The document discusses bipartite matching and describes how to find the maximum matching between two disjoint sets using the augmenting path algorithm. It explains that the algorithm works by searching for augmenting paths between unmatched vertices, reversing any that are found to increase the size of the matching, and repeating until no more augmenting paths exist. The time complexity of this algorithm is O(V×E) where V is the number of vertices and E is the number of edges.
The document discusses the following:
1. There will be a quiz on Jan 27 covering sections 1.4, 1.5, 1.7, and 1.8 and any issues with quiz 1 should be discussed asap.
2. Test 1 will be on Feb 1 in class with more details to come.
3. Matrix multiplication is defined only when the number of columns of the first matrix equals the number of rows of the second matrix.
In graph theory, a matching is a subset of a graph's edges such hat no two edges meet the same vertex. A matching is maximum if no other matching contains more edges. A trivial solution (exhaustive search) to the problem of finding a maximum matching has exponential complexity. We illustrate polynomial time solutions to the problem that were published between 1965 and 1991.
The document discusses graphs and their applications. It defines key graph terms like vertices, edges, directed/undirected graphs, paths, cycles, etc. It then describes algorithms for finding minimum spanning trees, Eulerian cycles, Hamiltonian paths, and approximations for the traveling salesman problem. Examples are provided to illustrate minimum spanning tree and Christofide's algorithms for TSP.
The Floyd-Warshall algorithm is used to find the shortest paths between all pairs of vertices in a weighted graph. It creates distance and sequence tables with rows and columns for each vertex. The distance table stores the weights of edges between vertices. The sequence table stores the path vertices to reach between each pair. It iterates through the tables, updating distances and paths at each iteration based on whether a new shortest path is found through a third vertex. When complete, the final distance table shows the shortest distance between each pair of vertices, and the sequence table shows the shortest path.
This document summarizes the Stepping Stone Method used in Operations Research to determine an optimal initial basic feasible solution for transportation problems. The method treats the transportation table as a pond that can only be crossed using occupied cells as stepping stones. It involves forming closed loops through unused and occupied cells, calculating improvement indices, and reallocating units to find an optimal solution. The example problem shows applying the method to find the optimal allocation of products from warehouses to destinations.
The document discusses minimum spanning trees in graphs. A minimum spanning tree is a subgraph that connects all vertices of a graph with the minimum total weight of edges. Two common algorithms for finding minimum spanning trees are Prim's algorithm and Kruskal's algorithm. Prim's algorithm grows a connected component by adding the smallest weighted edge that leaves the component. Kruskal's algorithm considers edges from lowest to highest weight and adds them if they do not cause cycles. The cut optimality and path optimality theorems relate to properties of minimum spanning trees.
This document discusses low-density parity-check (LDPC) codes and their decoding using belief propagation on factor graphs. It introduces LDPC codes and their representation by sparse parity-check matrices and Tanner graphs. It describes irregular and regular LDPC codes, degree distributions, code ensembles, and decoding using belief propagation on factor graphs and the sum-product algorithm. Examples of decoding a LDPC code over a binary-input additive white Gaussian noise channel are also presented.
The document describes three problems involving determining the optimal sequence of jobs through multiple machines to minimize the total elapsed time.
For the first problem involving two machines, the optimal sequence is job 2, 1, 6, 5, 4, 3 with a total elapsed time of 85 hours.
The second problem involving three machines is converted to two virtual machines, and the optimal sequence is job 3, 4, 2, 1, 5 with a total elapsed time of 51 hours.
The third problem involving four machines is also converted to two virtual machines, and the optimal sequence is job C, A, B, D with a total elapsed time of 82 hours.
The document discusses the following:
1. There will be a quiz on Jan 27 covering sections 1.4, 1.5, 1.7, and 1.8 and any issues with quiz 1 should be discussed asap.
2. Test 1 will be on Feb 1 in class with more details to come.
3. Matrix multiplication is defined only when the number of columns of the first matrix equals the number of rows of the second matrix.
In graph theory, a matching is a subset of a graph's edges such hat no two edges meet the same vertex. A matching is maximum if no other matching contains more edges. A trivial solution (exhaustive search) to the problem of finding a maximum matching has exponential complexity. We illustrate polynomial time solutions to the problem that were published between 1965 and 1991.
The document discusses graphs and their applications. It defines key graph terms like vertices, edges, directed/undirected graphs, paths, cycles, etc. It then describes algorithms for finding minimum spanning trees, Eulerian cycles, Hamiltonian paths, and approximations for the traveling salesman problem. Examples are provided to illustrate minimum spanning tree and Christofide's algorithms for TSP.
The Floyd-Warshall algorithm is used to find the shortest paths between all pairs of vertices in a weighted graph. It creates distance and sequence tables with rows and columns for each vertex. The distance table stores the weights of edges between vertices. The sequence table stores the path vertices to reach between each pair. It iterates through the tables, updating distances and paths at each iteration based on whether a new shortest path is found through a third vertex. When complete, the final distance table shows the shortest distance between each pair of vertices, and the sequence table shows the shortest path.
This document summarizes the Stepping Stone Method used in Operations Research to determine an optimal initial basic feasible solution for transportation problems. The method treats the transportation table as a pond that can only be crossed using occupied cells as stepping stones. It involves forming closed loops through unused and occupied cells, calculating improvement indices, and reallocating units to find an optimal solution. The example problem shows applying the method to find the optimal allocation of products from warehouses to destinations.
The document discusses minimum spanning trees in graphs. A minimum spanning tree is a subgraph that connects all vertices of a graph with the minimum total weight of edges. Two common algorithms for finding minimum spanning trees are Prim's algorithm and Kruskal's algorithm. Prim's algorithm grows a connected component by adding the smallest weighted edge that leaves the component. Kruskal's algorithm considers edges from lowest to highest weight and adds them if they do not cause cycles. The cut optimality and path optimality theorems relate to properties of minimum spanning trees.
This document discusses low-density parity-check (LDPC) codes and their decoding using belief propagation on factor graphs. It introduces LDPC codes and their representation by sparse parity-check matrices and Tanner graphs. It describes irregular and regular LDPC codes, degree distributions, code ensembles, and decoding using belief propagation on factor graphs and the sum-product algorithm. Examples of decoding a LDPC code over a binary-input additive white Gaussian noise channel are also presented.
The document describes three problems involving determining the optimal sequence of jobs through multiple machines to minimize the total elapsed time.
For the first problem involving two machines, the optimal sequence is job 2, 1, 6, 5, 4, 3 with a total elapsed time of 85 hours.
The second problem involving three machines is converted to two virtual machines, and the optimal sequence is job 3, 4, 2, 1, 5 with a total elapsed time of 51 hours.
The third problem involving four machines is also converted to two virtual machines, and the optimal sequence is job C, A, B, D with a total elapsed time of 82 hours.
This document introduces the topic of graph theory. It defines what graphs are, including vertices, edges, directed and undirected graphs. It provides examples of graphs like social networks, transportation maps, and more. It covers basic graph terminology such as degree, regular graphs, subgraphs, walks, paths and cycles. It also discusses graph classes like trees, complete graphs and bipartite graphs. Finally, it touches on some historical graph problems, complexity analysis, centrality analysis, facility location problems and applications of graph theory.
The document discusses shortest path problems and algorithms. It defines the shortest path problem as finding the minimum weight path between two vertices in a weighted graph. It presents the Bellman-Ford algorithm, which can handle graphs with negative edge weights but detects negative cycles. It also presents Dijkstra's algorithm, which only works for graphs without negative edge weights. Key steps of the algorithms include initialization, relaxation of edges to update distance estimates, and ensuring the shortest path property is satisfied.
1. The document introduces vectors and matrices as ways to collectively represent multiple quantities or relationships between quantities.
2. Vectors are used to represent positions, food orders, prices, and other grouped data. Matrices are used to represent ingredient amounts for different foods and connections between rooms in a floorplan.
3. All of the examples can be expressed using vectors and matrices, with the key information being the numbers in the vectors and matrices.
Dijkstra's algorithm finds the shortest paths between vertices in a graph with non-negative edge weights. It works by maintaining distances from the source vertex to all other vertices, initially setting all distances to infinity except the source which is 0. It then iteratively selects the unvisited vertex with the lowest distance, marks it as visited, and updates the distances to its neighbors if a shorter path is found through the selected vertex. This continues until all vertices are visited, at which point the distances will be the shortest paths from the source vertex.
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 6: Normal Probability Distribution
6.6: Normal as Approximation to Binomial
This document provides an introduction to networks and matrices. It defines key network terminology such as nodes, edges, degrees, paths, circuits, trees and graphs. It discusses using networks to model real-world systems and solve problems. Examples of network problems presented include the shortest path problem, shortest connection problem, Chinese postman problem and travelling salesman problem. The historical origins and applications of network theory are also summarized.
This presentation summarizes Kruskal's algorithm and Prim's algorithm for finding minimum spanning trees in graphs. Kruskal's algorithm works by selecting the shortest edge in a network that does not create a cycle, repeating until all vertices are connected. Prim's algorithm starts with any vertex and selects the shortest edge connected to vertices already in the tree, repeating until all vertices are connected. Both algorithms are demonstrated on a example network to connect five villages to a market town, finding the minimum total edge length of 18 using either algorithm by connecting edges of length 2, 3, 4, 4, and 5.
This document discusses iterative methods for solving systems of equations. It introduces the Jacobi iteration method and the Successive Over-Relaxation (SOR) method. SOR can accelerate the convergence compared to Jacobi by introducing an optimal relaxation parameter. Pseudocode is provided to implement SOR to iteratively solve a system of equations until the solution converges within a specified tolerance.
This document provides an overview of graph theory concepts including:
- The basics of graphs including definitions of vertices, edges, paths, cycles, and graph representations like adjacency matrices.
- Minimum spanning tree algorithms like Kruskal's and Prim's which find a spanning tree with minimum total edge weight.
- Graph coloring problems and their applications to scheduling problems.
- Other graph concepts covered include degree, Eulerian paths, planar graphs and graph isomorphism.
Graph isomorphism is the problem of determining whether two graphs are topologically identical. It is in NP but not known to be NP-complete. The refinement heuristic iteratively partitions vertices into equivalence classes to reduce possible mappings between graphs. For trees, there is a linear-time algorithm that labels vertices based on subtree structure to test rooted and unrooted tree isomorphism.
This document provides an overview of graph theory, including its history and key concepts. It defines a graph as a set of nodes connected by edges. Important early contributors are noted, such as Euler's work on the Seven Bridges of Königsberg problem. Exact definitions of graph types like simple, directed, and weighted graphs are given. Key graph concepts explained in less than three sentences include connectivity, degree, paths, cycles, and trees. The document also briefly discusses representing graphs through adjacency lists and matrices. It concludes by thanking the reader and listing references.
FPGA Implementation of FIR Filter using Various Algorithms: A RetrospectiveIJORCS
This Paper is a review study of FPGA implementation of Finite Impulse response (FIR) with low cost and high performance. The key observation of this paper is an elaborate analysis about hardware implementations of FIR filters using different algorithm i.e., Distributed Arithmetic (DA), DA-Offset Binary Coding (DA-OBC), Common Sub-expression Elimination (CSE) and sum-of-power-of-two (SOPOT) with less resources and without affecting the performance of the original FIR Filter.
Romberg's method is used to estimate definite integrals by applying Richardson extrapolation repeatedly to the trapezoidal rule or rectangular rule. This generates a triangular array that increases in accuracy. The method is an extension of trapezoidal and rectangular rules. It works by recursively calculating the integral using smaller step sizes to generate values in the triangular array. Convergence is reached when two successive values are very close. An example calculates a definite integral using Romberg's method in three cases with decreasing step sizes to populate the triangular array.
This document describes an improved Gilbert multiplier circuit and its applications. The proposed circuit modifies an existing 12T Gilbert cell by grounding the body terminals of nMOS transistors and connecting one body terminal to a control voltage. Simulation results show the proposed circuit has lower power consumption and faster speed than the existing design. Key applications discussed include frequency doubling, modulation, and demodulation. The circuit can double the frequency of an input signal. It can also modulate one sinusoidal or triangular signal onto another carrier signal. By adding capacitors, the circuit can be used for demodulation to reconstruct the transmitted signal.
This document summarizes the two phase simplex method for solving linear programming problems. In phase I, artificial variables are introduced to convert infeasible problems into feasible problems. The objective is to minimize the artificial variables. If the minimum is zero, the original problem is feasible and phase II begins. Phase II uses the original objective function and simplex method to find an optimal solution. An example problem is provided to illustrate the two phase method.
The document outlines Reed-Solomon error correction codes. It discusses how Reed-Solomon codes encode data using a generator polynomial to produce parity check symbols. The document then describes how Reed-Solomon codes can decode errors using syndrome calculation, error location polynomials, and finding the error positions and values through algorithms like Forney's method and Chien search. Reed-Solomon codes are widely used in applications like CDs, DVDs, wireless communications and digital television for their ability to efficiently correct both random and burst errors.
An ideal in a ring is called a principal ideal if it is generated by a single element of the ring. The First Fundamental Theorem of Ideals states that every ideal in a principal ideal domain is a principal ideal.
This document provides examples of constructing the dual problem of a linear programming primal problem and solving it using the two-phase simplex method. It first presents the rules for constructing the dual problem and then works through two examples. The first example derives the dual problem from the primal and solves it using the two-phase method. The second example shows how to find the optimal dual solution given the optimal primal solution using two methods - using the objective coefficients of the primal variables or using the inverse of the primal basic variable matrix.
The document discusses graphs and Eulerian circuits and paths. It defines what a graph is composed of and defines an Eulerian circuit and path. It states that a graph must be connected and have all vertices visited once for an Eulerian circuit, or have two odd vertices for an Eulerian path. Fleury's algorithm is described for finding an Euler circuit or path by traversing edges without crossing bridges twice. The algorithm works by choosing the next edge such that bridges are only crossed if necessary. Examples are given to demonstrate the algorithm. Applications mentioned include the Chinese postman problem and communicating networks.
The document describes Dinic's algorithm for finding the maximum flow in a flow network. It involves building a level graph of the network and finding blocking flows by finding augmenting paths from the source to the sink, then identifying bottlenecks on those paths and pushing flow along reverse paths from the bottlenecks to the sink in an iterative manner until no more augmenting paths exist. An example applying these steps on a sample network is provided to illustrate the algorithm.
This document describes an algorithm for determining if two trees are isomorphic by hashing the trees. It hashes each subtree and sorts the hashes to get a unique representation of the tree. It recursively hashes subtrees from leaves to root. The hash of the whole tree represents the tree and trees are isomorphic if they have the same hash value. The time complexity is O(N log N) where N is the number of vertices. Pseudocode and examples are provided.
This document introduces the topic of graph theory. It defines what graphs are, including vertices, edges, directed and undirected graphs. It provides examples of graphs like social networks, transportation maps, and more. It covers basic graph terminology such as degree, regular graphs, subgraphs, walks, paths and cycles. It also discusses graph classes like trees, complete graphs and bipartite graphs. Finally, it touches on some historical graph problems, complexity analysis, centrality analysis, facility location problems and applications of graph theory.
The document discusses shortest path problems and algorithms. It defines the shortest path problem as finding the minimum weight path between two vertices in a weighted graph. It presents the Bellman-Ford algorithm, which can handle graphs with negative edge weights but detects negative cycles. It also presents Dijkstra's algorithm, which only works for graphs without negative edge weights. Key steps of the algorithms include initialization, relaxation of edges to update distance estimates, and ensuring the shortest path property is satisfied.
1. The document introduces vectors and matrices as ways to collectively represent multiple quantities or relationships between quantities.
2. Vectors are used to represent positions, food orders, prices, and other grouped data. Matrices are used to represent ingredient amounts for different foods and connections between rooms in a floorplan.
3. All of the examples can be expressed using vectors and matrices, with the key information being the numbers in the vectors and matrices.
Dijkstra's algorithm finds the shortest paths between vertices in a graph with non-negative edge weights. It works by maintaining distances from the source vertex to all other vertices, initially setting all distances to infinity except the source which is 0. It then iteratively selects the unvisited vertex with the lowest distance, marks it as visited, and updates the distances to its neighbors if a shorter path is found through the selected vertex. This continues until all vertices are visited, at which point the distances will be the shortest paths from the source vertex.
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Chapter 6: Normal Probability Distribution
6.6: Normal as Approximation to Binomial
This document provides an introduction to networks and matrices. It defines key network terminology such as nodes, edges, degrees, paths, circuits, trees and graphs. It discusses using networks to model real-world systems and solve problems. Examples of network problems presented include the shortest path problem, shortest connection problem, Chinese postman problem and travelling salesman problem. The historical origins and applications of network theory are also summarized.
This presentation summarizes Kruskal's algorithm and Prim's algorithm for finding minimum spanning trees in graphs. Kruskal's algorithm works by selecting the shortest edge in a network that does not create a cycle, repeating until all vertices are connected. Prim's algorithm starts with any vertex and selects the shortest edge connected to vertices already in the tree, repeating until all vertices are connected. Both algorithms are demonstrated on a example network to connect five villages to a market town, finding the minimum total edge length of 18 using either algorithm by connecting edges of length 2, 3, 4, 4, and 5.
This document discusses iterative methods for solving systems of equations. It introduces the Jacobi iteration method and the Successive Over-Relaxation (SOR) method. SOR can accelerate the convergence compared to Jacobi by introducing an optimal relaxation parameter. Pseudocode is provided to implement SOR to iteratively solve a system of equations until the solution converges within a specified tolerance.
This document provides an overview of graph theory concepts including:
- The basics of graphs including definitions of vertices, edges, paths, cycles, and graph representations like adjacency matrices.
- Minimum spanning tree algorithms like Kruskal's and Prim's which find a spanning tree with minimum total edge weight.
- Graph coloring problems and their applications to scheduling problems.
- Other graph concepts covered include degree, Eulerian paths, planar graphs and graph isomorphism.
Graph isomorphism is the problem of determining whether two graphs are topologically identical. It is in NP but not known to be NP-complete. The refinement heuristic iteratively partitions vertices into equivalence classes to reduce possible mappings between graphs. For trees, there is a linear-time algorithm that labels vertices based on subtree structure to test rooted and unrooted tree isomorphism.
This document provides an overview of graph theory, including its history and key concepts. It defines a graph as a set of nodes connected by edges. Important early contributors are noted, such as Euler's work on the Seven Bridges of Königsberg problem. Exact definitions of graph types like simple, directed, and weighted graphs are given. Key graph concepts explained in less than three sentences include connectivity, degree, paths, cycles, and trees. The document also briefly discusses representing graphs through adjacency lists and matrices. It concludes by thanking the reader and listing references.
FPGA Implementation of FIR Filter using Various Algorithms: A RetrospectiveIJORCS
This Paper is a review study of FPGA implementation of Finite Impulse response (FIR) with low cost and high performance. The key observation of this paper is an elaborate analysis about hardware implementations of FIR filters using different algorithm i.e., Distributed Arithmetic (DA), DA-Offset Binary Coding (DA-OBC), Common Sub-expression Elimination (CSE) and sum-of-power-of-two (SOPOT) with less resources and without affecting the performance of the original FIR Filter.
Romberg's method is used to estimate definite integrals by applying Richardson extrapolation repeatedly to the trapezoidal rule or rectangular rule. This generates a triangular array that increases in accuracy. The method is an extension of trapezoidal and rectangular rules. It works by recursively calculating the integral using smaller step sizes to generate values in the triangular array. Convergence is reached when two successive values are very close. An example calculates a definite integral using Romberg's method in three cases with decreasing step sizes to populate the triangular array.
This document describes an improved Gilbert multiplier circuit and its applications. The proposed circuit modifies an existing 12T Gilbert cell by grounding the body terminals of nMOS transistors and connecting one body terminal to a control voltage. Simulation results show the proposed circuit has lower power consumption and faster speed than the existing design. Key applications discussed include frequency doubling, modulation, and demodulation. The circuit can double the frequency of an input signal. It can also modulate one sinusoidal or triangular signal onto another carrier signal. By adding capacitors, the circuit can be used for demodulation to reconstruct the transmitted signal.
This document summarizes the two phase simplex method for solving linear programming problems. In phase I, artificial variables are introduced to convert infeasible problems into feasible problems. The objective is to minimize the artificial variables. If the minimum is zero, the original problem is feasible and phase II begins. Phase II uses the original objective function and simplex method to find an optimal solution. An example problem is provided to illustrate the two phase method.
The document outlines Reed-Solomon error correction codes. It discusses how Reed-Solomon codes encode data using a generator polynomial to produce parity check symbols. The document then describes how Reed-Solomon codes can decode errors using syndrome calculation, error location polynomials, and finding the error positions and values through algorithms like Forney's method and Chien search. Reed-Solomon codes are widely used in applications like CDs, DVDs, wireless communications and digital television for their ability to efficiently correct both random and burst errors.
An ideal in a ring is called a principal ideal if it is generated by a single element of the ring. The First Fundamental Theorem of Ideals states that every ideal in a principal ideal domain is a principal ideal.
This document provides examples of constructing the dual problem of a linear programming primal problem and solving it using the two-phase simplex method. It first presents the rules for constructing the dual problem and then works through two examples. The first example derives the dual problem from the primal and solves it using the two-phase method. The second example shows how to find the optimal dual solution given the optimal primal solution using two methods - using the objective coefficients of the primal variables or using the inverse of the primal basic variable matrix.
The document discusses graphs and Eulerian circuits and paths. It defines what a graph is composed of and defines an Eulerian circuit and path. It states that a graph must be connected and have all vertices visited once for an Eulerian circuit, or have two odd vertices for an Eulerian path. Fleury's algorithm is described for finding an Euler circuit or path by traversing edges without crossing bridges twice. The algorithm works by choosing the next edge such that bridges are only crossed if necessary. Examples are given to demonstrate the algorithm. Applications mentioned include the Chinese postman problem and communicating networks.
The document describes Dinic's algorithm for finding the maximum flow in a flow network. It involves building a level graph of the network and finding blocking flows by finding augmenting paths from the source to the sink, then identifying bottlenecks on those paths and pushing flow along reverse paths from the bottlenecks to the sink in an iterative manner until no more augmenting paths exist. An example applying these steps on a sample network is provided to illustrate the algorithm.
This document describes an algorithm for determining if two trees are isomorphic by hashing the trees. It hashes each subtree and sorts the hashes to get a unique representation of the tree. It recursively hashes subtrees from leaves to root. The hash of the whole tree represents the tree and trees are isomorphic if they have the same hash value. The time complexity is O(N log N) where N is the number of vertices. Pseudocode and examples are provided.
Teaching Graph Algorithms in the Field - Bipartite Matching in optical datace...Kostas Katrinis
This document discusses using bipartite matching and optical MEMS switches to optimize data center networks. It begins with background on data center traffic trends and the need for higher capacity and lower cost networks. It then introduces a hybrid electrical-optical network architecture using MEMS optical switches to increase bandwidth between servers. Bipartite graphs and matchings are introduced as a way to model and maximize the use of optical switch ports. The Hungarian algorithm is presented as a method for finding maximum matchings in bipartite graphs, allowing the number of optical connections to be maximized. Pseudocode and examples are provided to illustrate how the algorithm works.
The document discusses graph traversal algorithms depth-first search (DFS) and breadth-first search (BFS). It provides examples of how DFS and BFS traverse graphs using a stack and queue respectively. Pseudocode for DFS and BFS algorithms is presented. The time complexity of both algorithms is O(V+E) where V is the number of nodes and E is the number of edges. Examples of DFS and BFS traversing a mouse maze are also shown.
The document discusses minimum cuts in graphs. It defines a minimum cut as a partition of the vertices of a graph into two disjoint subsets such that the number/weight of edges between the two subsets is minimized. For directed graphs, the cut-set of a cut consists only of edges going from one subset to the other. The maximum flow minimum cut theorem states that the maximum amount of flow passing from a source to sink node is equal to the capacity of the minimum s-t cut.
The document discusses Rust's ownership system and borrowing. It explains that variables own the memory for their values, and when a variable goes out of scope that memory is returned. References allow borrowing values without transferring ownership. References must live shorter than the values they reference. Mutable references also allow changing borrowed values, but there can only be one mutable reference at a time.
The document discusses special types of graph structures including complete graphs, cycles, wheels, and bipartite graphs. It provides examples of how to determine if a graph is bipartite by checking if all its cycles are of even length. The document also mentions subgraphs and spanning subgraphs, defining a subgraph as one whose edges are a subset of the original graph and a spanning subgraph as one that contains all the original vertices.
Graph terminologies & special type graphsNabeel Ahsen
The document discusses various graph terminologies and special types of graphs. It defines undirected and directed graphs, and describes degrees, handshaking theorem, and other properties. Special graph types covered include complete graphs, cycles, wheels, n-cubes, and bipartite graphs. It provides examples of constructing new graphs from existing ones and an application using a bipartite graph model for employee skills and job assignments.
The document discusses the knapsack problem and provides examples to illustrate the recursive solution approach. It describes the knapsack problem as selecting items to place in a knapsack with a weight capacity to maximize the total benefit without exceeding the weight limit. It shows that the optimal solution can be found using a recursive function f(w) that returns the maximum benefit for a knapsack of weight w. The examples demonstrate computing f(w) values to find the optimal solutions for sample knapsack problems.
Application of local search methods for solving a quadratic assignment proble...ertekg
Ertek, G., Aksu, B., Birbil, S. E., İkikat, M. C., Yıldırmaz, C. (2005). “Application of local search methods for solving a quadratic assignment problem: A case study”, Proceedings of Computers and Industrial Engineering Conference, 2005. Istanbul, Turkey.
Knapsack problem ==>>
Given some items, pack the knapsack to get
the maximum total value. Each item has some
weight and some value. Total weight that we can
carry is no more than some fixed number W.
So we must consider weights of items as well as
their values.
This document provides an introduction to greedy algorithms. It defines greedy algorithms as algorithms that make locally optimal choices at each step in the hope of finding a global optimum. The document then provides examples of problems that can be solved using greedy algorithms, including counting money, scheduling jobs, finding minimum spanning trees, and the traveling salesman problem. It also provides pseudocode for a general greedy algorithm and discusses some properties of greedy algorithms.
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 greedy method algorithm design paradigm. It can be used to solve optimization problems with the greedy-choice property, where choosing locally optimal decisions at each step leads to a globally optimal solution. Examples discussed include fractional knapsack problem, task scheduling, and making change problem. The greedy algorithm works by always making the choice that looks best at the moment, without considering future implications of that choice.
This document outlines greedy algorithms, their characteristics, and examples of their use. Greedy algorithms make locally optimal choices at each step in the hopes of finding a global optimum. They are simple to implement and fast, but may not always reach the true optimal solution. Examples discussed include coin changing, traveling salesman, minimum spanning trees using Kruskal's and Prim's algorithms, and Huffman coding.
The document discusses the 0/1 knapsack problem and the greedy algorithm approach. It describes the knapsack problem as selecting a subset of items with weights and values that fit within a knapsack capacity while maximizing the total value. The greedy algorithm works by selecting the highest value item at each step that fits within remaining capacity. The document provides an example problem of selecting boxes to fill a knapsack of 15kg capacity to maximize profit. It outlines the recurrence relation and time/space complexity of the greedy knapsack algorithm.
Greedy algorithms make locally optimal choices at each step to try to find a global optimum. They choose the best option available at each state. For the travelling salesman problem, the greedy approach is to always choose the nearest unvisited city from the current city to build the route. While greedy algorithms are simple to implement, they do not always find the true optimal solution, especially for large complex problems, as they only consider the best choice at each step rather than the overall route.
The document discusses the knapsack problem, which involves selecting a subset of items that fit within a knapsack of limited capacity to maximize the total value. There are two versions - the 0-1 knapsack problem where items can only be selected entirely or not at all, and the fractional knapsack problem where items can be partially selected. Solutions include brute force, greedy algorithms, and dynamic programming. Dynamic programming builds up the optimal solution by considering all sub-problems.
Rust is a systems programming language that provides memory safety without using a garbage collector. It achieves memory safety through rules of ownership, borrowing, and lifetimes that are checked at compile time. These rules prevent common memory bugs like memory leaks, dangling pointers, and use-after-free errors that are common in C and C++.
The document discusses rendering dropdown menus in the content process. It notes there are performance, styling, and event behavior issues with the current e10s-select approach. An alternative approach called content-select would render the dropdown as normal content in the content process to achieve better performance, stronger style support, and more accurate events without these issues. It provides an example of complex styling that can be applied to options in the dropdown.
The document discusses concurrency in C++ and the use of std::async and std::future. It recommends preferring task-based programming over thread-based due to easier management. It notes that the default launch policy for std::async allows asynchronous or synchronous execution, creating uncertainty. It advises specifying std::launch::async if asynchronicity is essential to ensure concurrent execution and avoid issues with thread-local variables and timeout-based waits.
Use C++ to Manipulate mozSettings in GeckoChih-Hsuan Kuo
If you want to manipulate mozSettings with JavaScript, you can reference to Settings API on MDN (https://developer.mozilla.org/en-US/docs/Web/API/Settings_API).
But if you want to manipulate it with C++, we can only reference to the codebase of Gecko. Now, let me show you some example.
The document describes how Necko handles data URLs in Firefox. It discusses:
1. The data URL protocol syntax and how Necko parses data URLs to determine the MIME type, encoding, and data.
2. How Necko gets the protocol handler for data URLs and uses the nsDataHandler to create an nsDataChannel for the URL.
3. How the nsDataChannel establishes a listener chain with the nsInputStreamPump and nsHtml5StreamParser/Listener to read the data in chunks and parse the HTML content.
4. The states the nsInputStreamPump goes through - START, TRANSFER, STOP - to read the data from the nsDataChannel and
The document discusses protocol handlers in Gecko. It explains that protocol handlers allow Gecko to interact with different URI schemes like http, ftp, file etc. It provides an overview of how the awesome bar, browser UI, DocShell and Necko components work together to handle protocol requests from inputting a URL in the awesome bar to creating a channel and loading content. It also briefly introduces channels and stream listeners in Necko which are used for asynchronous loading of content.
This document provides an introduction to Linux, including what it is, some popular Linux distributions like Ubuntu, and reasons why one might use Linux instead of Windows. It notes that while Linux's desktop market share is still lower than Windows, it is widely used in servers, Android devices, and supercomputers. The document discusses how to install Ubuntu and some basic features and software available in Ubuntu. It argues that Linux is more secure, customizable, and liberating for users compared to Windows. However, it acknowledges Linux still lacks versions of some essential software and may not support all hardware.
Python is an easy to read, easy to learn, easy to write programming language. It is interpreted rather than compiled, making it easy to execute without needing to compile code first. Python is a multi-paradigm language supporting object-oriented, imperative and functional programming. The Python logo depicts a python snake to represent the name of the language. The Zen of Python outlines design philosophies for the language, preferring simple and readable code.
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The document describes the closest pair problem and an algorithm to solve it in 3 steps:
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The document describes the bubble sort, merge sort, and quick sort algorithms. Bubble sort has a time complexity of O(n^2) as it may require up to n^2 comparisons in the worst case. Merge sort has a time complexity of O(nlogn) as it divides the list into halves recursively until single elements remain, then merges the sorted halves. Quick sort also has O(nlogn) time as it partitions the list around a pivot element, recursively sorts sublists, and merges the results.
Main Java[All of the Base Concepts}.docxadhitya5119
This is part 1 of my Java Learning Journey. This Contains Custom methods, classes, constructors, packages, multithreading , try- catch block, finally block and more.
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বাংলাদেশের অর্থনৈতিক সমীক্ষা ২০২৪ [Bangladesh Economic Review 2024 Bangla.pdf] কম্পিউটার , ট্যাব ও স্মার্ট ফোন ভার্সন সহ সম্পূর্ণ বাংলা ই-বুক বা pdf বই " সুচিপত্র ...বুকমার্ক মেনু 🔖 ও হাইপার লিংক মেনু 📝👆 যুক্ত ..
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LAND USE LAND COVER AND NDVI OF MIRZAPUR DISTRICT, UPRAHUL
This Dissertation explores the particular circumstances of Mirzapur, a region located in the
core of India. Mirzapur, with its varied terrains and abundant biodiversity, offers an optimal
environment for investigating the changes in vegetation cover dynamics. Our study utilizes
advanced technologies such as GIS (Geographic Information Systems) and Remote sensing to
analyze the transformations that have taken place over the course of a decade.
The complex relationship between human activities and the environment has been the focus
of extensive research and worry. As the global community grapples with swift urbanization,
population expansion, and economic progress, the effects on natural ecosystems are becoming
more evident. A crucial element of this impact is the alteration of vegetation cover, which plays a
significant role in maintaining the ecological equilibrium of our planet.Land serves as the foundation for all human activities and provides the necessary materials for
these activities. As the most crucial natural resource, its utilization by humans results in different
'Land uses,' which are determined by both human activities and the physical characteristics of the
land.
The utilization of land is impacted by human needs and environmental factors. In countries
like India, rapid population growth and the emphasis on extensive resource exploitation can lead
to significant land degradation, adversely affecting the region's land cover.
Therefore, human intervention has significantly influenced land use patterns over many
centuries, evolving its structure over time and space. In the present era, these changes have
accelerated due to factors such as agriculture and urbanization. Information regarding land use and
cover is essential for various planning and management tasks related to the Earth's surface,
providing crucial environmental data for scientific, resource management, policy purposes, and
diverse human activities.
Accurate understanding of land use and cover is imperative for the development planning
of any area. Consequently, a wide range of professionals, including earth system scientists, land
and water managers, and urban planners, are interested in obtaining data on land use and cover
changes, conversion trends, and other related patterns. The spatial dimensions of land use and
cover support policymakers and scientists in making well-informed decisions, as alterations in
these patterns indicate shifts in economic and social conditions. Monitoring such changes with the
help of Advanced technologies like Remote Sensing and Geographic Information Systems is
crucial for coordinated efforts across different administrative levels. Advanced technologies like
Remote Sensing and Geographic Information Systems
9
Changes in vegetation cover refer to variations in the distribution, composition, and overall
structure of plant communities across different temporal and spatial scales. These changes can
occur natural.
A workshop hosted by the South African Journal of Science aimed at postgraduate students and early career researchers with little or no experience in writing and publishing journal articles.
it describes the bony anatomy including the femoral head , acetabulum, labrum . also discusses the capsule , ligaments . muscle that act on the hip joint and the range of motion are outlined. factors affecting hip joint stability and weight transmission through the joint are summarized.
19. [2 - 5 - 1 - 9] is an augmenting path.
1 5
2 6
3 7
4 8
not matching edge 9
matching edge
20. [2 - 5 - 1 - 9] is an augmenting path.
1 5
2 6
3 7
4 8
not matching edge 9
matching edge
21. Reverse the augmenting path, cardinality will increase.
1 5
2 6
3 7
4 8
not matching edge 9
matching edge
22. Augmenting Path
• The length of an augmenting path
always is odd.
• Reversing an augmenting path will
increase the cardinality.
• If there is no augmenting path, the
cardinality is maximum.
23. Algorithm
1. Try to build augmenting paths from all
vertices in one side.
2. Travel on graph.
3. If an augmenting path exists, reverse
the augmenting path to increase
cardinality.
4. If no augmenting path exists, ignore
this vertex.
5. Repeat above step until there no
augmenting path exists.
24. [1 - 5] is an augmenting path.
1 5
2 6
3 7
4 8
current cardinality: 0
9
not matching edge
matching edge
25. [1 - 5] is an augmenting path.
1 5
2 6
3 7
4 8
current cardinality: 0
9
not matching edge
matching edge
26. Reverse it!
1 5
2 6
3 7
4 8
current cardinality: 1
9
not matching edge
matching edge
27. [2 - 5 - 1- 9] is an augmenting path.
1 5
2 6
3 7
4 8
current cardinality: 1
9
not matching edge
matching edge
28. [2 - 5 - 1- 9] is an augmenting path.
1 5
2 6
3 7
4 8
current cardinality: 1
9
not matching edge
matching edge
29. Reverse it!
1 5
2 6
3 7
4 8
current cardinality: 2
9
not matching edge
matching edge
30. [3 - 6] is an augmenting path.
1 5
2 6
3 7
4 8
current cardinality: 2
9
not matching edge
matching edge
31. [3 - 6] is an augmenting path.
1 5
2 6
3 7
4 8
current cardinality: 2
9
not matching edge
matching edge
32. Reverse it!
1 5
2 6
3 7
4 8
current cardinality: 3
9
not matching edge
matching edge
33. [4 - 5 - 2 - 6 - 3 - 7] is an augmenting path.
1 5
2 6
3 7
4 8
current cardinality: 3
9
not matching edge
matching edge
34. [4 - 5 - 2 - 6 - 3 - 7] is an augmenting path.
1 5
2 6
3 7
4 8
current cardinality: 3
9
not matching edge
matching edge
35. Reverse it!
1 5
2 6
3 7
4 8
current cardinality: 4
9
not matching edge
matching edge
37. Time Complexity
O(V × E)
V: number of vertices
E: number of edges
38. Source Code
// find an augmenting path
bool find_aug_path( int x ) {
for ( int i = 0; i < ( int )vertex[ x ].size(); ++i ) {
int next = vertex[ x ][ i ];
// not in augmenting path
if ( !visit[ next ] ) {
// setup this vertex in augmenting path
visit[ next ] = 1;
/*
* If this vertex is a unmatched vertex, reverse
augmenting path and return.
* If this vector is a matched vertex, try to reverse
augmenting path and continue find an unmatched vertex.
*/
if ( lnk2[ next ] == -1 || find_aug_path( lnk2[ next ] ) )
{
lnk1[ x ] = next, lnk2[ next ] = x;
return 1;
}
}
}
return 0;
}
42. weight adjustment
If add (subtract) some value on all edges connected with
vertex X, the maximum matching won’t be effected.
1 4 1 4
a a+d
b b+d
2 5 2 5
c c+d
3 6 3 6
matching edge not matching edge
43. vertex labeling
For convenient, add a variable on vertices to denote the
value add (subtract) on edges connected with them.
1 4 1 4
a a+d
b b+d
d 2
c
5 ≡ 2
c+d
5
3 6 3 6
matching edge not matching edge
66. Algorithm
1. Initial vertex labeling to fit l(x)+l(y)≥w(x,y)
2. Find augmenting paths composed with
admissible edges from all vertices in one side.
3. If no augmenting path exists, adjust vertex
labeling until the augmenting path found.
4. If augmenting path exists, continue to find next
augmenting path.
5. Repeat above step until there no augmenting
path exists.
6. Calculate the sum of weight on matching edges.