The document discusses various algorithms that use dynamic programming. It begins by defining dynamic programming as an approach that breaks problems down into optimal subproblems. It provides examples like knapsack and shortest path problems. It describes the characteristics of problems solved with dynamic programming as having optimal subproblems and overlapping subproblems. The document then discusses specific dynamic programming algorithms like matrix chain multiplication, string editing, longest common subsequence, shortest paths (Bellman-Ford and Floyd-Warshall). It provides explanations, recurrence relations, pseudocode and examples for these algorithms.
This document discusses advanced algorithm design and analysis techniques including dynamic programming, greedy algorithms, and amortized analysis. It provides examples of dynamic programming including matrix chain multiplication and longest common subsequence. Dynamic programming works by breaking problems down into overlapping subproblems and solving each subproblem only once. Greedy algorithms make locally optimal choices at each step to find a global optimum. Amortized analysis averages the costs of a sequence of operations to determine average-case performance.
The document discusses dynamic programming and its application to the matrix chain multiplication problem. It begins by explaining dynamic programming as a bottom-up approach to solving problems by storing solutions to subproblems. It then details the matrix chain multiplication problem of finding the optimal way to parenthesize the multiplication of a chain of matrices to minimize operations. Finally, it provides an example applying dynamic programming to the matrix chain multiplication problem, showing the construction of cost and split tables to recursively build the optimal solution.
This document discusses the concept of dynamic programming. It provides examples of dynamic programming problems including assembly line scheduling and matrix chain multiplication. The key steps of a dynamic programming problem are: (1) characterize the optimal structure of a solution, (2) define the problem recursively, (3) compute the optimal solution in a bottom-up manner by solving subproblems only once and storing results, and (4) construct an optimal solution from the computed information.
Dynamic programming is used to solve optimization problems by combining solutions to overlapping subproblems. It works by breaking down problems into subproblems, solving each subproblem only once, and storing the solutions in a table to avoid recomputing them. There are two key properties for applying dynamic programming: overlapping subproblems and optimal substructure. Some applications of dynamic programming include finding shortest paths, matrix chain multiplication, the traveling salesperson problem, and knapsack problems.
Dynamic programming (DP) involves breaking problems down into overlapping subproblems. It solves each subproblem only once, storing and reusing the results through a bottom-up approach. This avoids recomputing common subproblems as in naive recursive solutions. The document discusses DP through the example of matrix chain multiplication, explaining how to characterize optimal solutions, define recursive relationships between subproblems, and construct memory-efficient algorithms to solve problems optimally in polynomial time.
Dynamic programming is used to solve optimization problems by breaking them down into overlapping subproblems. It solves subproblems only once, storing the results in a table to lookup when the same subproblem occurs again, avoiding recomputing solutions. Key steps are characterizing optimal substructures, defining solutions recursively, computing solutions bottom-up, and constructing the overall optimal solution. Examples provided are matrix chain multiplication and longest common subsequence.
Matrix chain multiplication in design analysis of algorithmRajKumar323561
This document discusses the matrix chain multiplication problem and provides an algorithm to solve it using dynamic programming. Specifically:
- The problem is to find the most efficient way to multiply a sequence of matrices by determining the optimal parenthesization that minimizes the number of scalar multiplications.
- A dynamic programming approach is used where the problem is broken down into optimal subproblems and a bottom-up method is employed to compute the solution.
- Recursive formulas are defined to calculate the minimum number of multiplications (m) needed to compute matrix chain products of increasing length. Additional data (s) tracks the optimal splitting points.
- The algorithm fills a table from bottom to top and left to right using the
Learn about dynamic programming and how to design algorithMazenulIslamKhan
Dynamic Programming (DP): A 3000-Character Description
Dynamic Programming (DP) is a powerful algorithmic technique used to solve complex problems by breaking them down into simpler subproblems and solving each of those subproblems only once. It is especially useful for optimization problems, where the goal is to find the best possible solution from a set of feasible solutions. DP avoids the repeated calculation of the same subproblem by storing the results of solved subproblems in a table (usually an array or matrix) and reusing those results when needed. This approach is known as memoization when done recursively and tabulation when done iteratively.
The main idea behind dynamic programming is the principle of optimal substructure, which means that the solution to a problem can be composed of optimal solutions to its subproblems. Additionally, DP problems exhibit overlapping subproblems, meaning the same subproblems are solved multiple times during the execution of a naive recursive solution. By solving each unique subproblem just once and storing its result, dynamic programming reduces the time complexity significantly compared to a naive approach like brute-force recursion.
DP is commonly applied in a variety of domains such as computer science, operations research, bioinformatics, and economics. Some classic examples of dynamic programming problems include the Fibonacci sequence, Longest Common Subsequence (LCS), Longest Increasing Subsequence (LIS), Knapsack problem, Matrix Chain Multiplication, Edit Distance, and Coin Change problem. Each of these demonstrates how breaking down a problem and reusing computed results can lead to efficient solutions.
There are two main approaches to implementing DP:
1. Top-Down (Memoization): This involves writing a recursive function to solve the problem, but before computing the result of a subproblem, the function checks whether it has already been computed. If it has, the stored result is returned instead of recomputing it. This avoids redundant calculations.
2. Bottom-Up (Tabulation): This approach involves solving all related subproblems in a specific order and storing their results in a table. It starts from the smallest subproblems and combines their results to solve larger subproblems, ultimately reaching the final solution. This method usually uses iteration and avoids recursion.
One of the strengths of dynamic programming is its ability to transform exponential-time problems into polynomial-time ones. However, it requires careful problem formulation and identification of states and transitions between those states. A typical DP solution involves defining a state, figuring out the recurrence relation, and determining the base cases.
In summary, dynamic programming is a key technique for solving optimization problems with overlapping subproblems and optimal substructure. It requires a strategic approach to modeling the problem, but when applied correctly, it can yield solutions that are b
This document discusses dynamic programming and provides examples of problems that can be solved using dynamic programming, including assembly line scheduling and matrix chain multiplication. It explains the key aspects of a dynamic programming algorithm:
1) Characterizing the optimal substructure of a problem - how optimal solutions can be built from optimal solutions to subproblems.
2) Defining the problem recursively in terms of optimal solutions to subproblems.
3) Computing the optimal solution in a bottom-up manner by first solving subproblems and building up to the final solution.
Dynamic programming (DP) is a powerful technique for solving optimization problems by breaking them down into overlapping subproblems and storing the results of already solved subproblems. The document provides examples of how DP can be applied to problems like rod cutting, matrix chain multiplication, and longest common subsequence. It explains the key elements of DP, including optimal substructure (subproblems can be solved independently and combined to solve the overall problem) and overlapping subproblems (subproblems are solved repeatedly).
Dynamic programming is used to solve optimization problems by breaking them down into overlapping subproblems. It is applicable to problems that exhibit optimal substructure and overlapping subproblems. The matrix chain multiplication problem can be solved using dynamic programming in O(n^3) time by defining the problem recursively, computing the costs of subproblems in a bottom-up manner using dynamic programming, and tracing the optimal solution back from the computed information. Similarly, the longest common subsequence problem exhibits optimal substructure and can be solved using dynamic programming.
This document provides an introduction to linear and integer programming. It defines key concepts such as linear programs (LP), integer programs (IP), and mixed integer programs (MIP). It discusses the complexity of different optimization problem types and gives examples of LP and IP formulations. It also covers common techniques for solving LPs and IPs, including the simplex method, cutting plane methods, branch and bound, and heuristics like beam search.
Least Square Optimization and Sparse-Linear SolverJi-yong Kwon
The document discusses least-square optimization and sparse linear systems. It introduces least-square optimization as a technique to find approximate solutions when exact solutions do not exist. It provides an example of using least-squares to find the line of best fit through three points. The objective is to minimize the sum of squared distances between the line and points. Solving the optimization problem yields a set of linear equations that can be solved using techniques like pseudo-inverse or conjugate gradient. Sparse linear systems with many zero entries can be solved more efficiently than dense systems.
(1) Dynamic programming is an algorithm design technique that solves problems by breaking them down into smaller subproblems and storing the results of already solved subproblems. (2) It is applicable to problems where subproblems overlap and solving them recursively would result in redundant computations. (3) The key steps of a dynamic programming algorithm are to characterize the optimal structure, define the problem recursively in terms of optimal substructures, and compute the optimal solution bottom-up by solving subproblems only once.
Dynamic programming is an algorithm design technique that solves problems by breaking them down into smaller subproblems and storing the results of already solved subproblems. It is applicable when subproblems overlap and share common subsubproblems. The dynamic programming approach involves (1) characterizing the optimal structure of a solution, (2) recursively defining the optimal solution value, and (3) computing the optimal solution in a bottom-up manner by solving subproblems from smallest to largest. This allows for computing the optimal solution without resolving overlapping subproblems multiple times.
The document summarizes key concepts in design analysis and algorithms including:
1. Number theory problems like the Chinese Remainder Theorem and GCD algorithms. Approximate algorithms for set cover and vertex cover problems are also discussed.
2. The Chinese Remainder Theorem allows determining solutions based on remainders when numbers are divided. Pseudocode and a program demonstrate its use.
3. Modular arithmetic operations like addition, multiplication, and exponentiation along with their properties and programs are outlined.
The document discusses algorithms and data structures. It begins with two quotes about programming and algorithms. It then provides pseudocode for naive and optimized recursive Fibonacci algorithms, as well as an iterative dynamic programming version. It also covers dynamic programming approaches for calculating Fibonacci numbers, Catalan numbers, the chessboard traversal problem, the rod cutting problem, longest common subsequence, and assembly line traversal. The key ideas are introducing dynamic programming techniques like memoization and bottom-up iteration to improve the time complexity of recursive algorithms from exponential to polynomial.
The document discusses the matrix chain multiplication problem, which involves finding the most efficient way to multiply a sequence of matrices by determining the optimal parenthesization. It describes that there are multiple ways to multiply the matrices and lists an example of different possibilities. It then introduces a dynamic programming approach to solve this problem in polynomial time by treating it as the combination of optimal solutions to subproblems. The algorithm works by computing a minimum cost table and split table to track the optimal way to multiply the matrices.
This document provides an overview of dimensionality reduction techniques. It discusses how increasing dimensionality can negatively impact classification accuracy due to the curse of dimensionality. Dimensionality reduction aims to select an optimal set of features of lower dimensionality to improve accuracy. Feature extraction and feature selection are two common approaches. Principal component analysis (PCA) is described as a popular linear feature extraction method that projects data to a lower dimensional space while preserving as much variance as possible.
Design and Implementation of Parallel and Randomized Approximation AlgorithmsAjay Bidyarthy
This document summarizes the design and implementation of parallel and randomized approximation algorithms for solving matrix games, linear programs, and semi-definite programs. It presents solvers for these problems that provide approximate solutions in sublinear or near-linear time. It analyzes the performance and precision-time tradeoffs of the solvers compared to other algorithms. It also provides examples of applying the SDP solver to approximate the Lovasz theta function.
The document discusses the greedy method algorithmic approach. It provides an overview of greedy algorithms including that they make locally optimal choices at each step to find a global optimal solution. The document also provides examples of problems that can be solved using greedy methods like job sequencing, the knapsack problem, finding minimum spanning trees, and single source shortest paths. It summarizes control flow and applications of greedy algorithms.
This document discusses dynamic programming and provides examples for solving problems related to longest common subsequences and optimal binary search trees using dynamic programming. It begins with an introduction to dynamic programming as an algorithm design technique for optimization problems. It then provides steps for solving problems with dynamic programming, including characterizing the optimal structure, defining the problem recursively, computing optimal values in a table, and constructing the optimal solution. The document uses the problems of longest common subsequence and optimal binary search tree to demonstrate how to apply these steps with examples.
This document discusses dynamic programming and provides examples for solving problems related to longest common subsequences and optimal binary search trees using dynamic programming. It begins with defining the longest common subsequence problem and providing a naive recursive solution. It then shows that the problem exhibits optimal substructure and can be solved using dynamic programming by computing a table of values in a bottom-up manner. A similar approach is taken for the optimal binary search tree problem, characterizing its optimal substructure and computing an expected search cost table to find the optimal tree configuration.
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.
Learn about dynamic programming and how to design algorithMazenulIslamKhan
Dynamic Programming (DP): A 3000-Character Description
Dynamic Programming (DP) is a powerful algorithmic technique used to solve complex problems by breaking them down into simpler subproblems and solving each of those subproblems only once. It is especially useful for optimization problems, where the goal is to find the best possible solution from a set of feasible solutions. DP avoids the repeated calculation of the same subproblem by storing the results of solved subproblems in a table (usually an array or matrix) and reusing those results when needed. This approach is known as memoization when done recursively and tabulation when done iteratively.
The main idea behind dynamic programming is the principle of optimal substructure, which means that the solution to a problem can be composed of optimal solutions to its subproblems. Additionally, DP problems exhibit overlapping subproblems, meaning the same subproblems are solved multiple times during the execution of a naive recursive solution. By solving each unique subproblem just once and storing its result, dynamic programming reduces the time complexity significantly compared to a naive approach like brute-force recursion.
DP is commonly applied in a variety of domains such as computer science, operations research, bioinformatics, and economics. Some classic examples of dynamic programming problems include the Fibonacci sequence, Longest Common Subsequence (LCS), Longest Increasing Subsequence (LIS), Knapsack problem, Matrix Chain Multiplication, Edit Distance, and Coin Change problem. Each of these demonstrates how breaking down a problem and reusing computed results can lead to efficient solutions.
There are two main approaches to implementing DP:
1. Top-Down (Memoization): This involves writing a recursive function to solve the problem, but before computing the result of a subproblem, the function checks whether it has already been computed. If it has, the stored result is returned instead of recomputing it. This avoids redundant calculations.
2. Bottom-Up (Tabulation): This approach involves solving all related subproblems in a specific order and storing their results in a table. It starts from the smallest subproblems and combines their results to solve larger subproblems, ultimately reaching the final solution. This method usually uses iteration and avoids recursion.
One of the strengths of dynamic programming is its ability to transform exponential-time problems into polynomial-time ones. However, it requires careful problem formulation and identification of states and transitions between those states. A typical DP solution involves defining a state, figuring out the recurrence relation, and determining the base cases.
In summary, dynamic programming is a key technique for solving optimization problems with overlapping subproblems and optimal substructure. It requires a strategic approach to modeling the problem, but when applied correctly, it can yield solutions that are b
This document discusses dynamic programming and provides examples of problems that can be solved using dynamic programming, including assembly line scheduling and matrix chain multiplication. It explains the key aspects of a dynamic programming algorithm:
1) Characterizing the optimal substructure of a problem - how optimal solutions can be built from optimal solutions to subproblems.
2) Defining the problem recursively in terms of optimal solutions to subproblems.
3) Computing the optimal solution in a bottom-up manner by first solving subproblems and building up to the final solution.
Dynamic programming (DP) is a powerful technique for solving optimization problems by breaking them down into overlapping subproblems and storing the results of already solved subproblems. The document provides examples of how DP can be applied to problems like rod cutting, matrix chain multiplication, and longest common subsequence. It explains the key elements of DP, including optimal substructure (subproblems can be solved independently and combined to solve the overall problem) and overlapping subproblems (subproblems are solved repeatedly).
Dynamic programming is used to solve optimization problems by breaking them down into overlapping subproblems. It is applicable to problems that exhibit optimal substructure and overlapping subproblems. The matrix chain multiplication problem can be solved using dynamic programming in O(n^3) time by defining the problem recursively, computing the costs of subproblems in a bottom-up manner using dynamic programming, and tracing the optimal solution back from the computed information. Similarly, the longest common subsequence problem exhibits optimal substructure and can be solved using dynamic programming.
This document provides an introduction to linear and integer programming. It defines key concepts such as linear programs (LP), integer programs (IP), and mixed integer programs (MIP). It discusses the complexity of different optimization problem types and gives examples of LP and IP formulations. It also covers common techniques for solving LPs and IPs, including the simplex method, cutting plane methods, branch and bound, and heuristics like beam search.
Least Square Optimization and Sparse-Linear SolverJi-yong Kwon
The document discusses least-square optimization and sparse linear systems. It introduces least-square optimization as a technique to find approximate solutions when exact solutions do not exist. It provides an example of using least-squares to find the line of best fit through three points. The objective is to minimize the sum of squared distances between the line and points. Solving the optimization problem yields a set of linear equations that can be solved using techniques like pseudo-inverse or conjugate gradient. Sparse linear systems with many zero entries can be solved more efficiently than dense systems.
(1) Dynamic programming is an algorithm design technique that solves problems by breaking them down into smaller subproblems and storing the results of already solved subproblems. (2) It is applicable to problems where subproblems overlap and solving them recursively would result in redundant computations. (3) The key steps of a dynamic programming algorithm are to characterize the optimal structure, define the problem recursively in terms of optimal substructures, and compute the optimal solution bottom-up by solving subproblems only once.
Dynamic programming is an algorithm design technique that solves problems by breaking them down into smaller subproblems and storing the results of already solved subproblems. It is applicable when subproblems overlap and share common subsubproblems. The dynamic programming approach involves (1) characterizing the optimal structure of a solution, (2) recursively defining the optimal solution value, and (3) computing the optimal solution in a bottom-up manner by solving subproblems from smallest to largest. This allows for computing the optimal solution without resolving overlapping subproblems multiple times.
The document summarizes key concepts in design analysis and algorithms including:
1. Number theory problems like the Chinese Remainder Theorem and GCD algorithms. Approximate algorithms for set cover and vertex cover problems are also discussed.
2. The Chinese Remainder Theorem allows determining solutions based on remainders when numbers are divided. Pseudocode and a program demonstrate its use.
3. Modular arithmetic operations like addition, multiplication, and exponentiation along with their properties and programs are outlined.
The document discusses algorithms and data structures. It begins with two quotes about programming and algorithms. It then provides pseudocode for naive and optimized recursive Fibonacci algorithms, as well as an iterative dynamic programming version. It also covers dynamic programming approaches for calculating Fibonacci numbers, Catalan numbers, the chessboard traversal problem, the rod cutting problem, longest common subsequence, and assembly line traversal. The key ideas are introducing dynamic programming techniques like memoization and bottom-up iteration to improve the time complexity of recursive algorithms from exponential to polynomial.
The document discusses the matrix chain multiplication problem, which involves finding the most efficient way to multiply a sequence of matrices by determining the optimal parenthesization. It describes that there are multiple ways to multiply the matrices and lists an example of different possibilities. It then introduces a dynamic programming approach to solve this problem in polynomial time by treating it as the combination of optimal solutions to subproblems. The algorithm works by computing a minimum cost table and split table to track the optimal way to multiply the matrices.
This document provides an overview of dimensionality reduction techniques. It discusses how increasing dimensionality can negatively impact classification accuracy due to the curse of dimensionality. Dimensionality reduction aims to select an optimal set of features of lower dimensionality to improve accuracy. Feature extraction and feature selection are two common approaches. Principal component analysis (PCA) is described as a popular linear feature extraction method that projects data to a lower dimensional space while preserving as much variance as possible.
Design and Implementation of Parallel and Randomized Approximation AlgorithmsAjay Bidyarthy
This document summarizes the design and implementation of parallel and randomized approximation algorithms for solving matrix games, linear programs, and semi-definite programs. It presents solvers for these problems that provide approximate solutions in sublinear or near-linear time. It analyzes the performance and precision-time tradeoffs of the solvers compared to other algorithms. It also provides examples of applying the SDP solver to approximate the Lovasz theta function.
The document discusses the greedy method algorithmic approach. It provides an overview of greedy algorithms including that they make locally optimal choices at each step to find a global optimal solution. The document also provides examples of problems that can be solved using greedy methods like job sequencing, the knapsack problem, finding minimum spanning trees, and single source shortest paths. It summarizes control flow and applications of greedy algorithms.
This document discusses dynamic programming and provides examples for solving problems related to longest common subsequences and optimal binary search trees using dynamic programming. It begins with an introduction to dynamic programming as an algorithm design technique for optimization problems. It then provides steps for solving problems with dynamic programming, including characterizing the optimal structure, defining the problem recursively, computing optimal values in a table, and constructing the optimal solution. The document uses the problems of longest common subsequence and optimal binary search tree to demonstrate how to apply these steps with examples.
This document discusses dynamic programming and provides examples for solving problems related to longest common subsequences and optimal binary search trees using dynamic programming. It begins with defining the longest common subsequence problem and providing a naive recursive solution. It then shows that the problem exhibits optimal substructure and can be solved using dynamic programming by computing a table of values in a bottom-up manner. A similar approach is taken for the optimal binary search tree problem, characterizing its optimal substructure and computing an expected search cost table to find the optimal tree configuration.
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.
This paper proposes a shoulder inverse kinematics (IK) technique. Shoulder complex is comprised of the sternum, clavicle, ribs, scapula, humerus, and four joints.
Lidar for Autonomous Driving, LiDAR Mapping for Driverless Cars.pptxRishavKumar530754
LiDAR-Based System for Autonomous Cars
Autonomous Driving with LiDAR Tech
LiDAR Integration in Self-Driving Cars
Self-Driving Vehicles Using LiDAR
LiDAR Mapping for Driverless Cars
How to use nRF24L01 module with ArduinoCircuitDigest
Learn how to wirelessly transmit sensor data using nRF24L01 and Arduino Uno. A simple project demonstrating real-time communication with DHT11 and OLED display.
Sorting Order and Stability in Sorting.
Concept of Internal and External Sorting.
Bubble Sort,
Insertion Sort,
Selection Sort,
Quick Sort and
Merge Sort,
Radix Sort, and
Shell Sort,
External Sorting, Time complexity analysis of Sorting Algorithms.
"Boiler Feed Pump (BFP): Working, Applications, Advantages, and Limitations E...Infopitaara
A Boiler Feed Pump (BFP) is a critical component in thermal power plants. It supplies high-pressure water (feedwater) to the boiler, ensuring continuous steam generation.
⚙️ How a Boiler Feed Pump Works
Water Collection:
Feedwater is collected from the deaerator or feedwater tank.
Pressurization:
The pump increases water pressure using multiple impellers/stages in centrifugal types.
Discharge to Boiler:
Pressurized water is then supplied to the boiler drum or economizer section, depending on design.
🌀 Types of Boiler Feed Pumps
Centrifugal Pumps (most common):
Multistage for higher pressure.
Used in large thermal power stations.
Positive Displacement Pumps (less common):
For smaller or specific applications.
Precise flow control but less efficient for large volumes.
🛠️ Key Operations and Controls
Recirculation Line: Protects the pump from overheating at low flow.
Throttle Valve: Regulates flow based on boiler demand.
Control System: Often automated via DCS/PLC for variable load conditions.
Sealing & Cooling Systems: Prevent leakage and maintain pump health.
⚠️ Common BFP Issues
Cavitation due to low NPSH (Net Positive Suction Head).
Seal or bearing failure.
Overheating from improper flow or recirculation.
The role of the lexical analyzer
Specification of tokens
Finite state machines
From a regular expressions to an NFA
Convert NFA to DFA
Transforming grammars and regular expressions
Transforming automata to grammars
Language for specifying lexical analyzers
In tube drawing process, a tube is pulled out through a die and a plug to reduce its diameter and thickness as per the requirement. Dimensional accuracy of cold drawn tubes plays a vital role in the further quality of end products and controlling rejection in manufacturing processes of these end products. Springback phenomenon is the elastic strain recovery after removal of forming loads, causes geometrical inaccuracies in drawn tubes. Further, this leads to difficulty in achieving close dimensional tolerances. In the present work springback of EN 8 D tube material is studied for various cold drawing parameters. The process parameters in this work include die semi-angle, land width and drawing speed. The experimentation is done using Taguchi’s L36 orthogonal array, and then optimization is done in data analysis software Minitab 17. The results of ANOVA shows that 15 degrees die semi-angle,5 mm land width and 6 m/min drawing speed yields least springback. Furthermore, optimization algorithms named Particle Swarm Optimization (PSO), Simulated Annealing (SA) and Genetic Algorithm (GA) are applied which shows that 15 degrees die semi-angle, 10 mm land width and 8 m/min drawing speed results in minimal springback with almost 10.5 % improvement. Finally, the results of experimentation are validated with Finite Element Analysis technique using ANSYS.
Value Stream Mapping Worskshops for Intelligent Continuous SecurityMarc Hornbeek
This presentation provides detailed guidance and tools for conducting Current State and Future State Value Stream Mapping workshops for Intelligent Continuous Security.
"Feed Water Heaters in Thermal Power Plants: Types, Working, and Efficiency G...Infopitaara
A feed water heater is a device used in power plants to preheat water before it enters the boiler. It plays a critical role in improving the overall efficiency of the power generation process, especially in thermal power plants.
🔧 Function of a Feed Water Heater:
It uses steam extracted from the turbine to preheat the feed water.
This reduces the fuel required to convert water into steam in the boiler.
It supports Regenerative Rankine Cycle, increasing plant efficiency.
🔍 Types of Feed Water Heaters:
Open Feed Water Heater (Direct Contact)
Steam and water come into direct contact.
Mixing occurs, and heat is transferred directly.
Common in low-pressure stages.
Closed Feed Water Heater (Surface Type)
Steam and water are separated by tubes.
Heat is transferred through tube walls.
Common in high-pressure systems.
⚙️ Advantages:
Improves thermal efficiency.
Reduces fuel consumption.
Lowers thermal stress on boiler components.
Minimizes corrosion by removing dissolved gases.
Fluid mechanics is the branch of physics concerned with the mechanics of fluids (liquids, gases, and plasmas) and the forces on them. Originally applied to water (hydromechanics), it found applications in a wide range of disciplines, including mechanical, aerospace, civil, chemical, and biomedical engineering, as well as geophysics, oceanography, meteorology, astrophysics, and biology.
It can be divided into fluid statics, the study of various fluids at rest, and fluid dynamics.
Fluid statics, also known as hydrostatics, is the study of fluids at rest, specifically when there's no relative motion between fluid particles. It focuses on the conditions under which fluids are in stable equilibrium and doesn't involve fluid motion.
Fluid kinematics is the branch of fluid mechanics that focuses on describing and analyzing the motion of fluids, such as liquids and gases, without considering the forces that cause the motion. It deals with the geometrical and temporal aspects of fluid flow, including velocity and acceleration. Fluid dynamics, on the other hand, considers the forces acting on the fluid.
Fluid dynamics is the study of the effect of forces on fluid motion. It is a branch of continuum mechanics, a subject which models matter without using the information that it is made out of atoms; that is, it models matter from a macroscopic viewpoint rather than from microscopic.
Fluid mechanics, especially fluid dynamics, is an active field of research, typically mathematically complex. Many problems are partly or wholly unsolved and are best addressed by numerical methods, typically using computers. A modern discipline, called computational fluid dynamics (CFD), is devoted to this approach. Particle image velocimetry, an experimental method for visualizing and analyzing fluid flow, also takes advantage of the highly visual nature of fluid flow.
Fundamentally, every fluid mechanical system is assumed to obey the basic laws :
Conservation of mass
Conservation of energy
Conservation of momentum
The continuum assumption
For example, the assumption that mass is conserved means that for any fixed control volume (for example, a spherical volume)—enclosed by a control surface—the rate of change of the mass contained in that volume is equal to the rate at which mass is passing through the surface from outside to inside, minus the rate at which mass is passing from inside to outside. This can be expressed as an equation in integral form over the control volume.
The continuum assumption is an idealization of continuum mechanics under which fluids can be treated as continuous, even though, on a microscopic scale, they are composed of molecules. Under the continuum assumption, macroscopic (observed/measurable) properties such as density, pressure, temperature, and bulk velocity are taken to be well-defined at "infinitesimal" volume elements—small in comparison to the characteristic length scale of the system, but large in comparison to molecular length scale
1. Optimization Problems
• In which a set of choices must be made in
order to arrive at an optimal (min/max)
solution, subject to some constraints. (There
may be several solutions to achieve an
optimal value.)
• Two common techniques:
– Dynamic Programming (global)
– Greedy Algorithms (local)
2. Dynamic Programming
• Similar to divide-and-conquer, it breaks
problems down into smaller problems that are
solved recursively.
• In contrast, DP is applicable when the sub-
problems are not independent, i.e. when sub-
problems share sub-sub-problems. It solves
every sub-sub-problem just once and save the
results in a table to avoid duplicated
computation.
3. Elements of DP Algorithms
• Sub-structure: decompose problem into smaller sub-
problems. Express the solution of the original problem
in terms of solutions for smaller problems.
• Table-structure: Store the answers to the sub-problem
in a table, because sub-problem solutions may be used
many times.
• Bottom-up computation: combine solutions on smaller
sub-problems to solve larger sub-problems, and
eventually arrive at a solution to the complete problem.
4. Applicability to Optimization
Problems
• Optimal sub-structure (principle of optimality): for the global
problem to be solved optimally, each sub-problem should be
solved optimally. This is often violated due to sub-problem
overlaps. Often by being “less optimal” on one problem, we
may make a big savings on another sub-problem.
• Overlapping of sub-problems: Many NP-hard problems can
be formulated as DP problems, but these formulations are not
efficient, because the number of sub-problems is
exponentially large. Ideally, the number of sub-problems
should be at most a polynomial number.
5. Optimized Chain Operations
• Determine the optimal sequence for performing a series of
operations. (the general class of the problem is important
in compiler design for code optimization & in databases
for query optimization)
• For example: given a series of matrices: A1…An , we can
“parenthesize” this expression however we like, since matrix
multiplication is associative (but not commutative).
• Multiply a p x q matrix A times a q x r matrix B, the result will
be a p x r matrix C. (# of columns of A must be equal to # of
rows of B.)
6. Matrix Multiplication
• In particular for 1 i p and 1 j r,
C[i, j] = k = 1 to q A[i, k] B[k, j]
• Observe that there are pr total entries in C
and each takes O(q) time to compute, thus
the total time to multiply 2 matrices is pqr.
7. Chain Matrix Multiplication
• Given a sequence of matrices A1 A2…An , and
dimensions p0 p1…pn where Ai is of dimension pi-1
x pi , determine multiplication sequence that
minimizes the number of operations.
• This algorithm does not perform the
multiplication, it just figures out the best order in
which to perform the multiplication.
8. Example: CMM
• Consider 3 matrices: A1 be 5 x 4, A2 be 4 x 6,
and A3 be 6 x 2.
Mult[((A1 A2)A3)] = (5x4x6) + (5x6x2) = 180
Mult[(A1 (A2A3 ))] = (4x6x2) + (5x4x2) = 88
Even for this small example, considerable savings
can be achieved by reordering the evaluation
sequence.
9. DP Solution (I)
• Let Ai…j be the product of matrices i through j. Ai…j is a pi-1 x pj matrix. At the
highest level, we are multiplying two matrices together. That is, for any k, 1
k n-1,
A1…n = (A1…k)(Ak+1…n)
• The problem of determining the optimal sequence of multiplication is broken
up into 2 parts:
Q : How do we decide where to split the chain (what k)?
A : Consider all possible values of k.
Q : How do we parenthesize the subchains A1…k & Ak+1…n?
A : Solve by recursively applying the same scheme.
NOTE: this problem satisfies the “principle of optimality”.
• Next, we store the solutions to the sub-problems in a table and build the table
in a bottom-up manner.
10. DP Solution (II)
• For 1 i j n, let m[i, j] denote the minimum number
of multiplications needed to compute Ai…j .
• Example: Minimum number of multiplies for A3…7
• In terms of pi , the product A3…7 has
dimensions ____.
11. DP Solution (III)
• The optimal cost can be described be as follows:
– i = j the sequence contains only 1 matrix, so m[i, j] = 0.
– i < j This can be split by considering each k, i k < j,
as Ai…k (pi-1 x pk ) times Ak+1…j (pk x pj).
• This suggests the following recursive rule for computing
m[i, j]:
m[i, i] = 0
m[i, j] = mini k < j (m[i, k] + m[k+1, j] + pi-1pkpj ) for i < j
12. Computing m[i, j]
• For a specific k,
(Ai …Ak)(Ak+1 …Aj)
=
m[i, j] = mini k < j (m[i, k] + m[k+1, j] + pi-1pkpj )
13. Computing m[i, j]
• For a specific k,
(Ai …Ak)(Ak+1 …Aj)
= Ai…k(Ak+1 …Aj) (m[i, k] mults)
m[i, j] = mini k < j (m[i, k] + m[k+1, j] + pi-1pkpj )
14. Computing m[i, j]
• For a specific k,
(Ai …Ak)(Ak+1 …Aj)
= Ai…k(Ak+1 …Aj) (m[i, k] mults)
= Ai…k Ak+1…j (m[k+1, j] mults)
m[i, j] = mini k < j (m[i, k] + m[k+1, j] + pi-1pkpj )
15. Computing m[i, j]
• For a specific k,
(Ai …Ak)(Ak+1 …Aj)
= Ai…k(Ak+1 …Aj) (m[i, k] mults)
= Ai…k Ak+1…j (m[k+1, j] mults)
= Ai…j (pi-1 pk pj mults)
m[i, j] = mini k < j (m[i, k] + m[k+1, j] + pi-1pkpj )
16. Computing m[i, j]
• For a specific k,
(Ai …Ak)(Ak+1 …Aj)
= Ai…k(Ak+1 …Aj) (m[i, k] mults)
= Ai…k Ak+1…j (m[k+1, j] mults)
= Ai…j (pi-1 pk pj mults)
• For solution, evaluate for all k and take minimum.
m[i, j] = mini k < j (m[i, k] + m[k+1, j] + pi-1pkpj )
17. Matrix-Chain-Order(p)
1. n length[p] - 1
2. for i 1 to n // initialization: O(n) time
3. do m[i, i] 0
4. for L 2 to n // L = length of sub-chain
5. do for i 1 to n - L+1
6. do j i + L - 1
7. m[i, j]
8. for k i to j - 1
9. do q m[i, k] + m[k+1, j] + pi-1 pk pj
10. if q < m[i, j]
11. then m[i, j] q
12. s[i, j] k
13. return m and s
18. Extracting Optimum Sequence
• Leave a split marker indicating where the best split is (i.e.
the value of k leading to minimum values of m[i, j]). We
maintain a parallel array s[i, j] in which we store the value
of k providing the optimal split.
• If s[i, j] = k, the best way to multiply the sub-chain Ai…j is
to first multiply the sub-chain Ai…k and then the sub-chain
Ak+1…j , and finally multiply them together. Intuitively s[i, j]
tells us what multiplication to perform last. We only need
to store s[i, j] if we have at least 2 matrices & j > i.
21. Example: DP for CMM
• The initial set of dimensions are <5, 4, 6, 2, 7>: we are
multiplying A1 (5x4) times A2 (4x6) times A3 (6x2) times
A4 (2x7). Optimal sequence is (A1 (A2A3 )) A4.
22. Finding a Recursive Solution
• Figure out the “top-level” choice you
have to make (e.g., where to split the
list of matrices)
• List the options for that decision
• Each option should require smaller sub-
problems to be solved
• Recursive function is the minimum (or
max) over all the options
m[i, j] = mini k < j (m[i, k] + m[k+1, j] + pi-1pkpj )
26. Longest Common Subsequence
(LCS)
• Problem: Given sequences x[1..m] and
y[1..n], find a longest common
subsequence of both.
• Example: x=ABCBDAB and
y=BDCABA,
– BCA is a common subsequence and
– BCBA and BDAB are two LCSs
27. LCS
• Writing a recurrence equation
• The dynamic programming solution
28. Brute force solution
• Solution: For every subsequence of x,
check if it is a subsequence of y.
29. Writing the recurrence
equation
• Let Xi denote the ith prefix x[1..i] of x[1..m],
and
• X0 denotes an empty prefix
• We will first compute the length of an LCS of
Xm and Yn, LenLCS(m, n), and then use
information saved during the computation for
finding the actual subsequence
• We need a recursive formula for computing
LenLCS(i, j).
30. Writing the recurrence
equation
• If Xi and Yj end with the same character xi=yj,
the LCS must include the character. If it did
not we could get a longer LCS by adding the
common character.
• If Xi and Yj do not end with the same
character there are two possibilities:
– either the LCS does not end with xi,
– or it does not end with yj
• Let Zk denote an LCS of Xi and Yj
31. Xi and Yj end with xi=yj
Xi
x1 x2 … xi-1 xi
Yj
y1 y2 … yj-1 yj=xi
Zk
z1 z2…zk-1 zk =yj=xi
Zk is Zk -1 followed by zk = yj = xi where
Zk-1 is an LCS of Xi-1 and Yj -1 and
LenLCS(i, j)=LenLCS(i-1, j-1)+1
32. Xi and Yj end with xi yj
Xi
x1 x2 … xi-1 xi
Yj
y1 y2 … yj-1 yj
Zk
z1 z2…zk-1 zk yj
Zk is an LCS of Xi and Yj -1
Xi
x1 x2 … xi-1 x i
Yj
yj y1 y2 …yj-1 yj
Zk
z1 z2…zk-1 zk xi
Zk is an LCS of Xi -1 and Yj
LenLCS(i, j)=max{LenLCS(i, j-1), LenLCS(i-1, j)}
33. The recurrence equation
lenLCS i j
i j
lenLCS i j i j x y
lenLCS i j lenLCS i j
( , )
,
( , )
max{ ( , ), ( , )}
0 0 0
1 1 1
1 1
if or
if , > 0 and =
otherwise
i j
34. The dynamic programming
solution
• Initialize the first row and the first column of
the matrix LenLCS to 0
• Calculate LenLCS (1, j) for j = 1,…, n
• Then the LenLCS (2, j) for j = 1,…, n, etc.
• Store also in a table an arrow pointing to the
array element that was used in the
computation.
35. Example
yj B D C A
xj 0 0 0 0 0
A 0 0 0 0 1
B 0 1 1 1 1
C 0 1 1 2 2
B 0 1 1 2 2
To find an LCS follow the arrows, for each
diagonal arrow there is a member of the LCS