 terms, obtaining a time complexity of O(n2.81). His key insight was applying different equations on the sub-matrix multiplication formulas to minimize operations.
The document discusses various backtracking techniques including bounding functions, promising functions, and pruning to avoid exploring unnecessary paths. It provides examples of problems that can be solved using backtracking including n-queens, graph coloring, Hamiltonian circuits, sum-of-subsets, 0-1 knapsack. Search techniques for backtracking problems include depth-first search (DFS), breadth-first search (BFS), and best-first search combined with branch-and-bound pruning.
The document discusses the 0/1 knapsack problem and dynamic programming algorithm to solve it. The 0/1 knapsack problem involves selecting a subset of items to pack in a knapsack that maximizes the total value without exceeding the knapsack's weight capacity. The dynamic programming algorithm solves this by building up a table where each entry represents the maximum value for a given weight. It iterates through items, checking if including each item increases the maximum value for that weight.
This presentation discusses the knapsack problem and its two main versions: 0/1 and fractional. The 0/1 knapsack problem involves indivisible items that are either fully included or not included, and is solved using dynamic programming. The fractional knapsack problem allows items to be partially included, and is solved using a greedy algorithm. Examples are provided of solving each version using their respective algorithms. The time complexity of these algorithms is also presented. Real-world applications of the knapsack problem include cutting raw materials and selecting investments.
PPT on Analysis Of Algorithms.
The ppt includes Algorithms,notations,analysis,analysis of algorithms,theta notation, big oh notation, omega notation, notation graphs
The document discusses the analysis of algorithms. It begins by defining an algorithm and describing different types. It then covers analyzing algorithms in terms of correctness, time efficiency, space efficiency, and optimality through theoretical and empirical analysis. The document discusses analyzing time efficiency by determining the number of repetitions of basic operations as a function of input size. It provides examples of input size, basic operations, and formulas for counting operations. It also covers analyzing best, worst, and average cases and establishes asymptotic efficiency classes. The document then analyzes several examples of non-recursive and recursive algorithms.
This is a short presentation on Vertex Cover Problem for beginners in the field of Graph Theory...
Download the presentation for a better experience...
The document discusses various activation functions used in deep learning neural networks including sigmoid, tanh, ReLU, LeakyReLU, ELU, softmax, swish, maxout, and softplus. For each activation function, the document provides details on how the function works and lists pros and cons. Overall, the document provides an overview of common activation functions and considerations for choosing an activation function for different types of deep learning problems.
The document discusses various search algorithms including greedy search, A* search, and their application to problems like the knapsack problem. It provides an example of using a greedy approach to solve the fractional knapsack problem by selecting items to pack based on their value per unit weight. It also describes how A* search works by evaluating nodes using an f(n) function combining the actual cost to reach a node and the estimated cost to the goal.
This document discusses the N-Queens problem, which involves placing N chess queens on an N×N chessboard so that no two queens attack each other. It provides an overview of the problem statement, history, backtracking algorithm used to solve it, data flow diagram, pseudocode, sample outputs, and advantages of the backtracking approach. The backtracking algorithm places queens column-by-column, checking for valid placements and backtracking when it reaches an invalid configuration. The time complexity increases exponentially with board size N as the number of possible solutions grows.
The document discusses three classes of decision problems:
1) P problems that can be solved quickly in polynomial time.
2) NP problems where a "YES" answer has a proof checkable in polynomial time.
3) co-NP problems where a "NO" answer has a proof checkable in polynomial time.
It then defines NP-Complete problems as the hardest problems in NP, and explains that 3SAT is a famous NP-Complete problem involving finding a variable assignment that satisfies a Boolean formula of clauses with 3 variables each. The document provides methods for proving other problems like Clique and Independent Set are also NP-Complete by reducing 3SAT to them in polynomial time.
The document discusses time and space complexity analysis of algorithms. Time complexity measures the number of steps to solve a problem based on input size, with common orders being O(log n), O(n), O(n log n), O(n^2). Space complexity measures memory usage, which can be reused unlike time. Big O notation describes asymptotic growth rates to compare algorithm efficiencies, with constant O(1) being best and exponential O(c^n) being worst.
Introduction to Recurrent Neural NetworkKnoldus Inc.
The document provides an introduction to recurrent neural networks (RNNs). It discusses how RNNs differ from feedforward neural networks in that they have internal memory and can use their output from the previous time step as input. This allows RNNs to process sequential data like time series. The document outlines some common RNN types and explains the vanishing gradient problem that can occur in RNNs due to multiplication of small gradient values over many time steps. It discusses solutions to this problem like LSTMs and techniques like weight initialization and gradient clipping.
this is a briefer overview about the Big O Notation. Big O Notaion are useful to check the Effeciency of an algorithm and to check its limitation at higher value. with big o notation some examples are also shown about its cases and some functions in c++ are also described.
BackTracking Algorithm: Technique and ExamplesFahim Ferdous
This slides gives a strong overview of backtracking algorithm. How it came and general approaches of the techniques. Also some well-known problem and solution of backtracking algorithm.
Algorithm And analysis Lecture 03& 04-time complexity.Tariq Khan
This document discusses algorithm efficiency and complexity analysis. It defines key terms like algorithms, asymptotic complexity, Big O notation, and different complexity classes. It provides examples of analyzing time complexity for different algorithms like loops, nested loops, and recursive functions. The document explains that Big O notation allows analyzing algorithms independent of machine or input by focusing on the highest order term as the problem size increases. Overall, the document introduces methods for measuring an algorithm's efficiency and analyzing its time and space complexity asymptotically.
The document discusses gradient descent methods for unconstrained convex optimization problems. It introduces gradient descent as an iterative method to find the minimum of a differentiable function by taking steps proportional to the negative gradient. It describes the basic gradient descent update rule and discusses convergence conditions such as Lipschitz continuity, strong convexity, and condition number. It also covers techniques like exact line search, backtracking line search, coordinate descent, and steepest descent methods.
NP completeness. Classes P and NP are two frequently studied classes of problems in computer science. Class P is the set of all problems that can be solved by a deterministic Turing machine in polynomial time.
The document discusses the divide-and-conquer algorithm design paradigm. It explains that a problem is divided into smaller subproblems, the subproblems are solved independently, and then the solutions are combined. Recurrence equations can be used to analyze the running time of divide-and-conquer algorithms. The document provides examples of solving recurrences using methods like the recursion tree method and the master theorem.
Local beam search begins with k randomly generated states and generates the successors of those states at each step. It selects the k best successors based on a heuristic and abandons the others, repeating this process to quickly focus on the most promising searches. A stochastic version chooses successors probabilistically based on their goodness. The search assumes a fixed beam width and performs breadth-first exploration while only keeping the best new nodes at each level according to the heuristic.
Problem-Solving Strategies in Artificial Intelligence" delves into the core techniques and methods employed by AI systems to address complex problems. This exploration covers the two main categories of search strategies: uninformed and informed, revealing how they navigate the solution space. It also investigates the use of heuristics, which provide a shortcut for guiding the search, and local search algorithms' role in tackling optimization problems. The description offers insights into the critical concepts and strategies that power AI's ability to find solutions efficiently and effectively in various domains.
In "Problem-Solving Strategies in Artificial Intelligence," we dive deeper into the foundational techniques and methodologies that AI systems rely on to tackle challenging problems. This comprehensive exploration begins with an in-depth examination of search strategies. Uninformed search strategies, often referred to as blind searches, are dissected, along with informed search strategies that harness domain-specific knowledge and heuristics to guide the search process more intelligently.
The role of heuristics in AI problem-solving is thoroughly investigated. These problem-solving techniques employ domain-specific rules of thumb to estimate the quality of potential solutions, aiding in decision-making and prioritization. The famous A* search algorithm, which combines actual cost and heuristic estimation, is highlighted as a prime example of informed search.
Local search algorithms, another critical component, are discussed in the context of optimization problems. These algorithms excel in finding the best solution within a local neighborhood of the current solution and are particularly valuable for various optimization challenges. You'll explore methods like hill climbing and simulated annealing, which are vital for optimizing solutions in constrained problem spaces.
This insightful exploration provides a comprehensive understanding of the problem-solving strategies employed in AI, offering a solid foundation for those seeking to apply AI techniques to real-world challenges and further the field of artificial intelligence.
Talk on Optimization for Deep Learning, which gives an overview of gradient descent optimization algorithms and highlights some current research directions.
This document discusses constraint satisfaction problems (CSPs). It defines CSPs as problems with variables that must satisfy constraints. CSPs can represent many real-world problems and are solved through constraint satisfaction methods. The document outlines CSP components like variables, domains, and constraints. It also describes representing problems as CSPs, solving CSPs through backtracking search, and the role of heuristics like minimum remaining values in improving the search process.
The document discusses backtracking methodology for solving problems. It involves representing solutions as n-tuples and searching the solution space tree using depth-first search. Nodes are marked as live, dead, or the current node being expanded (E-node). Bounding functions help prune portions of the tree and avoid searching invalid or non-optimal subtrees. The 4-Queens problem is used as an example to illustrate backtracking.
Backtracking and branch and bound are algorithms used to solve problems with large search spaces. Backtracking uses depth-first search and prunes subtrees that don't lead to viable solutions. Branch and bound uses breadth-first search and pruning, maintaining partial solutions in a priority queue. Both techniques systematically eliminate possibilities to find optimal solutions faster than exhaustive search. Examples where they can be applied include maze pathfinding, the eight queens problem, sudoku, and the traveling salesman problem.
The document discusses the analysis of algorithms. It begins by defining an algorithm and describing different types. It then covers analyzing algorithms in terms of correctness, time efficiency, space efficiency, and optimality through theoretical and empirical analysis. The document discusses analyzing time efficiency by determining the number of repetitions of basic operations as a function of input size. It provides examples of input size, basic operations, and formulas for counting operations. It also covers analyzing best, worst, and average cases and establishes asymptotic efficiency classes. The document then analyzes several examples of non-recursive and recursive algorithms.
This is a short presentation on Vertex Cover Problem for beginners in the field of Graph Theory...
Download the presentation for a better experience...
The document discusses various activation functions used in deep learning neural networks including sigmoid, tanh, ReLU, LeakyReLU, ELU, softmax, swish, maxout, and softplus. For each activation function, the document provides details on how the function works and lists pros and cons. Overall, the document provides an overview of common activation functions and considerations for choosing an activation function for different types of deep learning problems.
The document discusses various search algorithms including greedy search, A* search, and their application to problems like the knapsack problem. It provides an example of using a greedy approach to solve the fractional knapsack problem by selecting items to pack based on their value per unit weight. It also describes how A* search works by evaluating nodes using an f(n) function combining the actual cost to reach a node and the estimated cost to the goal.
This document discusses the N-Queens problem, which involves placing N chess queens on an N×N chessboard so that no two queens attack each other. It provides an overview of the problem statement, history, backtracking algorithm used to solve it, data flow diagram, pseudocode, sample outputs, and advantages of the backtracking approach. The backtracking algorithm places queens column-by-column, checking for valid placements and backtracking when it reaches an invalid configuration. The time complexity increases exponentially with board size N as the number of possible solutions grows.
The document discusses three classes of decision problems:
1) P problems that can be solved quickly in polynomial time.
2) NP problems where a "YES" answer has a proof checkable in polynomial time.
3) co-NP problems where a "NO" answer has a proof checkable in polynomial time.
It then defines NP-Complete problems as the hardest problems in NP, and explains that 3SAT is a famous NP-Complete problem involving finding a variable assignment that satisfies a Boolean formula of clauses with 3 variables each. The document provides methods for proving other problems like Clique and Independent Set are also NP-Complete by reducing 3SAT to them in polynomial time.
The document discusses time and space complexity analysis of algorithms. Time complexity measures the number of steps to solve a problem based on input size, with common orders being O(log n), O(n), O(n log n), O(n^2). Space complexity measures memory usage, which can be reused unlike time. Big O notation describes asymptotic growth rates to compare algorithm efficiencies, with constant O(1) being best and exponential O(c^n) being worst.
Introduction to Recurrent Neural NetworkKnoldus Inc.
The document provides an introduction to recurrent neural networks (RNNs). It discusses how RNNs differ from feedforward neural networks in that they have internal memory and can use their output from the previous time step as input. This allows RNNs to process sequential data like time series. The document outlines some common RNN types and explains the vanishing gradient problem that can occur in RNNs due to multiplication of small gradient values over many time steps. It discusses solutions to this problem like LSTMs and techniques like weight initialization and gradient clipping.
this is a briefer overview about the Big O Notation. Big O Notaion are useful to check the Effeciency of an algorithm and to check its limitation at higher value. with big o notation some examples are also shown about its cases and some functions in c++ are also described.
BackTracking Algorithm: Technique and ExamplesFahim Ferdous
This slides gives a strong overview of backtracking algorithm. How it came and general approaches of the techniques. Also some well-known problem and solution of backtracking algorithm.
Algorithm And analysis Lecture 03& 04-time complexity.Tariq Khan
This document discusses algorithm efficiency and complexity analysis. It defines key terms like algorithms, asymptotic complexity, Big O notation, and different complexity classes. It provides examples of analyzing time complexity for different algorithms like loops, nested loops, and recursive functions. The document explains that Big O notation allows analyzing algorithms independent of machine or input by focusing on the highest order term as the problem size increases. Overall, the document introduces methods for measuring an algorithm's efficiency and analyzing its time and space complexity asymptotically.
The document discusses gradient descent methods for unconstrained convex optimization problems. It introduces gradient descent as an iterative method to find the minimum of a differentiable function by taking steps proportional to the negative gradient. It describes the basic gradient descent update rule and discusses convergence conditions such as Lipschitz continuity, strong convexity, and condition number. It also covers techniques like exact line search, backtracking line search, coordinate descent, and steepest descent methods.
NP completeness. Classes P and NP are two frequently studied classes of problems in computer science. Class P is the set of all problems that can be solved by a deterministic Turing machine in polynomial time.
The document discusses the divide-and-conquer algorithm design paradigm. It explains that a problem is divided into smaller subproblems, the subproblems are solved independently, and then the solutions are combined. Recurrence equations can be used to analyze the running time of divide-and-conquer algorithms. The document provides examples of solving recurrences using methods like the recursion tree method and the master theorem.
Local beam search begins with k randomly generated states and generates the successors of those states at each step. It selects the k best successors based on a heuristic and abandons the others, repeating this process to quickly focus on the most promising searches. A stochastic version chooses successors probabilistically based on their goodness. The search assumes a fixed beam width and performs breadth-first exploration while only keeping the best new nodes at each level according to the heuristic.
Problem-Solving Strategies in Artificial Intelligence" delves into the core techniques and methods employed by AI systems to address complex problems. This exploration covers the two main categories of search strategies: uninformed and informed, revealing how they navigate the solution space. It also investigates the use of heuristics, which provide a shortcut for guiding the search, and local search algorithms' role in tackling optimization problems. The description offers insights into the critical concepts and strategies that power AI's ability to find solutions efficiently and effectively in various domains.
In "Problem-Solving Strategies in Artificial Intelligence," we dive deeper into the foundational techniques and methodologies that AI systems rely on to tackle challenging problems. This comprehensive exploration begins with an in-depth examination of search strategies. Uninformed search strategies, often referred to as blind searches, are dissected, along with informed search strategies that harness domain-specific knowledge and heuristics to guide the search process more intelligently.
The role of heuristics in AI problem-solving is thoroughly investigated. These problem-solving techniques employ domain-specific rules of thumb to estimate the quality of potential solutions, aiding in decision-making and prioritization. The famous A* search algorithm, which combines actual cost and heuristic estimation, is highlighted as a prime example of informed search.
Local search algorithms, another critical component, are discussed in the context of optimization problems. These algorithms excel in finding the best solution within a local neighborhood of the current solution and are particularly valuable for various optimization challenges. You'll explore methods like hill climbing and simulated annealing, which are vital for optimizing solutions in constrained problem spaces.
This insightful exploration provides a comprehensive understanding of the problem-solving strategies employed in AI, offering a solid foundation for those seeking to apply AI techniques to real-world challenges and further the field of artificial intelligence.
Talk on Optimization for Deep Learning, which gives an overview of gradient descent optimization algorithms and highlights some current research directions.
This document discusses constraint satisfaction problems (CSPs). It defines CSPs as problems with variables that must satisfy constraints. CSPs can represent many real-world problems and are solved through constraint satisfaction methods. The document outlines CSP components like variables, domains, and constraints. It also describes representing problems as CSPs, solving CSPs through backtracking search, and the role of heuristics like minimum remaining values in improving the search process.
The document discusses backtracking methodology for solving problems. It involves representing solutions as n-tuples and searching the solution space tree using depth-first search. Nodes are marked as live, dead, or the current node being expanded (E-node). Bounding functions help prune portions of the tree and avoid searching invalid or non-optimal subtrees. The 4-Queens problem is used as an example to illustrate backtracking.
Backtracking and branch and bound are algorithms used to solve problems with large search spaces. Backtracking uses depth-first search and prunes subtrees that don't lead to viable solutions. Branch and bound uses breadth-first search and pruning, maintaining partial solutions in a priority queue. Both techniques systematically eliminate possibilities to find optimal solutions faster than exhaustive search. Examples where they can be applied include maze pathfinding, the eight queens problem, sudoku, and the traveling salesman problem.
The document discusses the 8-Queens problem, which aims to place 8 queens on an 8x8 chessboard so that no two queens attack each other. It outlines the objective and constraints of ensuring no two queens are in the same row, column, or diagonal. Possible solutions are presented, along with a backtracking algorithm to systematically place queens one by one until a solution is found or no options remain.
This document discusses the 0/1 knapsack problem and how it can be solved using backtracking. It begins with an introduction to backtracking and the difference between backtracking and branch and bound. It then discusses the knapsack problem, giving the definitions of the profit vector, weight vector, and knapsack capacity. It explains how the problem is to find the combination of items that achieves the maximum total value without exceeding the knapsack capacity. The document constructs state space trees to demonstrate solving the knapsack problem using backtracking and fixed tuples. It concludes with examples problems and references.
Backtracking is a technique for solving problems by incrementally building candidates to the solutions, and abandoning each partial candidate ("backtracking") as soon as it is determined that the candidate cannot possibly be completed to a valid solution. It is useful for problems with constraints or complex conditions that are difficult to test incrementally. The key steps are: 1) systematically generate potential solutions; 2) test if a solution is complete and satisfies all constraints; 3) if not, backtrack and vary the previous choice. Backtracking has been used to solve problems like the N-queens puzzle, maze generation, Sudoku puzzles, and finding Hamiltonian cycles in graphs.
Backtracking and branch and bound are algorithms for solving problems systematically by trying options in an orderly manner. Backtracking uses depth-first search and prunes subtrees that don't lead to solutions. Branch and bound uses breadth-first search and pruning, maintaining upper and lower bounds to eliminate options. Both aim to avoid exhaustive search by eliminating non-promising options early. Examples that can use these techniques include maze navigation, the eight queens problem, and Sudoku puzzles.
This document discusses solving Sudoku puzzles using constraint satisfaction techniques. It presents the objectives, which are to examine Sudoku as a constraint satisfaction problem and determine if puzzle symmetry affects solving time. It then provides details on constraint satisfaction problems, Sudoku rules, proposed solutions using backtracking search and forward checking, evaluation of solving times for different puzzle types and analysis of results. Symmetric puzzles were found to solve faster than asymmetric ones due to providing more constraints.
The document discusses the 8 queens problem and how backtracking can be used to solve it. The 8 queens problem aims to place 8 queens on a chessboard so that no two queens attack each other. Backtracking is an algorithm that builds candidate solutions incrementally and abandons partial solutions ("backtracks") that cannot be completed. It explains that backtracking works by placing queens in columns, removing placements that lead to conflicts, and backtracking to try other placements. The document also provides the number of solutions for placing different numbers of queens on boards of corresponding sizes.
Backtracking is an algorithmic technique for solving problems recursively by trying to build candidates for the solutions incrementally, and abandoning each partial candidate ("backtracking") as soon as it is determined that the candidate cannot possibly be completed to a valid solution. It is useful for constraint satisfaction problems and involves systematically trying choices until finding one that "works". The eight queens problem, which involves placing eight queens on a chessboard so that none attack each other, is a classic example solved using backtracking.
The document outlines Module 5 which covers backtracking, branch and bound, and NP problems. It discusses backtracking techniques like N-Queens problem and graph coloring. Branch and bound is presented as a more intelligent variation of backtracking to solve optimization problems. Examples covered are assignment problem, travelling salesperson problem (TSP), and 0/1 knapsack. NP-complete and NP-hard problems are also introduced.
The document discusses backtracking as a general method for solving problems that involve systematically trying possibilities and abandoning partial solutions when they cannot possibly lead to a complete solution. It provides examples of applications of backtracking including the n-queens problem, sum of subsets problem, graph coloring, and finding Hamiltonian cycles in graphs. The general backtracking algorithm and terminology used are described. Specific algorithms for solving the n-queens problem and sum of subsets problem using backtracking are also presented.
Data Analysis and Algorithms Lecture 1: IntroductionTayyabSattar5
This document outlines a course on design and analysis of algorithms. It covers topics like algorithm complexity analysis using growth functions, classic algorithm problems like the traveling salesperson problem, and algorithm design techniques like divide-and-conquer, greedy algorithms, and dynamic programming. Example algorithms and problems are provided for each topic. Reference books on algorithms are also listed.
Analysis & Design of Algorithms
Backtracking
N-Queens Problem
Hamiltonian circuit
Graph coloring
A presentation on unit Backtracking from the ADA subject of Engineering.
This document describes a project on solving the 8 queens problem using object-oriented programming in C++. It includes an introduction to the 8 queens puzzle, a methodology section on the backtracking algorithm used, pseudocode for the algorithm, analysis of the time complexity, a flowchart, results and discussion of the 12 fundamental solutions, and the source code. It was completed by 5 students under the guidance of a professor to fulfill the requirements for a bachelor's degree in computer science and engineering.
Backtracking is an algorithmic-technique for solving problems recursively by trying to build a solution incrementally, one piece at a time, removing those solutions that fail to satisfy the constraints of the problem at any point of time
This document discusses backtracking as an algorithm design technique. It provides examples of problems that can be solved using backtracking, including the 8 queens problem, sum of subsets, graph coloring, Hamiltonian cycles, and the knapsack problem. It also provides pseudocode for general backtracking algorithms and algorithms for specific problems solved through backtracking.
Undecidable Problems - COPING WITH THE LIMITATIONS OF ALGORITHM POWERmuthukrishnavinayaga
This document discusses algorithms and their analysis. It begins by defining key properties of algorithms like their lower, upper, and tight bounds. It then discusses different techniques for determining algorithm lower bounds such as trivial, information theoretical, adversary, and reduction arguments. Decision trees are presented as a model for representing algorithms that use comparisons. Lower bounds proofs are given for sorting and searching algorithms. The document also covers polynomial time versus non-polynomial time problems, as well as NP-complete problems. Specific algorithms are analyzed like knapsack, traveling salesman, and approximation algorithms.
The document discusses various algorithms that can be solved using backtracking. It begins by defining backtracking as a general algorithm design technique for problems that involve searching for solutions satisfying constraints. It then provides examples of problems that can be solved using backtracking, including the 8 queens problem, sum of subsets, graph coloring, and finding Hamiltonian cycles in a graph. For each problem, it outlines the key steps and provides pseudocode for the backtracking algorithm.
Similar to module5_backtrackingnbranchnbound_2022.pdf (20)
The document provides an introduction to unsupervised learning and reinforcement learning. It then discusses eigen values and eigen vectors, showing how to calculate them from a matrix. It provides examples of covariance matrices and using Gaussian elimination to solve for eigen vectors. Finally, it discusses principal component analysis and different clustering algorithms like K-means clustering.
Cross validation is a technique for evaluating machine learning models by splitting the dataset into training and validation sets and training the model multiple times on different splits, to reduce variance. K-fold cross validation splits the data into k equally sized folds, where each fold is used once for validation while the remaining k-1 folds are used for training. Leave-one-out cross validation uses a single observation from the dataset as the validation set. Stratified k-fold cross validation ensures each fold has the same class proportions as the full dataset. Grid search evaluates all combinations of hyperparameters specified as a grid, while randomized search samples hyperparameters randomly within specified ranges. Learning curves show training and validation performance as a function of training set size and can diagnose underfitting
This document provides an overview of supervised machine learning algorithms for classification, including logistic regression, k-nearest neighbors (KNN), support vector machines (SVM), and decision trees. It discusses key concepts like evaluation metrics, performance measures, and use cases. For logistic regression, it covers the mathematics behind maximum likelihood estimation and gradient descent. For KNN, it explains the algorithm and discusses distance metrics and a numerical example. For SVM, it outlines the concept of finding the optimal hyperplane that maximizes the margin between classes.
The document discusses the greedy method and its applications. It begins by defining the greedy approach for optimization problems, noting that greedy algorithms make locally optimal choices at each step in hopes of finding a global optimum. Some applications of the greedy method include the knapsack problem, minimum spanning trees using Kruskal's and Prim's algorithms, job sequencing with deadlines, and finding the shortest path using Dijkstra's algorithm. The document then focuses on explaining the fractional knapsack problem and providing a step-by-step example of solving it using a greedy approach. It also provides examples and explanations of Kruskal's algorithm for finding minimum spanning trees.
The document describes various divide and conquer algorithms including binary search, merge sort, quicksort, and finding maximum and minimum elements. It begins by explaining the general divide and conquer approach of dividing a problem into smaller subproblems, solving the subproblems independently, and combining the solutions. Several examples are then provided with pseudocode and analysis of their divide and conquer implementations. Key algorithms covered in the document include binary search (log n time), merge sort (n log n time), and quicksort (n log n time on average).
What is an Algorithm
Time Complexity
Space Complexity
Asymptotic Notations
Recursive Analysis
Selection Sort
Insertion Sort
Recurrences
Substitution Method
Master Tree Method
Recursion Tree Method
This document provides an outline for a machine learning syllabus. It includes 14 modules covering topics like machine learning terminology, supervised and unsupervised learning algorithms, optimization techniques, and projects. It lists software and hardware requirements for the course. It also discusses machine learning applications, issues, and the steps to build a machine learning model.
The document discusses problem-solving agents and their approach to solving problems. Problem-solving agents (1) formulate a goal based on the current situation, (2) formulate the problem by defining relevant states and actions, and (3) search for a solution by exploring sequences of actions that lead to the goal state. Several examples of problems are provided, including the 8-puzzle, robotic assembly, the 8 queens problem, and the missionaries and cannibals problem. For each problem, the relevant states, actions, goal tests, and path costs are defined.
The simplex method is a linear programming algorithm that can solve problems with more than two decision variables. It works by generating a series of solutions, called tableaus, where each tableau corresponds to a corner point of the feasible solution space. The algorithm starts at the initial tableau, which corresponds to the origin. It then shifts to adjacent corner points, moving in the direction that optimizes the objective function. This process of generating new tableaus continues until an optimal solution is found.
The document discusses functions and the pigeonhole principle. It defines what a function is, how functions can be represented graphically and with tables and ordered pairs. It covers one-to-one, onto, and bijective functions. It also discusses function composition, inverse functions, and the identity function. The pigeonhole principle states that if n objects are put into m containers where n > m, then at least one container must hold more than one object. Examples are given to illustrate how to apply the principle to problems involving months, socks, and selecting numbers.
The document discusses relations and their representations. It defines a binary relation as a subset of A×B where A and B are nonempty sets. Relations can be represented using arrow diagrams, directed graphs, and zero-one matrices. A directed graph represents the elements of A as vertices and draws an edge from vertex a to b if aRb. The zero-one matrix representation assigns 1 to the entry in row a and column b if (a,b) is in the relation, and 0 otherwise. The document also discusses indegrees, outdegrees, composite relations, and properties of relations like reflexivity.
This document discusses logic and propositional logic. It covers the following topics:
- The history and applications of logic.
- Different types of statements and their grammar.
- Propositional logic including symbols, connectives, truth tables, and semantics.
- Quantifiers, universal and existential quantification, and properties of quantifiers.
- Normal forms such as disjunctive normal form and conjunctive normal form.
- Inference rules and the principle of mathematical induction, illustrated with examples.
1. Set theory is an important mathematical concept and tool that is used in many areas including programming, real-world applications, and computer science problems.
2. The document introduces some basic concepts of set theory including sets, members, operations on sets like union and intersection, and relationships between sets like subsets and complements.
3. Infinite sets are discussed as well as different types of infinite sets including countably infinite and uncountably infinite sets. Special sets like the empty set and power sets are also covered.
The document discusses uncertainty and probabilistic reasoning. It describes sources of uncertainty like partial information, unreliable information, and conflicting information from multiple sources. It then discusses representing and reasoning with uncertainty using techniques like default logic, rules with probabilities, and probability theory. The key approaches covered are conditional probability, independence, conditional independence, and using Bayes' rule to update probabilities based on new evidence.
The document outlines the objectives, outcomes, and learning outcomes of a course on artificial intelligence. The objectives include conceptualizing ideas and techniques for intelligent systems, understanding mechanisms of intelligent thought and action, and understanding advanced representation and search techniques. Outcomes include developing an understanding of AI building blocks, choosing appropriate problem solving methods, analyzing strengths and weaknesses of AI approaches, and designing models for reasoning with uncertainty. Learning outcomes include knowledge, intellectual skills, practical skills, and transferable skills in artificial intelligence.
Planning involves representing an initial state, possible actions, and a goal state. A planning agent uses a knowledge base to select action sequences that transform the initial state into a goal state. STRIPS is a common planning representation that uses predicates to describe states and logical operators to represent actions and their effects. A STRIPS planning problem specifies the initial state, goal conditions, and set of operators. A solution is a sequence of ground operator instances that produces the goal state from the initial state.
The document discusses knowledge-based agents and how they use inference to derive new representations of the world from their knowledge base in order to determine what actions to take. It provides the example of an agent exploring a cave, or "Wumpus world", where the goal is to locate gold and exit without being killed by the Wumpus monster or falling into pits. The agent uses its percepts and knowledge base along with inference rules to deduce its next action at each step.
Online train ticket booking system project.pdfKamal Acharya
Rail transport is one of the important modes of transport in India. Now a days we
see that there are railways that are present for the long as well as short distance
travelling which makes the life of the people easier. When compared to other
means of transport, a railway is the cheapest means of transport. The maintenance
of the railway database also plays a major role in the smooth running of this
system. The Online Train Ticket Management System will help in reserving the
tickets of the railways to travel from a particular source to the destination.
Sachpazis_Consolidation Settlement Calculation Program-The Python Code and th...Dr.Costas Sachpazis
Consolidation Settlement Calculation Program-The Python Code
By Professor Dr. Costas Sachpazis, Civil Engineer & Geologist
This program calculates the consolidation settlement for a foundation based on soil layer properties and foundation data. It allows users to input multiple soil layers and foundation characteristics to determine the total settlement.
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2. The backtracking method
• For problems where no. of choices grow exponentially with problem size
• A given problem has a set of constraints and possibly an objective function
• The solution optimizes an objective function, and/or is feasible.
• We can represent the solution space for the problem using a state space tree
– The root of the tree represents 0 choices,
– Nodes at depth 1 represent first choice
– Nodes at depth 2 represent the second choice, etc.
– In this tree a path from root to a leaf represents a candidate solution
• Definition: We call a node nonpromising if it cannot lead to a feasible (or
optimal) solution, otherwise it is promising
• Main idea: Backtracking consists of doing a DFS of the state space tree,
checking whether each node is promising and if the node is nonpromising
backtracking to the node’s parent
3. Backtracking
• For some problems, the only way to solve is to check all
possibilities.
• Backtracking is a systematic way to go through all the possible
configurations of a search space.
• Many problems solved by backtracking satisfy a set of
constraints which may divided into two categories: explicit and
implicit.
Explicit constraints are rules that restrict each to take on values only
from a given set
Implicit constraints are rules which relate to each other.
i
x
i
x
4. Backtracking
• Find whether there is any feasible solution? Decision Problem
• What is best solution? Optimization Problem
• List all feasible solutions? Enumeration Problem
5. • Problem State is each node in the tree
• State Space is all possible paths from root to other nodes
• Solution State is all possible paths from root to solution
• Answer State satisfies implicit constraints
• A live node is a node which has been generated but whose all children
have not been generated.
• An E-node (i.e., expanding node) is a live node whose children are
currently being generated.
• A dead node is a generated node which is not to be expanded further or
all of whose children have been generated.
• Two ways to generate problem states:
• Breadth First Generation (queue of live nodes)
• Depth First Generation (stack of live nodes)
Generating Problem States
6. • Depth First Generation (stack of live nodes)
• When a new child C of the current E-node R is generated, this child
becomes the new E-node.
• Then R will become the new E-node again when the subtree C has
been fully explored.
• Corresponds to a depth first search of the problem states.
Generating Problem States
7. • Breadth First Generation (queue of live nodes)
• The E-node remains the E-node until it is dead.
• Bounding functions are used in both to kill live nodes without generating all
of their children.
• At the end of the process, an answer node (or all answer nodes) are generated.
• The depth search generation method with bounding function is called
backtracking.
• The breadth first generation method is used in the branch-and-bound
method.
Generating Problem States
8. Improving Backtracking: Search Pruning
• Search pruning will help us to reduce the search space and hence get a
solution faster.
• The idea is to avoid those paths that may not lead to a solution as early
as possible by finding contradictions so that we can backtrack
immediately without the need to build a hopeless solution vector.
9. Greedy vs Backtracking
• Method of obtaining
optimum solution, without
revising previously
generated solutions
• State space tree not created
• Less efficient
• Eg. Job sequencing with
deadlines, Optimal storage
on tapes
• Method of obtaining
optimum solution, may
require revision of previously
generated solutions
• State space tree created
• Efficient to obtain optimum
solution
• Eg. 8 Queen problem, Sum of
subset problem
12. Eight Queen Problem
• Attempts to place 8 queens on a chessboard in such a
way that no queen can attack any other.
• A queen can attack another queen if it exists in the same
row, column or diagonal as the queen.
• This problem can be solved by trying to place the first
queen, then the second queen so that it cannot attack the
first, and then the third so that it is not conflicting with
previously placed queens.
• The solution is a vector of length 8 (x1, x2, x3, ... x8)
xi corresponds to the column where we should place the ith queen.
• The solution is to build a partial solution element by element until it is complete.
• We should backtrack in case we reach to a partial solution of length k, that we couldn't
expand any more.
13. Eight Queen Problem: Algorithm
putQueen(row)
{ for every position col on the same row
if position col is available
place the next queen in position col
if (row<8)
putQueen(row+1);
else success;
}
• Define an 8 by 8 array of 1s and 0s to represent the chessboard
• Note that the search space is very huge:
16,772, 216 (88) possibilities.
• Is there a way to reduce search space?
Yes Search Pruning.
14. • Since each queen (1-8) must be on a different row, we can assume queen i is
on row i.
• All solutions to the 8-queens problem can be represented as an 8-tuple
where queen i is on column j.
• The explicit constraints are
• The implicit constraints are that no two ’s can be the same (as queens must
be on different columns) and no two queens can be on the same diagonal.
• This implies that all solutions are permutations of the 8-tuple (1,2,…,8),
and reduces the solution space from tuples to 8! tuples.
1
0
,
0 or
x
x i
i
)
,...,
,
( 8
2
1 x
x
x
.
8
1
},
8
,...,
2
,
1
{
i
Si
8
8
i
x
Eight Queen Problem
15. • Generalizing earlier discussion, solution space contains all n! permutations of (1,2,…,n).
• The tree below shows possible organization for n=4.
• Tree is called a permutation tree (nodes are numbered as in depth first search).
• Edges labeled by possible values of
• The solution space is all paths from the root node to a leaf node.
• There are 4!=24 leaf nodes in tree.
44
42
39
37
33
31
28
26
23
21
17
15
12
10
7
5
1
1
x
2
x
2 3
1 4
60
58
55
53
49
47 63 65
43
41
38
36
32
30
27
25
22
20
16
14
11
9
6
4 59
57
54
52
48
46 62 64
40
35
29
24
19
13
8
3 56
51
45 61
34
18
2 50
2 3 4
2
3
1
4
1 1
2
3
4
2
3
4 2 3
4
2
3
4
2
3
4 3
4
3 4
1 2 3
4
1 2
3
4
1 2
3
4
1 2
4
1
2 3
2
1
2
3
2
1
3
1 2
1 2
1
1
1
i
x
Four Queen Problem
17. Backtracking Algorithm for n-Queens
problem
• Let represent where the ith queen is placed (in row i and
column on an n by n chessboard.
• Observe that two queens on the same diagonal that runs from “upper
left” to “lower right” have the same “row-column” value.
• Also two queens on the same diagonal from “upper right” to “lower
left” have the same “row+column” value
)
,...,
,
( 2
1 n
x
x
x
i
x
18. • Then two queens at (i,j) and (k,l) are on the same diagonal
• iff i-j=k-l or i+j=k+l
• iff i-k=j-l or j-l = k-i
• iff |j-l|=|i-k| .
• Algorithm PLACE(k,i) returns true if the kth queen can be placed in column i and
runs in O(k) time (see next slide)
• Using PLACE, the recursive version of the general backtracking method can be
used to give a precise solution to the n-queens problem
• Array x[ ] is global in the Algorithm invoked by NQUEENS(1, n).
Backtracking Algorithm for n-Queens
problem
19. bool Place(int k, int i)
// Returns true if a queen can be placed in kth row and ith column. Otherwise it returns false.
// x[] is a global array whose first (k-1) values have been set.
// abs(r) returns the absolute value of r.
{
for (int j = 1; j < k; j++)
if ((x[j] == i) // Two in the same column
|| (abs(x[j]-i) == abs(j-k))) // or in the same diagonal
return(false);
return(true);
}
void NQueens(int k, int n)
// Using backtracking, this procedure prints all possible placements of n queens on an n x n
// chessboard so that they are nonattacking
{
for (int i=1; i<=n; i++) {
if (Place(k, i)) {
x[k] = i; //if no conflict place queen
if (k==n) { for (int j=1;j<=n;j++) //dead end
cout << x[j] << ' '; cout << endl; //print board configuration
}
else NQueens(k+1, n); //try next queen next position
}
}
}
Backtracking Algorithm for n-Queens
problem
Complexity: n!
jth queen at x[j] and kth queen at i
22. Sum of Subset Problem
• Given positive numbers and m, find all subsets of ,
whose sum is m.
• If n=4, ={11, 13, 24, 7} and m=31, the desired solution sets
are (11, 13, 7) and (24, 7).
• If the solution vectors are given using the indices of the xi values used, then
the solution vectors are (1,2,4) and (3,4).
• In general, all solutions are k-tuples with and different solutions
may have different values of k.
• The explicit constraints on the solution space are that each
• The implicit constraints are that (so each item will occur
only once) and that the sum of the corresponding ’s be m.
i
w
)
,...,
,
( 2
1 k
x
x
x
n
k
1
}.
,...,
2
,
1
{ n
xi
,
1
,
1 n
i
x
x i
i
}
,
,
,
{ 4
3
2
1 w
w
w
w
,
1
, n
i
wi
1 1
(1)
k n
i i i
i i k
wk w m
m
w
k
w k
k
i
i
i
1
1
)
2
(
hold
)
2
(
and
)
1
(
iff
)
,...,
,
( 2
1 true
x
x
x
B k
k
The boundary function used uses both of the preceding conditions:
Bounding function
23. • The next figure gives the tree that corresponds to this variable tuple formation.
• An edge from a level i node to a level i+1 node represents a value for
• The solution space is all paths from the root node to any node in the tree.
• Possible paths include empty path, (1), (1,2), (1,2,3), (1,2,3,4), (1,2,4), (1,3,4), …
• The leftmost subtree gives all subsets containing , the next subtree gives all
subsets containing but not , etc.
.
i
x
1
w
2
w 1
w
16
1
1
x
2
x
2 3
1
4
15
14
13
12
11
10
9
8
7
6
4
3
2 5
2 3 4
4
3
3
4
4
4
4
4
3
x
4
x
Nodes are numbered as in breadth first search
First
Formulation
24. • Another formulation of this problem represents each solution by an n-tuple
with
• Here if is not chosen and if is chosen
• Given the earlier instance of (11,13,24,7) and m=31, the solutions (11,13,7) and
(24,7) are represented by (1,1,0,1) and (0,0,1,1).
• Here, all solutions have a fixed tuple size. The tree on next slide corresponds to this
formulation (nodes are numbered as in D-search).
• Edge from a level i node to a level i+1 node is labeled with the value of
• All paths from the root to a leaf give solution space.
• The left subtree gives all subsets containing and the right subtree gives all subsets
not containing .
)
,...,
,
( 2
1 n
x
x
x
.
1
},
1
,
0
{ n
i
xi
0
i
x i
w 1
i
x
i
w
)
1
0
( or
xi
1
w
1
w
Sum of Subset Problem : Second Formulation
26. Sum of subset Problem
i1
i2
i3
yes no
0
0
0
0
2
2
2
6
6
12 8
4
4
10 6
yes
yes
no
no
no
no
no
no
The sum of the included integers is stored at the node.
yes
yes yes
yes
State SpaceTree for n= 3 items
w1 = 2, w2 = 4, w3 = 6 and S = 6
Solutions:
{2,4} and {6}
27. A Depth First Search Solution
Pruned State Space Tree
0
0
0
3
3
3
7
7
12 8
4
4
9
5
3
4 4
0
0
0
5 5
0 0 0
0
6
13 7 - backtrack
1
2
3
4 5
6 7
8
10
9
11
12
15
14
13
Nodes numbered in “call” order
w1 = 3, w2 = 4, w3 = 5, w4 = 6;
S = 13
28. sumOfSubsets ( s,k,r ) //sum, index, remaining sum
//generate left child until s+w[k]≤m
if (s+w[k]=m)
write(x(1:k)) //subset found
else if (s+w[k]+w[k+1]] ≤ m)
sumOfSubsets ( s+w[k],k+1,r-w[k] )
//generate right child
if (s+r-w[k]≥ m) and (s+w[k+1]] ≤ m)
x[k]=0
sumOfSubsets ( s,k+1,r-w[k] )
Initial call sumOfSubsets(0, 0, )
n
i
i
w
1
Sum of subset Algorithm
Complexity: 2n
29. Given n=6,M=30 and
W(1…6)=(5,10,12,13,15,18).
Ist solution is A -> 1 1 0 0 1 0
IInd solution is B -> 1 0 1 1 0 0
III rd solution is C -> 0 0 1 0 0 1
15,5,33
0,1,73
5,2,68
15,3,58
27,4,46 15,4,46
5,3,58
17,4,46 5,4,4
0,2,68
10,3,587
10,4,46
0,3,58
C
A
B 5,5,33 10,5,33
20,6,18
Xi=1
Xi=0 Xi=1
Xi=0
Xi=0
Xi=1 Xi=0
Xi=1
Xi=0
Xi=0
Xi=0
Xi=1
Xi=1
Xi=1
Xi=1
Xi=1
Xi=0
Xi=0
Xi=0
B 27,5,33
B
27,6,18
B
B
31. GRAPH / MAP coloring
• Graph Coloring is an assignment of colors
• Proper Coloring is if no two adjacent vertices have the same color
• Optimization Problem
• Chromatic no: smallest no. of colors used to color graph
• The Four Color Theorem states that any map on a plane can be
colored with no more than four colors, so that no two countries with a
common border are the same color
32. Origin of the problem
m-coloring decision problem: whether the graph can be colored or not
m-coloring optimization problem: min # of colors to color graph
chromatic problem
More than 1 possible solution
33. Algo
• Number out each vertex (V0, V1, V2, … Vn-1)
• Number out the available colors (C0, C1, C2, … Cm-1)
• Start assigning Ci to each Vi. While assigning the colors note that two
adjacent vertices should not be colored by the same color. And least
no. of colors should be used.
• While assigning the appropriate color, just backtrack and change the
color if two adjacent colors are getting the same.
Complexity: mn
34.
35. Applications of Graph Theory
• Computer N/W security (Minimum Vertex Cover)
• Timetabling Problem (Vertex Coloring of Bipartite MultiGraph)
• GSM Mobile Phone Networks (Map Coloring)
• Represent Natural Language Parsing Diagrams
• Pattern recognition
• Molecules in chemistry and physics
36. Branch and Bound
• Branch and bound is used to find optimal solutions to optimization
problems.
• Is applied where Greedy and Dynamic fail.
• Is indeed much slower. Might require exponential time in worst case.
• BnB uses state space tree where in all children of node generated
before expanding any of its child.
37. Backtracking vs Branch and Bound
• Follows DFS approach
• Solves Decision Problems
• While finding solution, bad
choices can be made
• State space tree is searched until
solution is found
• Space required is O(ht. of tree)
• Applications: N Queens, Sum of
subset
• Follows DFS / BFS / Best First
Search approach
• Solves optimization problems
• Proceeds on a better solution, so
possibility of bad solution is
ruled out
• State space tree is searched
completely since solution can be
found elsewhere
• Space required is O(No. of
leaves)
• Applications: 15 puzzle, TSP
38. The Branch and Bound Steps
• In BnB, a state space tree is built and all children of E nodes (a live
node whose children are currently been generated) are generated
before any other node can become live node.
• For exploring new nodes either BFS (FIFO queue) or D-search (LIFO
stack) is used or replace the FIFO queue with a priority queue (least-
cost (or max priority)).
– The search for an answer node can often be speeded up using an
“intelligent” ranking function for the live nodes though it requires
additional computational effort.
• In this method, a space tree of possible solutions is generated. Then
partitioning (branching) is done at each node. LB and UB computed
at each node, leading to selection of answer node.
• Bounding functions (when LB>=UB) avoid generation of trees
(fathoming) not containing answer node.
39. • A branch-and-bound algorithm computes a number (bound) at a node
to determine whether the node is promising.
• The number is a bound on the value of the solution that could be
obtained by expanding beyond the node.
• If that bound is no better than the value of the best solution found so
far, the node is nonpromising. Otherwise, it is promising.
• Besides using the bound to determine whether a node is promising, we
can compare the bounds of promising nodes and visit the children of
the one with the best bound.
• This approach is called best-first search with branch-and-bound
pruning. The implementation of this approach is a modification of the
breadth-first search with branch-and-bound pruning.
The Branch and Bound variants
40. Branch and Bound Algorithm
Algorithm BnB()
E←new(node) //E is node pointer pointing root node
while(true)
if(E is a final leaf)
write(path from E to root) //E is optimal sol.
return
Expand(E)
if(H is empty) //H is heap for all live nodes
write(“there is no solution”) //E is optimal sol.
return
E← delete_top(H)
Algorithm Expand(E)
Generate all children of E
Compute approximate cost value of each child
Insert each child into heap H
41. Least Cost search
• In BnB, basic idea is selection of E-node, so that we reach to answer
node quickly.
• Using FIFO and LIFO BnB selection of E-node is complicated (since
expansion of all live nodes required before leading to an answer) and
blind.
• Best First Search and D-search are special cases of LC-search.
• In LC-search an intelligent ranking function (smallest c^(x)) is
formed. c^(x)=f(x)+ ĝ(x) (where f(x) is cost of reaching x from root,
ĝ(x) is an estimate of additional effort needed to reach an answer node
from x).
• LC-search terminates only when either an answer node is found or the
entire state space tree has been generated and searched.
• Note that termination is only guaranteed for finite space trees.