The branch and bound method searches a tree model of the solution space for discrete optimization problems. It generates nodes that define problem states, with solution states satisfying the problem constraints. Bounding functions are used to prune subtrees without optimal solutions to avoid full exploration. The method searches the state space tree using techniques like first-in-first-out (FIFO) or least cost search, which selects the next node to expand based on estimated distance to an optimal solution.
Branch and Bound is a state space search algorithm that involves generating all children of a node before exploring any children. It uses lower bounds to prune parts of the search tree that cannot produce better solutions than what has already been found. The algorithm is demonstrated on problems like the 8-puzzle and Travelling Salesman Problem. For TSP, it works by reducing the cost matrix at each node to calculate lower bounds, and exploring the child with the lowest estimated total cost.
The document compares and contrasts the Backtracking and Branch & Bound algorithms. Backtracking searches the entire state space tree using depth-first search until it finds a solution, while realizing when it has made an incorrect choice. Branch & Bound may search the tree using depth-first or breadth-first search, and prunes branches when it finds a better solution than exploring that branch could provide. The document also provides an example of applying the Branch & Bound algorithm to the Traveling Salesman Problem and explains how to compute the costs of nodes in this problem.
Branch and bound is a state space search method that generates all children of a node before expanding any children. It associates a cost or profit with each node and uses a min or max heap to select the next node to expand. For the travelling salesman problem, it constructs a permutation tree representing all possible routes and uses lower bounds and reduced cost matrices at each node to prune the search space and find an optimal solution.
The document discusses the branch and bound algorithm for solving the 15-puzzle problem. It describes the key components of branch and bound including live nodes, e-nodes, and dead nodes. It also defines the cost function used to evaluate nodes as the sum of the path length and number of misplaced tiles. The algorithm generates all possible child nodes from the current node and prunes the search tree by comparing node costs to avoid exploring subtrees without solutions.
This document discusses using the Branch and Bound technique to solve the traveling salesman problem and water jug problem. Branch and Bound is a method for solving discrete and combinatorial optimization problems by breaking the problem into smaller subsets, calculating bounds on the objective function, and discarding subsets that cannot produce better solutions than the best found so far. The document provides examples of applying Branch and Bound to find the optimal path between states for the water jug problem and the shortest route between cities for the traveling salesman problem.
The branch-and-bound method is used to solve optimization problems by traversing a state space tree. It computes a bound at each node to determine if the node is promising. Better approaches traverse nodes breadth-first and choose the most promising node using a bounding heuristic. The traveling salesperson problem is solved using branch-and-bound by finding an initial tour, defining a bounding heuristic as the actual cost plus minimum remaining cost, and expanding promising nodes in best-first order until finding the minimal tour.
The document discusses using the branch and bound technique to solve optimization problems like the traveling salesman problem and water jug problem. It explains that branch and bound systematically breaks problems into smaller subsets, calculates bounds on objective functions, and discards subsets that cannot produce better solutions than what is currently known. For the water jug problem, it provides an example path and explains how branch and bound finds the optimal 6-state solution. For traveling salesman, it notes the problem is to minimize travel distance by visiting all cities once and returning home.
The document discusses two principal approaches to solving intractable problems: exact algorithms that guarantee an optimal solution but may not run in polynomial time, and approximation algorithms that can find a suboptimal solution in polynomial time. It focuses on exact algorithms like exhaustive search, backtracking, and branch-and-bound. Backtracking constructs a state space tree and prunes non-promising nodes to reduce search space. Branch-and-bound uses bounding functions to determine if nodes are promising or not, pruning those that are not. The traveling salesman problem is used as an example to illustrate branch-and-bound, discussing different bounding functions that can be used.
Branch and Bound is a state space search algorithm that involves generating all children of a node before exploring any children. It uses lower bounds to prune parts of the search tree that cannot produce better solutions than what has already been found. The algorithm is demonstrated on problems like the 8-puzzle and Travelling Salesman Problem. For TSP, it works by reducing the cost matrix at each node to calculate lower bounds, and exploring the child with the lowest estimated total cost.
The document compares and contrasts the Backtracking and Branch & Bound algorithms. Backtracking searches the entire state space tree using depth-first search until it finds a solution, while realizing when it has made an incorrect choice. Branch & Bound may search the tree using depth-first or breadth-first search, and prunes branches when it finds a better solution than exploring that branch could provide. The document also provides an example of applying the Branch & Bound algorithm to the Traveling Salesman Problem and explains how to compute the costs of nodes in this problem.
Branch and bound is a state space search method that generates all children of a node before expanding any children. It associates a cost or profit with each node and uses a min or max heap to select the next node to expand. For the travelling salesman problem, it constructs a permutation tree representing all possible routes and uses lower bounds and reduced cost matrices at each node to prune the search space and find an optimal solution.
The document discusses the branch and bound algorithm for solving the 15-puzzle problem. It describes the key components of branch and bound including live nodes, e-nodes, and dead nodes. It also defines the cost function used to evaluate nodes as the sum of the path length and number of misplaced tiles. The algorithm generates all possible child nodes from the current node and prunes the search tree by comparing node costs to avoid exploring subtrees without solutions.
This document discusses using the Branch and Bound technique to solve the traveling salesman problem and water jug problem. Branch and Bound is a method for solving discrete and combinatorial optimization problems by breaking the problem into smaller subsets, calculating bounds on the objective function, and discarding subsets that cannot produce better solutions than the best found so far. The document provides examples of applying Branch and Bound to find the optimal path between states for the water jug problem and the shortest route between cities for the traveling salesman problem.
The branch-and-bound method is used to solve optimization problems by traversing a state space tree. It computes a bound at each node to determine if the node is promising. Better approaches traverse nodes breadth-first and choose the most promising node using a bounding heuristic. The traveling salesperson problem is solved using branch-and-bound by finding an initial tour, defining a bounding heuristic as the actual cost plus minimum remaining cost, and expanding promising nodes in best-first order until finding the minimal tour.
The document discusses using the branch and bound technique to solve optimization problems like the traveling salesman problem and water jug problem. It explains that branch and bound systematically breaks problems into smaller subsets, calculates bounds on objective functions, and discards subsets that cannot produce better solutions than what is currently known. For the water jug problem, it provides an example path and explains how branch and bound finds the optimal 6-state solution. For traveling salesman, it notes the problem is to minimize travel distance by visiting all cities once and returning home.
The document discusses two principal approaches to solving intractable problems: exact algorithms that guarantee an optimal solution but may not run in polynomial time, and approximation algorithms that can find a suboptimal solution in polynomial time. It focuses on exact algorithms like exhaustive search, backtracking, and branch-and-bound. Backtracking constructs a state space tree and prunes non-promising nodes to reduce search space. Branch-and-bound uses bounding functions to determine if nodes are promising or not, pruning those that are not. The traveling salesman problem is used as an example to illustrate branch-and-bound, discussing different bounding functions that can be used.
This document discusses dynamic programming and algorithms for solving all-pair shortest path problems. It begins by explaining dynamic programming as an optimization technique that works bottom-up by solving subproblems once and storing their solutions, rather than recomputing them. It then presents Floyd's algorithm for finding shortest paths between all pairs of nodes in a graph. The algorithm iterates through nodes, updating the shortest path lengths between all pairs that include that node by exploring paths through it. Finally, it discusses solving multistage graph problems using forward and backward methods that work through the graph stages in different orders.
Analysis and design of algorithms part 4Deepak John
Complexity Theory - Introduction. P and NP. NP-Complete problems. Approximation algorithms. Bin packing, Graph coloring. Traveling salesperson Problem.
This document summarizes a lecture on algorithms and graph traversal techniques. It discusses:
1) Breadth-first search (BFS) and depth-first search (DFS) algorithms for traversing graphs. BFS uses a queue while DFS uses a stack.
2) Applications of BFS and DFS, including finding connected components, minimum spanning trees, and bi-connected components.
3) Identifying articulation points to determine biconnected components in a graph.
4) The 0/1 knapsack problem and approaches for solving it using greedy algorithms, backtracking, and branch and bound search.
This document discusses dynamic programming and greedy algorithms. It begins by defining dynamic programming as a technique for solving problems with overlapping subproblems. Examples provided include computing the Fibonacci numbers and binomial coefficients. Greedy algorithms are introduced as constructing solutions piece by piece through locally optimal choices. Applications discussed are the change-making problem, minimum spanning trees using Prim's and Kruskal's algorithms, and single-source shortest paths. Floyd's algorithm for all pairs shortest paths and optimal binary search trees are also summarized.
Backtracking is a general algorithm for finding all (or some) solutions to some computational problems, notably constraint satisfaction problems, that incrementally builds candidates to the solutions, and abandons each partial candidate c ("backtracks") as soon as it determines that c cannot possibly be completed to a valid solution.
This document discusses hashing techniques for indexing and retrieving elements in a data structure. It begins by defining hashing and its components like hash functions, collisions, and collision handling. It then describes two common collision handling techniques - separate chaining and open addressing. Separate chaining uses linked lists to handle collisions while open addressing resolves collisions by probing to find alternate empty slots using techniques like linear probing and quadratic probing. The document provides examples and explanations of how these hashing techniques work.
Skiena algorithm 2007 lecture16 introduction to dynamic programmingzukun
This document summarizes a lecture on dynamic programming. It begins by introducing dynamic programming as a powerful tool for solving optimization problems on ordered items like strings. It then contrasts greedy algorithms, which make locally optimal choices, with dynamic programming, which systematically searches all possibilities while storing results. The document provides examples of computing Fibonacci numbers and binomial coefficients using dynamic programming by storing partial results rather than recomputing them. It outlines three key steps to applying dynamic programming: formulating a recurrence, bounding subproblems, and specifying an evaluation order.
Cs6402 design and analysis of algorithms may june 2016 answer keyappasami
The document discusses algorithms and complexity analysis. It provides Euclid's algorithm for computing greatest common divisor, compares the orders of growth of n(n-1)/2 and n^2, and describes the general strategy of divide and conquer methods. It also defines problems like the closest pair problem, single source shortest path problem, and assignment problem. Finally, it discusses topics like state space trees, the extreme point theorem, and lower bounds.
Graph Traversal Algorithms - Depth First Search TraversalAmrinder Arora
This document discusses graph traversal techniques, specifically depth-first search (DFS) and breadth-first search (BFS). It provides pseudocode for DFS and explains key properties like edge classification, time complexity of O(V+E), and applications such as finding connected components and articulation points.
Iterative improvement is an algorithm design technique for solving optimization problems. It starts with a feasible solution and repeatedly makes small changes to the current solution to find a solution with a better objective function value until no further improvements can be found. The simplex method, Ford-Fulkerson algorithm, and local search heuristics are examples that use this technique. The maximum flow problem can be solved using the iterative Ford-Fulkerson algorithm, which finds augmenting paths in a flow network to incrementally increase the flow from the source to the sink until no more augmenting paths exist.
This document discusses various heuristic search algorithms including A*, iterative-deepening A*, and recursive best-first search. It begins by introducing the concept of using evaluation functions to guide best-first search and preferentially expand nodes with lower heuristic values. It then presents the general graph search algorithm and describes how A* specifically reorders nodes using an evaluation function that considers path cost and estimated cost to the goal. Consistency conditions for the heuristic function are discussed which guarantee A* finds optimal solutions.
The document discusses using graphs to model real-world problems and algorithms for solving graph problems. It introduces basic graph terminology like vertices, edges, adjacency matrix/list representations. It then summarizes algorithms for finding shortest paths like Dijkstra's and Bellman-Ford. It also discusses using depth-first search to solve problems like detecting cycles, topological sorting of vertices, and calculating longest paths in directed acyclic graphs.
This document provides an overview of brute force and divide-and-conquer algorithms. It discusses various brute force algorithms like computing an, string matching, closest pair problem, convex hull problems, and exhaustive search algorithms like the traveling salesman problem and knapsack problem. It also analyzes the time efficiency of these brute force algorithms. The document then discusses the divide-and-conquer approach and provides examples like merge sort, quicksort, and matrix multiplication. It provides pseudocode and analysis for mergesort. In summary, the document covers brute force and divide-and-conquer techniques for solving algorithmic problems.
The document discusses backtracking algorithms. It begins by defining backtracking as a methodical way to try different sequences of decisions to solve a problem until a solution is found. It then provides examples of backtracking for finding a maze path and coloring a map. The key aspects of a backtracking algorithm are that it uses depth-first search to recursively explore choices, pruning paths that do not lead to solutions.
Backtracking is a general algorithm for finding all (or some) solutions to some computational problems, notably constraint satisfaction problems, that incrementally builds candidates to the solutions, and abandons each partial candidate c ("backtracks") as soon as it determines that c cannot possibly be completed to a valid solution.
This document discusses lower bounds and limitations of algorithms. It begins by defining lower bounds and providing examples of problems where tight lower bounds have been established, such as sorting requiring Ω(nlogn) comparisons. It then discusses methods for establishing lower bounds, including trivial bounds, decision trees, adversary arguments, and problem reduction. The document explores different classes of problems based on complexity, such as P, NP, and NP-complete problems. It concludes by examining approaches for tackling difficult combinatorial problems that are NP-hard, such as using exact algorithms, approximation algorithms, and local search heuristics.
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.
This is the second lecture in the CS 6212 class. Covers asymptotic notation and data structures. Also outlines the coming lectures wherein we will study the various algorithm design techniques.
Types Of Recursion in C++, Data Stuctures by DHEERAJ KATARIADheeraj Kataria
The document discusses recursion and provides examples of recursion including nested recursion. It explains recursion using functions like the Ackermann function and Fibonacci numbers. It discusses excessive recursion and how it can lead to many repeated computations. The document also covers backtracking as a method for systematically trying sequences of decisions to find a solution. It provides the example of solving the eight queens problem using backtracking and recursion.
Design and analysis of computer algorithms (dave mount, 1999) Krishna Chaytaniah
El documento describe los beneficios de la colaboración entre empresas y universidades para el desarrollo de nuevos productos y servicios. Las empresas pueden acceder a los conocimientos y recursos de las universidades, mientras que estas obtienen fondos para investigación y oportunidades para que los estudiantes obtengan experiencia práctica. Un ejemplo exitoso es la asociación entre la empresa Cambridge y la universidad para el desarrollo de tecnologías que han generado nuevos empleos e innovaciones.
The branch and bound method searches a tree model of the solution space for discrete optimization problems. It uses bounding functions to prune subtrees that cannot contain optimal solutions, avoiding evaluating all possible solutions. The method generates the tree in a depth-first manner, and selects the next node to expand (the E-node) based on cost estimates. Common strategies are FIFO, LIFO, and least cost search. The traveling salesman problem can be solved using branch and bound by formulating the solution space as a tree and applying row and column reduction techniques to the cost matrix at each node to identify prunable branches.
This document discusses dynamic programming and algorithms for solving all-pair shortest path problems. It begins by explaining dynamic programming as an optimization technique that works bottom-up by solving subproblems once and storing their solutions, rather than recomputing them. It then presents Floyd's algorithm for finding shortest paths between all pairs of nodes in a graph. The algorithm iterates through nodes, updating the shortest path lengths between all pairs that include that node by exploring paths through it. Finally, it discusses solving multistage graph problems using forward and backward methods that work through the graph stages in different orders.
Analysis and design of algorithms part 4Deepak John
Complexity Theory - Introduction. P and NP. NP-Complete problems. Approximation algorithms. Bin packing, Graph coloring. Traveling salesperson Problem.
This document summarizes a lecture on algorithms and graph traversal techniques. It discusses:
1) Breadth-first search (BFS) and depth-first search (DFS) algorithms for traversing graphs. BFS uses a queue while DFS uses a stack.
2) Applications of BFS and DFS, including finding connected components, minimum spanning trees, and bi-connected components.
3) Identifying articulation points to determine biconnected components in a graph.
4) The 0/1 knapsack problem and approaches for solving it using greedy algorithms, backtracking, and branch and bound search.
This document discusses dynamic programming and greedy algorithms. It begins by defining dynamic programming as a technique for solving problems with overlapping subproblems. Examples provided include computing the Fibonacci numbers and binomial coefficients. Greedy algorithms are introduced as constructing solutions piece by piece through locally optimal choices. Applications discussed are the change-making problem, minimum spanning trees using Prim's and Kruskal's algorithms, and single-source shortest paths. Floyd's algorithm for all pairs shortest paths and optimal binary search trees are also summarized.
Backtracking is a general algorithm for finding all (or some) solutions to some computational problems, notably constraint satisfaction problems, that incrementally builds candidates to the solutions, and abandons each partial candidate c ("backtracks") as soon as it determines that c cannot possibly be completed to a valid solution.
This document discusses hashing techniques for indexing and retrieving elements in a data structure. It begins by defining hashing and its components like hash functions, collisions, and collision handling. It then describes two common collision handling techniques - separate chaining and open addressing. Separate chaining uses linked lists to handle collisions while open addressing resolves collisions by probing to find alternate empty slots using techniques like linear probing and quadratic probing. The document provides examples and explanations of how these hashing techniques work.
Skiena algorithm 2007 lecture16 introduction to dynamic programmingzukun
This document summarizes a lecture on dynamic programming. It begins by introducing dynamic programming as a powerful tool for solving optimization problems on ordered items like strings. It then contrasts greedy algorithms, which make locally optimal choices, with dynamic programming, which systematically searches all possibilities while storing results. The document provides examples of computing Fibonacci numbers and binomial coefficients using dynamic programming by storing partial results rather than recomputing them. It outlines three key steps to applying dynamic programming: formulating a recurrence, bounding subproblems, and specifying an evaluation order.
Cs6402 design and analysis of algorithms may june 2016 answer keyappasami
The document discusses algorithms and complexity analysis. It provides Euclid's algorithm for computing greatest common divisor, compares the orders of growth of n(n-1)/2 and n^2, and describes the general strategy of divide and conquer methods. It also defines problems like the closest pair problem, single source shortest path problem, and assignment problem. Finally, it discusses topics like state space trees, the extreme point theorem, and lower bounds.
Graph Traversal Algorithms - Depth First Search TraversalAmrinder Arora
This document discusses graph traversal techniques, specifically depth-first search (DFS) and breadth-first search (BFS). It provides pseudocode for DFS and explains key properties like edge classification, time complexity of O(V+E), and applications such as finding connected components and articulation points.
Iterative improvement is an algorithm design technique for solving optimization problems. It starts with a feasible solution and repeatedly makes small changes to the current solution to find a solution with a better objective function value until no further improvements can be found. The simplex method, Ford-Fulkerson algorithm, and local search heuristics are examples that use this technique. The maximum flow problem can be solved using the iterative Ford-Fulkerson algorithm, which finds augmenting paths in a flow network to incrementally increase the flow from the source to the sink until no more augmenting paths exist.
This document discusses various heuristic search algorithms including A*, iterative-deepening A*, and recursive best-first search. It begins by introducing the concept of using evaluation functions to guide best-first search and preferentially expand nodes with lower heuristic values. It then presents the general graph search algorithm and describes how A* specifically reorders nodes using an evaluation function that considers path cost and estimated cost to the goal. Consistency conditions for the heuristic function are discussed which guarantee A* finds optimal solutions.
The document discusses using graphs to model real-world problems and algorithms for solving graph problems. It introduces basic graph terminology like vertices, edges, adjacency matrix/list representations. It then summarizes algorithms for finding shortest paths like Dijkstra's and Bellman-Ford. It also discusses using depth-first search to solve problems like detecting cycles, topological sorting of vertices, and calculating longest paths in directed acyclic graphs.
This document provides an overview of brute force and divide-and-conquer algorithms. It discusses various brute force algorithms like computing an, string matching, closest pair problem, convex hull problems, and exhaustive search algorithms like the traveling salesman problem and knapsack problem. It also analyzes the time efficiency of these brute force algorithms. The document then discusses the divide-and-conquer approach and provides examples like merge sort, quicksort, and matrix multiplication. It provides pseudocode and analysis for mergesort. In summary, the document covers brute force and divide-and-conquer techniques for solving algorithmic problems.
The document discusses backtracking algorithms. It begins by defining backtracking as a methodical way to try different sequences of decisions to solve a problem until a solution is found. It then provides examples of backtracking for finding a maze path and coloring a map. The key aspects of a backtracking algorithm are that it uses depth-first search to recursively explore choices, pruning paths that do not lead to solutions.
Backtracking is a general algorithm for finding all (or some) solutions to some computational problems, notably constraint satisfaction problems, that incrementally builds candidates to the solutions, and abandons each partial candidate c ("backtracks") as soon as it determines that c cannot possibly be completed to a valid solution.
This document discusses lower bounds and limitations of algorithms. It begins by defining lower bounds and providing examples of problems where tight lower bounds have been established, such as sorting requiring Ω(nlogn) comparisons. It then discusses methods for establishing lower bounds, including trivial bounds, decision trees, adversary arguments, and problem reduction. The document explores different classes of problems based on complexity, such as P, NP, and NP-complete problems. It concludes by examining approaches for tackling difficult combinatorial problems that are NP-hard, such as using exact algorithms, approximation algorithms, and local search heuristics.
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.
This is the second lecture in the CS 6212 class. Covers asymptotic notation and data structures. Also outlines the coming lectures wherein we will study the various algorithm design techniques.
Types Of Recursion in C++, Data Stuctures by DHEERAJ KATARIADheeraj Kataria
The document discusses recursion and provides examples of recursion including nested recursion. It explains recursion using functions like the Ackermann function and Fibonacci numbers. It discusses excessive recursion and how it can lead to many repeated computations. The document also covers backtracking as a method for systematically trying sequences of decisions to find a solution. It provides the example of solving the eight queens problem using backtracking and recursion.
Design and analysis of computer algorithms (dave mount, 1999) Krishna Chaytaniah
El documento describe los beneficios de la colaboración entre empresas y universidades para el desarrollo de nuevos productos y servicios. Las empresas pueden acceder a los conocimientos y recursos de las universidades, mientras que estas obtienen fondos para investigación y oportunidades para que los estudiantes obtengan experiencia práctica. Un ejemplo exitoso es la asociación entre la empresa Cambridge y la universidad para el desarrollo de tecnologías que han generado nuevos empleos e innovaciones.
The branch and bound method searches a tree model of the solution space for discrete optimization problems. It uses bounding functions to prune subtrees that cannot contain optimal solutions, avoiding evaluating all possible solutions. The method generates the tree in a depth-first manner, and selects the next node to expand (the E-node) based on cost estimates. Common strategies are FIFO, LIFO, and least cost search. The traveling salesman problem can be solved using branch and bound by formulating the solution space as a tree and applying row and column reduction techniques to the cost matrix at each node to identify prunable branches.
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.
The document discusses algorithm analysis and design. It begins with an introduction to analyzing algorithms including average case analysis and solving recurrences. It then provides definitions of algorithms both informal and formal. Key aspects of algorithm study and specification methods like pseudocode are outlined. Selection sort and tower of Hanoi problems are presented as examples and analyzed for time and space complexity. Average case analysis is discussed assuming all inputs are equally likely.
1) The document describes the divide-and-conquer algorithm design paradigm. It splits problems into smaller subproblems, solves the subproblems recursively, and then combines the solutions to solve the original problem.
2) Binary search is provided as an example algorithm that uses divide-and-conquer. It divides the search space in half at each step to quickly determine if an element is present.
3) Finding the maximum and minimum elements in an array is another problem solved using divide-and-conquer. It recursively finds the max and min of halves of the array and combines the results.
Managerial economics provides tools and techniques to help managers make effective decisions. It bridges traditional economics and business practices. Managerial economics analyzes issues like supply, price, production, and profit to seek solutions to managerial problems. It aims to maximize returns and minimize costs. While macroeconomics looks at overall economic conditions, managerial economics focuses on decisions at the individual firm level.
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.
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.
Cs6402 design and analysis of algorithms impartant part b questions appasamiappasami
This document outlines important questions from 5 units that could appear on the CS64O2 DESIGN AND ANALYSIS OF ALGORITHMS exam. Unit I covers algorithm problem types, problem solving fundamentals, asymptotic notations, and analyzing recursive/non-recursive algorithms. Unit II discusses closest pair, convex hull, TSP, KP, sorting algorithms, and binary search. Unit III is about dynamic programming, graph algorithms, OBST, and HMT. Unit IV is linear programming problems. Unit V covers complexity classes, backtracking, branch and bound, and approximation algorithms for NP-hard problems like TSP and KP. The document provides guidance on questions to study for the exam.
This document discusses graph coloring and its applications. It begins by defining graph coloring as assigning labels or colors to elements of a graph such that no two adjacent elements have the same color. It then provides examples of vertex coloring, edge coloring, and face coloring. The document also discusses the chromatic number and chromatic polynomial. It describes several real-world applications of graph coloring, including frequency assignment in cellular networks.
The document describes the backtracking method for solving problems that require finding optimal solutions. Backtracking involves building a solution one component at a time and using bounding functions to prune partial solutions that cannot lead to an optimal solution. It then provides examples of applying backtracking to solve the 8 queens problem by placing queens on a chessboard with no attacks. The general backtracking method and a recursive backtracking algorithm are also outlined.
The document discusses solving the 8 queens problem using backtracking. It begins by explaining backtracking as an algorithm that builds partial candidates for solutions incrementally and abandons any partial candidate that cannot be completed to a valid solution. It then provides more details on the 8 queens problem itself - the goal is to place 8 queens on a chessboard so that no two queens attack each other. Backtracking is well-suited for solving this problem by attempting to place queens one by one and backtracking when an invalid placement is found.
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.
Branch and bound is a method for systematically searching a solution space that uses bounding functions to avoid generating subtrees that do not contain an answer node. It has a branching function that determines the order of exploring nodes and a bounding function that prunes the search tree efficiently. Branch and bound refers to methods where all children of the current node are generated before exploring other nodes. It generalizes breadth-first and depth-first search strategies. The method uses estimates of node costs and upper bounds to prune portions of the search space that cannot contain optimal solutions.
Here are the portions of the state space tree generated by LCBB and FIFOBB for the given knapsack problems:
a) n=5, (p1,p2,p3,p4,p5)=(10,15,6,8,4), (w1,w2,w3,w4,w5)=(4,6,3,4,2) and m=12
LCBB:
1
2 3 4
7 8 9 10
5 6
FIFOBB:
1
2
3
7
4
5 6
8 9
b) n=5, (p1,p2,p3,
The document discusses trees and graphs data structures. It begins with introducing different types of trees like binary trees, binary search trees, threaded binary trees, and their various traversal algorithms like inorder, preorder and postorder traversals. It then discusses tree operations like copying trees, testing for equality. It also covers the satisfiability problem and how binary trees can be used to represent logical expressions to solve this problem. Finally, it discusses threaded binary trees where null links in a binary tree are replaced with threads, and how this allows for efficient inorder traversal in linear time.
Kakuro: Solving the Constraint Satisfaction ProblemVarad Meru
This work was done as a part of the project for the course CS 271: Introduction to Artificial Intelligence (http://www.ics.uci.edu/~kkask/Fall-2014%20CS271/index.html), taught in Fall 2014.
The document discusses binary search trees and their properties. It explains that a binary search tree is a binary tree where every node's left subtree contains values less than the node's value and the right subtree contains greater values. Operations like search, insert, delete can be done in O(h) time where h is the height of the tree. The height is O(log n) for balanced trees but can be O(n) for unbalanced trees. The document also provides examples of using a binary search tree to sort a set of numbers in O(n log n) time by building the BST and doing an inorder traversal.
A derivative free high ordered hybrid equation solverZac Darcy
Generally a range of equation solvers for estimating the solution of an equation contain the derivative of
first or higher order. Such solvers are difficult to apply in the instances of complicated functional
relationship. The equation solver proposed in this paper meant to solve many of the involved complicated
problems and establishing a process tending towards a higher ordered by alloying the already proved
conventional methods like Newton-Raphson method (N-R), Regula Falsi method (R-F) & Bisection method
(BIS). The present method is good to solve those nonlinear and transcendental equations that cannot be
solved by the basic algebra. Comparative analysis are also made with the other racing formulas of this
group and the result shows that present method is faster than all such methods of the class.
A Derivative Free High Ordered Hybrid Equation Solver Zac Darcy
Generally a range of equation solvers for estimating the solution of an equation contain the derivative of
first or higher order. Such solvers are difficult to apply in the instances of complicated functional
relationship. The equation solver proposed in this paper meant to solve many of the involved complicated
problems and establishing a process tending towards a higher ordered by alloying the already proved
conventional methods like Newton-Raphson method (N-R), Regula Falsi method (R-F) & Bisection method
(BIS). The present method is good to solve those nonlinear and transcendental equations that cannot be
solved by the basic algebra. Comparative analysis are also made with the other racing formulas of this
group and the result shows that present method is faster than all such methods of the class.
A DERIVATIVE FREE HIGH ORDERED HYBRID EQUATION SOLVERZac Darcy
Generally a range of equation solvers for estimating the solution of an equation contain the derivative of
first or higher order. Such solvers are difficult to apply in the instances of complicated functional
relationship. The equation solver proposed in this paper meant to solve many of the involved complicated
problems and establishing a process tending towards a higher ordered by alloying the already proved
conventional methods like Newton-Raphson method (N-R), Regula Falsi method (R-F) & Bisection method
(BIS). The present method is good to solve those nonlinear and transcendental equations that cannot be
solved by the basic algebra. Comparative analysis are also made with the other racing formulas of this
group and the result shows that present method is faster than all such methods of the class.
In this article, we propose a new approach to solve intuitionistic fuzzy assignment
problem. Classical assignment problem deals with deterministic cost. In practical
situations it is not easy to determine the parameters. The parameters can be modeled
to fuzzy or intuitionistic fuzzy parameters. This paper develops an approach based on
diagonal optimal algorithm to solve an intuitionistic fuzzy assignment problem. A new
ranking procedure based on combined arithmetic mean is used to order the
intuitionistic fuzzy numbers so that Diagonal optimal algorithm [22] can be applied to
solve the intuitionistic fuzzy assignment problem. To illustrate the effectiveness of the
algorithm numerical examples were given..
In this paper, we investigate transportation problem in which supplies and demands are intuitionistic fuzzy numbers. Intuitionistic Fuzzy Vogel’s Approximation Method is proposed to find an initial basic feasible solution. Intuitionistic Fuzzy Modified Distribution Method is proposed to find the optimal solution in terms of triangular intuitionistic fuzzy numbers. The solution procedure is illustrated with suitable numerical example.
DESIGN AND IMPLEMENTATION OF BINARY NEURAL NETWORK LEARNING WITH FUZZY CLUSTE...cscpconf
In this paper, Design and Implementation of Binary Neural Network Learning with Fuzzy
Clustering (DIBNNFC), is proposed to classify semisupervised data, it is based on the
concept of binary neural network and geometrical expansion. Parameters are updated
according to the geometrical location of the training samples in the input space, and each
sample in the training set is learned only once. It’s a semisupervised based approach, the
training samples are semi-labelled i.e. for some samples, labels are known and for some
samples data labels are not known. The method starts with classification, which is done by
using the concept of ETL algorithm. In classification process various classes are formed.
These classes classify samples in to two classes after that considers each class as a region and calculates the average of the entire region separately. This average is centres of the region which is used for the purpose of clustering by using FCM algorithm. Once clustering process over labelling of semi supervised data is done, then whole samples would be classify by (DIBNNFC). The method proposes here is exhaustively tested with different benchmark datasets and it is found that, on increasing value of training parameters number of hidden neurons and training time both are getting decrease. The result reported, using real character recognition data set and result will compare with existing semi-supervised classifier, the proposed approach learned with semi-supervised leads to higher classification accuracy.
1. Hash tables are good for random access of elements but not sequential access. When records need to be accessed sequentially, hashing can be problematic because elements are stored in random locations instead of consecutively.
2. To find the successor of a node in a binary search tree, we take the right child. This operation has a runtime complexity of O(1).
3. When comparing operations like insertion, deletion, and searching between different data structures, arrays generally have the best performance for insertion and searching, while linked lists have better performance for deletion and allow for easy insertion/deletion anywhere. Binary search trees fall between these two.
(a) There are three ways to traverse a binary tree pre-order, in-or.docxajoy21
The document discusses three topics: (1) in-order tree traversal, (2) B-tree data structures, and (3) open addressing for hash collisions. It provides code for in-order traversal and explains that B-trees allow for efficient searching, insertion and deletion of large blocks of sorted data, keeping the tree balanced. Open addressing resolves collisions by probing alternate array locations using techniques like linear, quadratic or double hashing until finding an empty slot.
This document summarizes a research paper that proposes a new method to accelerate the nearest neighbor search step of the k-means clustering algorithm. The k-means algorithm is computationally expensive due to calculating distances between data points and cluster centers. The proposed method uses geometric relationships between data points and centers to reject centers that are unlikely to be the nearest neighbor, without decreasing clustering accuracy. Experimental results showed the method significantly reduced the number of distance computations required.
A Mixed Binary-Real NSGA II Algorithm Ensuring Both Accuracy and Interpretabi...IJECEIAES
In this work, a Neuro-Fuzzy Controller network, called NFC that implements a Mamdani fuzzy inference system is proposed. This network includes neurons able to perform fundamental fuzzy operations. Connections between neurons are weighted through binary and real weights. Then a mixed binaryreal Non dominated Sorting Genetic Algorithm II (NSGA II) is used to perform both accuracy and interpretability of the NFC by minimizing two objective functions; one objective relates to the number of rules, for compactness, while the second is the mean square error, for accuracy. In order to preserve interpretability of fuzzy rules during the optimization process, some constraints are imposed. The approach is tested on two control examples: a single input single output (SISO) system and a multivariable (MIMO) system.
The document discusses greedy algorithms and provides examples. It begins with an overview of greedy algorithms and their properties. It then provides a sample problem (traveling salesman) and shows how a greedy approach can provide an iterative solution. The document notes advantages and disadvantages of greedy algorithms and provides additional examples, including optimal binary tree merging and the knapsack problem. It concludes with describing algorithms for optimal solutions to these problems.
The document provides details on using the bisection method to find the root of a nonlinear equation. It begins with an overview of the bisection method and how it iteratively searches for the root between two initial guesses. It then describes how to check if the initial guesses validly bracket a root by evaluating the product of the function values at the guesses. If the product is negative, a root is known to exist between the guesses. The method is then explained step-by-step, showing how the range is bisected in each iteration until converging on the root. An example problem is provided and solved over 12 iterations to demonstrate the convergence of the bisection method.
for sbi so Ds c c++ unix rdbms sql cn osalisha230390
This document contains 35 questions related to data structures and algorithms. It covers topics like data structures used in different areas like databases, networks and hierarchies. Other topics covered include trees, graphs, sorting, hashing and file structures. Sample problems are given related to these topics to test understanding.
Program Computing Project 4 builds upon CP3 to develop a program to .docxdessiechisomjj4
Program Computing Project 4 builds upon CP3 to develop a program to perform truss analysis. A truss consists of straight, slender bars pinned together at their end points. Truss members are considered to be two force, axial members. Thus, the force caused by each truss member - and the internal force in each member - acts only along it’s axis. In other words, the direction of each member force is known and only the magnitudes must be determined. To analyze a truss we study the forces acting at each individual pin joint. This is known as the Method of Joints. We will call each pin joint a node and the slender bars connecting the nodes will be called members. The previous project computed a unit vector to describe the vector direction of every member of a truss structure. To analyze the structure a few other key inputs must be included like the support reactions and external loads applied to the structure. With all of this information, you will need to make the correct changes to the provided planar (2-D) truss template program to be able to analyze a space (3-D) truss. What you need to do For a planar truss, every node has 2 degrees of freedom, the e1 and e2 directions. Therefore, for every planar truss problem, the total number of degrees of freedom (DOF) in the structure is equal to 2 times the number of nodes. We will consider the first degree of freedom for each node as the component acting in the e1 direction. So for any given node, i, the corresponding degree of freedom is (2·i)-1. For the same node, i, the corresponding value for the second degree of freedom, the component in the e2 direction, is 2-i. This numbering notation can be modified for a space truss. The difference with the space truss is that every node has 3 degrees of freedom, one degree for each of the e1, e2 and e3 directions. The degree of freedom indices are extremely crucial in understanding how to set up the matrices for the truss analysis. For this computing project, you will first need to understand the planar truss program and the inputs that are needed for that program. The first input is the spatial coordinates (x, y, z) of the nodal locations for a truss. It is convenient to label each node with a unique number (also known as the “node number”). Each row of the nodal coordinate array should contain the x and y coordinates of the node. We will use the matrix name of “x” for all nodal coordinates. Please note that “nNode” is an integer value that corresponds to the number of nodes in the truss and must be adjusted for every new truss problem. For Node 1 this matrix array input looks like: x(1,:) = [0,0]; Once the coordinates of the nodes are in the program, you will need to input how those nodes are connected by the members of the truss. In order to describe how the members connect the nodes you will also need to label each member with a “member number”. This connectivity array should contain only the nodes that are joined by a member, with each row containing firs.
This document discusses various interpolation methods:
- Newton's divided differences method uses finite differences to determine polynomial coefficients that fit scattered data points. Lagrange polynomials provide an alternative formulation.
- Spline interpolation smooths transitions by fitting different lower order polynomials to intervals between data points, maintaining continuity of derivatives at knots.
- Inverse interpolation finds the independent variable value corresponding to a given dependent variable value, using normal interpolation and root finding.
This document describes how to solve the traveling salesperson problem (TSP) using dynamic programming. It defines g(i,S) as the length of the shortest path from vertex i through vertices in S to vertex 1. It shows that g(1,V-{1}) gives the optimal tour length, and that g(i,S) can be calculated using equation (2) by considering neighboring vertices. This runs in O(n2^n) time but requires O(n2^n) space, which is infeasible for large n.
The document describes the travelling salesperson problem (TSP) and different approaches to solve it, including linear programming (LP) and dynamic programming (DP). It presents two LP formulations - one using a boolean decision variable matrix and constraints, and another using decision variables to represent the tour order. It also provides a DP formulation that recursively calculates the shortest tour length between subsets of remaining cities. An example problem is worked through step-by-step to demonstrate the DP approach.
Huffman coding is a method of compressing data by assigning variable-length codes to characters based on their frequency of occurrence. It creates a binary tree structure where the characters with the lowest frequencies are assigned the longest binary codes. This allows data to be compressed more efficiently compared to a fixed-length coding scheme. Huffman coding is proven to be an optimal prefix code, meaning no other prefix coding method can compress the data more than Huffman coding for a given frequency distribution.
The document discusses the technique of dynamic programming, including its use of storing solutions to subproblems to avoid redundant computations. Several examples are provided to illustrate dynamic programming, including matrix multiplication, Fibonacci numbers, and the traveling salesman problem. The principle of optimality is explained as the idea that optimal solutions to subproblems must be contained within an optimal solution to the overall problem.
The document discusses virtual memory and demand paging. It describes how virtual memory allows a program's logical address space to be larger than physical memory by paging portions of programs into and out of RAM as needed. When a program tries to access a page not in memory, a page fault occurs which the operating system handles by finding a free frame, paging out the contents of that frame if dirty, and paging in the requested page. Page replacement algorithms are needed when there are no free frames to select a page to replace.
1) The document describes the divide-and-conquer algorithm design paradigm. It can be applied to problems where the input can be divided into smaller subproblems, the subproblems can be solved independently, and the solutions combined to solve the original problem.
2) Binary search is provided as an example divide-and-conquer algorithm. It works by recursively dividing the search space in half and only searching the subspace containing the target value.
3) Finding the maximum and minimum elements in an array is also solved using divide-and-conquer. The array is divided into two halves, the max/min found for each subarray, and the overall max/min determined by comparing the subsolutions.
This document summarizes key concepts about virtual memory from the textbook chapter:
- Virtual memory allows processes to have a logical address space larger than physical memory by paging portions of processes into and out of frames in physical memory on demand.
- Demand paging brings pages into memory only when they are needed, reducing I/O compared to loading the whole process at once. It can cause page faults when accessing pages not yet loaded.
- Page replacement algorithms select pages to swap out to disk when free frames are needed, aiming to minimize future page faults. The choice of replacement algorithm impacts memory performance.
This document discusses the design and components of a solar tracking system. It describes how single-axis and dual-axis trackers work to follow the sun's movement and maximize solar panel efficiency. The system uses light dependent resistors to sense the sun's position and a microcontroller to command stepper motors that adjust the panel orientation accordingly. It aims to automatically point the solar panel towards the sun to produce the most electricity from sunlight.
A SYSTEMATIC RISK ASSESSMENT APPROACH FOR SECURING THE SMART IRRIGATION SYSTEMSIJNSA Journal
The smart irrigation system represents an innovative approach to optimize water usage in agricultural and landscaping practices. The integration of cutting-edge technologies, including sensors, actuators, and data analysis, empowers this system to provide accurate monitoring and control of irrigation processes by leveraging real-time environmental conditions. The main objective of a smart irrigation system is to optimize water efficiency, minimize expenses, and foster the adoption of sustainable water management methods. This paper conducts a systematic risk assessment by exploring the key components/assets and their functionalities in the smart irrigation system. The crucial role of sensors in gathering data on soil moisture, weather patterns, and plant well-being is emphasized in this system. These sensors enable intelligent decision-making in irrigation scheduling and water distribution, leading to enhanced water efficiency and sustainable water management practices. Actuators enable automated control of irrigation devices, ensuring precise and targeted water delivery to plants. Additionally, the paper addresses the potential threat and vulnerabilities associated with smart irrigation systems. It discusses limitations of the system, such as power constraints and computational capabilities, and calculates the potential security risks. The paper suggests possible risk treatment methods for effective secure system operation. In conclusion, the paper emphasizes the significant benefits of implementing smart irrigation systems, including improved water conservation, increased crop yield, and reduced environmental impact. Additionally, based on the security analysis conducted, the paper recommends the implementation of countermeasures and security approaches to address vulnerabilities and ensure the integrity and reliability of the system. By incorporating these measures, smart irrigation technology can revolutionize water management practices in agriculture, promoting sustainability, resource efficiency, and safeguarding against potential security threats.
Introduction- e - waste – definition - sources of e-waste– hazardous substances in e-waste - effects of e-waste on environment and human health- need for e-waste management– e-waste handling rules - waste minimization techniques for managing e-waste – recycling of e-waste - disposal treatment methods of e- waste – mechanism of extraction of precious metal from leaching solution-global Scenario of E-waste – E-waste in India- case studies.
KuberTENes Birthday Bash Guadalajara - K8sGPT first impressionsVictor Morales
K8sGPT is a tool that analyzes and diagnoses Kubernetes clusters. This presentation was used to share the requirements and dependencies to deploy K8sGPT in a local environment.
Presentation of IEEE Slovenia CIS (Computational Intelligence Society) Chapte...University of Maribor
Slides from talk presenting:
Aleš Zamuda: Presentation of IEEE Slovenia CIS (Computational Intelligence Society) Chapter and Networking.
Presentation at IcETRAN 2024 session:
"Inter-Society Networking Panel GRSS/MTT-S/CIS
Panel Session: Promoting Connection and Cooperation"
IEEE Slovenia GRSS
IEEE Serbia and Montenegro MTT-S
IEEE Slovenia CIS
11TH INTERNATIONAL CONFERENCE ON ELECTRICAL, ELECTRONIC AND COMPUTING ENGINEERING
3-6 June 2024, Niš, Serbia
Low power architecture of logic gates using adiabatic techniquesnooriasukmaningtyas
The growing significance of portable systems to limit power consumption in ultra-large-scale-integration chips of very high density, has recently led to rapid and inventive progresses in low-power design. The most effective technique is adiabatic logic circuit design in energy-efficient hardware. This paper presents two adiabatic approaches for the design of low power circuits, modified positive feedback adiabatic logic (modified PFAL) and the other is direct current diode based positive feedback adiabatic logic (DC-DB PFAL). Logic gates are the preliminary components in any digital circuit design. By improving the performance of basic gates, one can improvise the whole system performance. In this paper proposed circuit design of the low power architecture of OR/NOR, AND/NAND, and XOR/XNOR gates are presented using the said approaches and their results are analyzed for powerdissipation, delay, power-delay-product and rise time and compared with the other adiabatic techniques along with the conventional complementary metal oxide semiconductor (CMOS) designs reported in the literature. It has been found that the designs with DC-DB PFAL technique outperform with the percentage improvement of 65% for NOR gate and 7% for NAND gate and 34% for XNOR gate over the modified PFAL techniques at 10 MHz respectively.
Advanced control scheme of doubly fed induction generator for wind turbine us...IJECEIAES
This paper describes a speed control device for generating electrical energy on an electricity network based on the doubly fed induction generator (DFIG) used for wind power conversion systems. At first, a double-fed induction generator model was constructed. A control law is formulated to govern the flow of energy between the stator of a DFIG and the energy network using three types of controllers: proportional integral (PI), sliding mode controller (SMC) and second order sliding mode controller (SOSMC). Their different results in terms of power reference tracking, reaction to unexpected speed fluctuations, sensitivity to perturbations, and resilience against machine parameter alterations are compared. MATLAB/Simulink was used to conduct the simulations for the preceding study. Multiple simulations have shown very satisfying results, and the investigations demonstrate the efficacy and power-enhancing capabilities of the suggested control system.
Using recycled concrete aggregates (RCA) for pavements is crucial to achieving sustainability. Implementing RCA for new pavement can minimize carbon footprint, conserve natural resources, reduce harmful emissions, and lower life cycle costs. Compared to natural aggregate (NA), RCA pavement has fewer comprehensive studies and sustainability assessments.
Recycled Concrete Aggregate in Construction Part III
Unit 5 jwfiles
1. UNIT – V
BRANCH AND BOUND -- THE METHOD
The design technique known as branch and bound is very similar to backtracking
(seen in unit 4) in that it searches a tree model of the solution space and is applicable to a
wide variety of discrete combinatorial problems.
Each node in the combinatorial tree generated in the last Unit defines a problem state.
All paths from the root to other nodes define the state space of the problem.
Solution states are those problem states 's' for which the path from the root to 's'
defines a tuple in the solution space. The leaf nodes in the combinatorial tree are the
solution states.
Answer states are those solution states 's' for which the path from the root to 's'
defines a tuple that is a member of the set of solutions (i.e.,it satisfies the implicit
constraints) of the problem.
The tree organization of the solution space is referred to as the state space tree.
A node which has been generated and all of whose children have not yet been
generated is called a live node.
The live node whose children are currently being generated is called the E-node (node
being expanded).
A dead node is a generated node, which is not to be expanded further or all of whose
children have been generated.
Bounding functions are used to kill live nodes without generating all their children.
Depth first node generation with bounding function is called backtracking. State
generation methods in which the E-node remains the E-node until it is dead lead to
branch-and-bound method.
The term branch-and-bound refers to all state space search methods in which all
children of the E-node are generated before any other live node can become the E-node.
In branch-and-bound terminology breadth first search(BFS)- like state space search
will be called FIFO (First In First Output) search as the list of live nodes is a first -in-first
-out list(or queue).
A D-search (depth search) state space search will be called LIFO (Last In First Out)
search, as the list of live nodes is a list-in-first-out list (or stack).
1
2. Bounding functions are used to help avoid the generation of sub trees that do not
contain an answer node.
The branch-and-bound algorithms search a tree model of the solution space to get the
solution. However, this type of algorithms is oriented more toward optimization. An
algorithm of this type specifies a real -valued cost function for each of the nodes that
appear in the search tree.
Usually, the goal here is to find a configuration for which the cost function is
minimized. The branch-and-bound algorithms are rarely simple. They tend to be quite
complicated in many cases.
Example 8.1[4-queens] Let us see how a FIFO branch-and-bound algorithm would search
the state space tree (figure 7.2) for the 4-queens problem.
Initially, there is only one live node, node1. This represents the case in which no
queen has been placed on the chessboard. This node becomes the E-node.
It is expanded and its children, nodes2, 18, 34 and 50 are generated.
These nodes represent a chessboard with queen1 in row 1and columns 1, 2, 3, and 4
respectively.
The only live nodes 2, 18, 34, and 50.If the nodes are generated in this order, then the
next E-node are node 2.
2
3. It is expanded and the nodes 3, 8, and 13 are generated. Node 3 is immediately killed
using the bounding function. Nodes 8 and 13 are added to the queue of live nodes.
Node 18 becomes the next E-node. Nodes 19, 24, and 29 are generated. Nodes 19 and
24 are killed as a result of the bounding functions. Node 29 is added to the queue of live
nodes.
Now the E-node is node 34.Figure 8.1 shows the portion of the tree of Figure 7.2 that
is generated by a FIFO branch-and-bound search. Nodes that are killed as a result of the
bounding functions are a "B" under them.
Numbers inside the nodes correspond to the numbers in Figure 7.2.Numbers outside
the nodes give the order in which the nodes are generated by FIFO branch-and-bound.
At the time the answer node, node 31, is reached, the only live nodes remaining are
nodes 38 and 54.
Least Cost (LC) Search:
In both LIFO and FIFO branch-and-bound the selection rule for the next E-node
is rather rigid and in a sense blind. The selection rule for the next E-node does not give
any preference to a node that has a very good chance of getting the search to an answer
node quickly.
Thus, in Example 8.1, when node 30 is generated, it should have become obvious to
the search algorithm that this node will lead to answer node in one move. However, the
rigid FIFO rule first requires the expansion of all live nodes generated before node 30
was expanded.
3
4. The search for an answer node can often be speeded by using an "intelligent" ranking
function (.) for live nodes. The next E-node is selected on the basis of this ranking
function.
If in the 4-queens example we use a ranking function that assigns node 30 a better
rank than all other live nodes, then node 30 will become E-node, following node 29.The
remaining live nodes will never become E-nodes as the expansion of node 30 results in
the generation of an answer node (node 31).
The ideal way to assign ranks would be on the basis of the additional computational
effort (or cost) needed to reach an answer node from the live node. For any node x, this
cost could be
(1) The number of nodes on the sub-tree x that need to be generated before any
answer node is generated or, more simply,
(2) The number of levels the nearest answer node (in the sub-tree x) is from x
Using cost measure (2), the cost of the root of the tree of Figure 8.1 is 4 (node 31 is
four levels from node 1).The costs of nodes 18 and 34,29 and 35,and 30 and 38 are
respectively 3, 2, and 1.The costs of all remaining nodes on levels 2, 3, and 4 are
respectively greater than 3, 2, and 1.
Using these costs as a basis to select the next E-node, the E-nodes are nodes 1, 18, 29,
and 30 (in that order).The only other nodes to get generated are nodes 2, 34, 50, 19, 24,
32, and 31.
The difficulty of using the ideal cost function is that computing the cost of a node
usually involves a search of the sub-tree x for an answer node. Hence, by the time the
cost of a node is determined, that sub-tree has been searched and there is no need to
explore x again. For this reason, search algorithms usually rank nodes only based on an
estimate (.) of their cost.
Let (x) be an estimate of the additional effort needed to reach an answer node from x.
node x is assigned a rank using a function (.) such that (x) =f (h(x)) + (x), where h(x)
is the cost of reaching x from the root and f(.) is any non-decreasing function.
A search strategy that uses a cost function (x) =f (h(x)) + (x), to select the next e-
node would always choose for its next e-node a live node with least (.).Hence, such a
strategy is called an LC-search (least cost search).
Cost function c (.) is defined as, if x is an answer node, then c(x) is the cost (level,
computational difficulty, etc.) of reaching x from the root of the state space tree. If x is
not an answer node, then c(x) =infinity, providing the sub-tree x contains no answer
node; otherwise c(x) is equals the cost of a minimum cost answer node in the sub-tree x.
4
5. It should be easy to see that (.) with f (h(x)) =h(x) is an approximation to c (.). From
now on (x) is referred to as the cost of x.
Bounding:
A branch -and-bound searches the state space tree using any search mechanism in
which all the children of the E-node are generated before another node becomes the E-
node.
We assume that each answer node x has a cost c(x) associated with it and that a
minimum-cost answer node is to be found. Three common search strategies are FIFO,
LIFO, and LC.
A cost function (.) such that (x) <=c(x) is used to provide lower bounds on
solutions obtainable from any node x. If upper is an upper bound on the cost of a
minimum-cost solution, then all live nodes x with (x)>upper may be killed as all
answer nodes reachable from x have cost c(x)>= (x)>upper. The starting value for upper
can be set to infinity.
Clearly, so long as the initial value for upper is no less than the cost of a minimum-
cost answer node, the above rule to kill live nodes will not result in the killing of a live
node that can reach a minimum-cost answer node .Each time a new answer is found ,the
value of upper can be updated.
As an example optimization problem, consider the problem of job scheduling with
deadlines. We generalize this problem to allow jobs with different processing times. We
are given n jobs and one processor. Each job i has associated with it a three tuple
( ).job i requires units of processing time .if its processing is not completed by
the deadline , and then a penalty is incurred.
The objective is to select a subset j of the n jobs such that all jobs in j can be
completed by their deadlines. Hence, a penalty can be incurred only on those jobs not in
j. The subset j should be such that the penalty incurred is minimum among all possible
subsets j. such a j is optimal.
Consider the following instances: n=4,( , , )=(5,1,1),( , , )=(10,3,2),(
, , )=(6,2,1),and( , , )=(3,1,1).The solution space for this instances consists
of all possible subsets of the job index set{1,2,3,4}. The solution space can be organized
into a tree by means of either of the two formulations used for the sum of subsets
problem.
5
6. Figure 8.7
Figure 8.6 corresponds to the variable tuple size formulations while figure 8.7
corresponds to the fixed tuple size formulation. In both figures square nodes represent
infeasible subsets. In figure 8.6 all non-square nodes are answer nodes. Node 9 represents
an optimal solution and is the only minimum-cost answer node .For this node j= {2,3}
and the penalty (cost) is 8. In figure 8.7 only non-square leaf nodes are answer nodes.
Node 25 represents the optimal solution and is also a minimum-cost answer node. This
node corresponds to j={2,3} and a penalty of 8. The costs of the answer nodes of figure
8.7 are given below the nodes.
6
7. TRAVELLING SALESMAN PROBLEM
INTRODUCTION:
It is algorithmic procedures similar to backtracking in which a new branch is
chosen and is there (bound there) until new branch is choosing for advancing.
This technique is implemented in the traveling salesman problem [TSP] which are
asymmetric (Cij <>Cij) where this technique is an effective procedure.
STEPS INVOLVED IN THIS PROCEDURE ARE AS FOLLOWS:
STEP 0: Generate cost matrix C [for the given graph g]
STEP 1: [ROW REDUCTION]
For all rows do step 2
STEP: Find least cost in a row and negate it with rest of the
elements.
STEP 3: [COLUMN REDUCTION]
Use cost matrix- Row reduced one for all columns do STEP 4.
STEP 4: Find least cost in a column and negate it with rest of the elements.
STEP 5: Preserve cost matrix C [which row reduced first and then column reduced]
for the i th
time.
STEP 6: Enlist all edges (i, j) having cost = 0.
STEP 7: Calculate effective cost of the edges. ∑ (i, j)=least cost in the i th
row
excluding (i, j) + least cost in the j th
column excluding (i, j).
STEP 8: Compare all effective cost and pick up the largest l. If two or more have
same cost then arbitrarily choose any one among them.
STEP 9: Delete (i, j) means delete ith
row and jth
column change (j, i) value to
infinity. (Used to avoid infinite loop formation) If (i,j) not present, leave it.
STEP 10: Repeat step 1 to step 9 until the resultant cost matrix having order of 2*2
and reduce it. (Both R.R and C.C)
STEP 11: Use preserved cost matrix Cn, Cn-1… C1
7
8. Choose an edge [i, j] having value =0, at the first time for a preserved
matrix and leave that matrix.
STEP 12: Use result obtained in Step 11 to generate a complete tour.
EXAMPLE: Given graph G
22 5
27 25
9 19
25
8
1
50
10 31 17 15
6 40
30
7
6
24
MATRIX:
1 2 3 4 5
1
2
3
4
5
8
α 25 40 31 27
5 α 17 30 25
19 15 α 6 1
9 50 24 α 6
22 8 7 10 α
1
5 2
4 3
16. 1
2
3
4
5
STEP 12:
i) Use C4 =
2 3
3
5
Pick up an edge (I, j) =0 having least index
Here (3,2) =0
Hence, T (3,2)
Use C3 =
2 3 4
1
3
5
Pick up an edge (i, j) =0 having least index
Here (1,4) =0
Hence, T(3,2), (1,4)
Use C2=
2 3 4 5
16
α 0 15 3 2
0 α 12 22 20
18 14 α 2 0
3 44 18 α 0
15 1 0 0 α
0 α
0 0
α 12 0
11 α 0
0 0 α
17. 1
3
4
5
Pick up an edge (i, j) with least cost index.
Here (1,5) not possible because already chosen index i (i=j)
(3,5) not possible as already chosen index.
(4,5) 0
Hence, T (3,2), (1,4), (4,5)
Use C1 =
1 2 3 4 5
1
2
3
4
5
Pick up an edge (i, j) with least index
(1,2) Not possible
(2,1) Choose it
HENCE T (3,2), (1,4), (4,5), (2,1)
SOLUTION:
17
α 13 1 0
13 α 2 0
43 18 α 0
0 0 0 α
α 0 15 3 2
0 α 12 22 20
18 14 α 2 0
3 44 18 α 0
15 1 0 0 α
18. From the above list
3—2—1—4—5
This result now, we have to return to the same city where we started (Here 3).
Final result:
3—2—1—4—5—3
Cost is 15+15+31+6+7=64
18