Dynamic programming is a technique for solving problems with overlapping subproblems and optimal substructure. It works by breaking problems down into smaller subproblems and storing the results in a table to avoid recomputing them. Examples where it can be applied include the knapsack problem, longest common subsequence, and computing Fibonacci numbers efficiently through bottom-up iteration rather than top-down recursion. The technique involves setting up recurrences relating larger instances to smaller ones, solving the smallest instances, and building up the full solution using the stored results.
Algorithms Lecture 2: Analysis of Algorithms IMohamed Loey
This document discusses analysis of algorithms and time complexity. It explains that analysis of algorithms determines the resources needed to execute algorithms. The time complexity of an algorithm quantifies how long it takes. There are three cases to analyze - worst case, average case, and best case. Common notations for time complexity include O(1), O(n), O(n^2), O(log n), and O(n!). The document provides examples of algorithms and determines their time complexity in different cases. It also discusses how to combine complexities of nested loops and loops in algorithms.
This Presentation will Use to develop your knowledge and doubts in Knapsack problem. This Slide also include Memory function part. Use this Slides to Develop your knowledge on Knapsack and Memory function
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.
(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.
The document discusses the knapsack problem and greedy algorithms. It defines the knapsack problem as an optimization problem where given constraints and an objective function, the goal is to find the feasible solution that maximizes or minimizes the objective. It describes the knapsack problem has having two versions: 0-1 where items are indivisible, and fractional where items can be divided. The fractional knapsack problem can be solved using a greedy approach by sorting items by value to weight ratio and filling the knapsack accordingly until full.
This document discusses greedy algorithms and dynamic programming techniques for solving optimization problems. It covers the activity selection problem, which can be solved greedily by always selecting the shortest remaining activity. It also discusses the knapsack problem and how the fractional version can be solved greedily while the 0-1 version requires dynamic programming due to its optimal substructure but non-greedy nature. Dynamic programming builds up solutions by combining optimal solutions to overlapping subproblems.
how to calclute time complexity of algortihmSajid Marwat
This document discusses algorithm analysis and complexity. It defines key terms like asymptotic complexity, Big-O notation, and time complexity. It provides examples of analyzing simple algorithms like a sum function to determine their time complexity. Common analyses include looking at loops, nested loops, and sequences of statements. The goal is to classify algorithms according to their complexity, which is important for large inputs and machine-independent. Algorithms are classified based on worst, average, and best case analyses.
Algorithms Lecture 2: Analysis of Algorithms IMohamed Loey
This document discusses analysis of algorithms and time complexity. It explains that analysis of algorithms determines the resources needed to execute algorithms. The time complexity of an algorithm quantifies how long it takes. There are three cases to analyze - worst case, average case, and best case. Common notations for time complexity include O(1), O(n), O(n^2), O(log n), and O(n!). The document provides examples of algorithms and determines their time complexity in different cases. It also discusses how to combine complexities of nested loops and loops in algorithms.
This Presentation will Use to develop your knowledge and doubts in Knapsack problem. This Slide also include Memory function part. Use this Slides to Develop your knowledge on Knapsack and Memory function
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.
(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.
The document discusses the knapsack problem and greedy algorithms. It defines the knapsack problem as an optimization problem where given constraints and an objective function, the goal is to find the feasible solution that maximizes or minimizes the objective. It describes the knapsack problem has having two versions: 0-1 where items are indivisible, and fractional where items can be divided. The fractional knapsack problem can be solved using a greedy approach by sorting items by value to weight ratio and filling the knapsack accordingly until full.
This document discusses greedy algorithms and dynamic programming techniques for solving optimization problems. It covers the activity selection problem, which can be solved greedily by always selecting the shortest remaining activity. It also discusses the knapsack problem and how the fractional version can be solved greedily while the 0-1 version requires dynamic programming due to its optimal substructure but non-greedy nature. Dynamic programming builds up solutions by combining optimal solutions to overlapping subproblems.
how to calclute time complexity of algortihmSajid Marwat
This document discusses algorithm analysis and complexity. It defines key terms like asymptotic complexity, Big-O notation, and time complexity. It provides examples of analyzing simple algorithms like a sum function to determine their time complexity. Common analyses include looking at loops, nested loops, and sequences of statements. The goal is to classify algorithms according to their complexity, which is important for large inputs and machine-independent. Algorithms are classified based on worst, average, and best case analyses.
The document discusses the divide and conquer algorithm design technique. It begins by explaining the basic approach of divide and conquer which is to (1) divide the problem into subproblems, (2) conquer the subproblems by solving them recursively, and (3) combine the solutions to the subproblems into a solution for the original problem. It then provides merge sort as a specific example of a divide and conquer algorithm for sorting a sequence. It explains that merge sort divides the sequence in half recursively until individual elements remain, then combines the sorted halves back together to produce the fully sorted sequence.
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.
This presentation discusses the knapsack problem and its two main versions: 0/1 and fractional. The 0/1 knapsack problem involves indivisible items that are either fully included or not included, and is solved using dynamic programming. The fractional knapsack problem allows items to be partially included, and is solved using a greedy algorithm. Examples are provided of solving each version using their respective algorithms. The time complexity of these algorithms is also presented. Real-world applications of the knapsack problem include cutting raw materials and selecting investments.
The document discusses brute force and exhaustive search approaches to solving problems. It provides examples of how brute force can be applied to sorting, searching, and string matching problems. Specifically, it describes selection sort and bubble sort as brute force sorting algorithms. For searching, it explains sequential search and brute force string matching. It also discusses using brute force to solve the closest pair, convex hull, traveling salesman, knapsack, and assignment problems, noting that brute force leads to inefficient exponential time algorithms for TSP and knapsack.
This document provides an overview of dynamic programming. It begins by explaining that dynamic programming is a technique for solving optimization problems by breaking them down into overlapping subproblems and storing the results of solved subproblems in a table to avoid recomputing them. It then provides examples of problems that can be solved using dynamic programming, including Fibonacci numbers, binomial coefficients, shortest paths, and optimal binary search trees. The key aspects of dynamic programming algorithms, including defining subproblems and combining their solutions, are also outlined.
The document discusses greedy algorithms and their application to optimization problems. It provides examples of problems that can be solved using greedy approaches, such as fractional knapsack and making change. However, it notes that some problems like 0-1 knapsack and shortest paths on multi-stage graphs cannot be solved optimally with greedy algorithms. The document also describes various greedy algorithms for minimum spanning trees, single-source shortest paths, and fractional knapsack problems.
The branch-and-bound method is used to solve optimization problems by traversing a state space tree. It computes a bound at each node to determine if the node is promising. Better approaches traverse nodes breadth-first and choose the most promising node using a bounding heuristic. The traveling salesperson problem is solved using branch-and-bound by finding an initial tour, defining a bounding heuristic as the actual cost plus minimum remaining cost, and expanding promising nodes in best-first order until finding the minimal tour.
This document discusses optimal binary search trees and provides an example problem. It begins with basic definitions of binary search trees and optimal binary search trees. It then shows an example problem with keys 1, 2, 3 and calculates the cost as 17. The document explains how to use dynamic programming to find the optimal binary search tree for keys 10, 12, 16, 21 with frequencies 4, 2, 6, 3. It provides the solution matrix and explains that the minimum cost is 2 with the optimal tree as 10, 12, 16, 21.
This document discusses algorithms for finding minimum and maximum elements in an array, including simultaneous minimum and maximum algorithms. It introduces dynamic programming as a technique for improving inefficient divide-and-conquer algorithms by storing results of subproblems to avoid recomputing them. Examples of dynamic programming include calculating the Fibonacci sequence and solving an assembly line scheduling problem to minimize total time.
Knapsack problem using dynamic programmingkhush_boo31
The document describes the 0-1 knapsack problem and provides an example of solving it using dynamic programming. Specifically, it defines the 0-1 knapsack problem, provides the formula for solving it dynamically using a 2D array C, walks through populating the C array and backtracking to find the optimal solution for a sample problem instance, and analyzes the time complexity of the dynamic programming algorithm.
This document provides an introduction to the Master Theorem, which can be used to determine the asymptotic runtime of recursive algorithms. It presents the three main conditions of the Master Theorem and examples of applying it to solve recurrence relations. It also notes some pitfalls in using the Master Theorem and briefly introduces a fourth condition for cases where the non-recursive term is polylogarithmic rather than polynomial.
Dynamic programming is a recursive optimization technique used to solve problems with interrelated decisions. It breaks the problem down into sequential steps, where each step builds on the solutions to previous steps. The optimal solution is determined by working through each step in order. Dynamic programming has advantages like computational savings over complete enumeration and providing insight into problem nature. However, it also has disadvantages like requiring more expertise, lacking general algorithms, and facing dimensionality problems for applications with multiple states.
This document contains a presentation on solving the coin change problem using greedy and dynamic programming algorithms. It introduces the coin change problem and provides an example. It then describes the greedy algorithm approach and how it works for some cases but fails to find an optimal solution in other cases when coin values are not uniform. The document next explains dynamic programming, its four step process, and how it can be applied to the coin change problem to always find an optimal solution using a bottom-up approach and storing results of subproblems to build the final solution.
The document discusses the problem of determining the optimal way to fully parenthesize the product of a chain of matrices to minimize the number of scalar multiplications. It presents a dynamic programming approach to solve this problem in four steps: 1) characterize the structure of an optimal solution, 2) recursively define the cost of an optimal solution, 3) compute the costs using tables, 4) construct the optimal solution from the tables. An example is provided to illustrate computing the costs table and finding the optimal parenthesization of a chain of 6 matrices.
Dynamic programming is used to solve optimization problems by breaking them down into subproblems. It solves each subproblem only once, storing the results in a table to lookup when the subproblem recurs. This avoids recomputing solutions and reduces computation. The key is determining the optimal substructure of problems. It involves characterizing optimal solutions recursively, computing values in a bottom-up table, and tracing back the optimal solution. An example is the 0/1 knapsack problem to maximize profit fitting items in a knapsack of limited capacity.
This document discusses dynamic programming and provides examples of how it can be applied to optimize algorithms to solve problems with overlapping subproblems. It summarizes dynamic programming, provides examples for the Fibonacci numbers, binomial coefficients, and knapsack problems, and analyzes the time and space complexity of algorithms developed using dynamic programming approaches.
The document summarizes optimization techniques for training deep neural networks, including gradient descent, mini-batch gradient descent, and Newton's method. While Newton's method converges the fastest, it is rarely used due to high computational cost and instability. Limited memory BFGS provides an approximate Newton method that is more efficient and stable by using a low-dimensional approximation of the Hessian matrix inverse. Stochastic gradient descent converges more slowly than other methods but is widely used in practice due to its ability to train large networks on massive datasets.
The document discusses the divide and conquer algorithm design technique. It begins by explaining the basic approach of divide and conquer which is to (1) divide the problem into subproblems, (2) conquer the subproblems by solving them recursively, and (3) combine the solutions to the subproblems into a solution for the original problem. It then provides merge sort as a specific example of a divide and conquer algorithm for sorting a sequence. It explains that merge sort divides the sequence in half recursively until individual elements remain, then combines the sorted halves back together to produce the fully sorted sequence.
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.
This presentation discusses the knapsack problem and its two main versions: 0/1 and fractional. The 0/1 knapsack problem involves indivisible items that are either fully included or not included, and is solved using dynamic programming. The fractional knapsack problem allows items to be partially included, and is solved using a greedy algorithm. Examples are provided of solving each version using their respective algorithms. The time complexity of these algorithms is also presented. Real-world applications of the knapsack problem include cutting raw materials and selecting investments.
The document discusses brute force and exhaustive search approaches to solving problems. It provides examples of how brute force can be applied to sorting, searching, and string matching problems. Specifically, it describes selection sort and bubble sort as brute force sorting algorithms. For searching, it explains sequential search and brute force string matching. It also discusses using brute force to solve the closest pair, convex hull, traveling salesman, knapsack, and assignment problems, noting that brute force leads to inefficient exponential time algorithms for TSP and knapsack.
This document provides an overview of dynamic programming. It begins by explaining that dynamic programming is a technique for solving optimization problems by breaking them down into overlapping subproblems and storing the results of solved subproblems in a table to avoid recomputing them. It then provides examples of problems that can be solved using dynamic programming, including Fibonacci numbers, binomial coefficients, shortest paths, and optimal binary search trees. The key aspects of dynamic programming algorithms, including defining subproblems and combining their solutions, are also outlined.
The document discusses greedy algorithms and their application to optimization problems. It provides examples of problems that can be solved using greedy approaches, such as fractional knapsack and making change. However, it notes that some problems like 0-1 knapsack and shortest paths on multi-stage graphs cannot be solved optimally with greedy algorithms. The document also describes various greedy algorithms for minimum spanning trees, single-source shortest paths, and fractional knapsack problems.
The branch-and-bound method is used to solve optimization problems by traversing a state space tree. It computes a bound at each node to determine if the node is promising. Better approaches traverse nodes breadth-first and choose the most promising node using a bounding heuristic. The traveling salesperson problem is solved using branch-and-bound by finding an initial tour, defining a bounding heuristic as the actual cost plus minimum remaining cost, and expanding promising nodes in best-first order until finding the minimal tour.
This document discusses optimal binary search trees and provides an example problem. It begins with basic definitions of binary search trees and optimal binary search trees. It then shows an example problem with keys 1, 2, 3 and calculates the cost as 17. The document explains how to use dynamic programming to find the optimal binary search tree for keys 10, 12, 16, 21 with frequencies 4, 2, 6, 3. It provides the solution matrix and explains that the minimum cost is 2 with the optimal tree as 10, 12, 16, 21.
This document discusses algorithms for finding minimum and maximum elements in an array, including simultaneous minimum and maximum algorithms. It introduces dynamic programming as a technique for improving inefficient divide-and-conquer algorithms by storing results of subproblems to avoid recomputing them. Examples of dynamic programming include calculating the Fibonacci sequence and solving an assembly line scheduling problem to minimize total time.
Knapsack problem using dynamic programmingkhush_boo31
The document describes the 0-1 knapsack problem and provides an example of solving it using dynamic programming. Specifically, it defines the 0-1 knapsack problem, provides the formula for solving it dynamically using a 2D array C, walks through populating the C array and backtracking to find the optimal solution for a sample problem instance, and analyzes the time complexity of the dynamic programming algorithm.
This document provides an introduction to the Master Theorem, which can be used to determine the asymptotic runtime of recursive algorithms. It presents the three main conditions of the Master Theorem and examples of applying it to solve recurrence relations. It also notes some pitfalls in using the Master Theorem and briefly introduces a fourth condition for cases where the non-recursive term is polylogarithmic rather than polynomial.
Dynamic programming is a recursive optimization technique used to solve problems with interrelated decisions. It breaks the problem down into sequential steps, where each step builds on the solutions to previous steps. The optimal solution is determined by working through each step in order. Dynamic programming has advantages like computational savings over complete enumeration and providing insight into problem nature. However, it also has disadvantages like requiring more expertise, lacking general algorithms, and facing dimensionality problems for applications with multiple states.
This document contains a presentation on solving the coin change problem using greedy and dynamic programming algorithms. It introduces the coin change problem and provides an example. It then describes the greedy algorithm approach and how it works for some cases but fails to find an optimal solution in other cases when coin values are not uniform. The document next explains dynamic programming, its four step process, and how it can be applied to the coin change problem to always find an optimal solution using a bottom-up approach and storing results of subproblems to build the final solution.
The document discusses the problem of determining the optimal way to fully parenthesize the product of a chain of matrices to minimize the number of scalar multiplications. It presents a dynamic programming approach to solve this problem in four steps: 1) characterize the structure of an optimal solution, 2) recursively define the cost of an optimal solution, 3) compute the costs using tables, 4) construct the optimal solution from the tables. An example is provided to illustrate computing the costs table and finding the optimal parenthesization of a chain of 6 matrices.
Dynamic programming is used to solve optimization problems by breaking them down into subproblems. It solves each subproblem only once, storing the results in a table to lookup when the subproblem recurs. This avoids recomputing solutions and reduces computation. The key is determining the optimal substructure of problems. It involves characterizing optimal solutions recursively, computing values in a bottom-up table, and tracing back the optimal solution. An example is the 0/1 knapsack problem to maximize profit fitting items in a knapsack of limited capacity.
This document discusses dynamic programming and provides examples of how it can be applied to optimize algorithms to solve problems with overlapping subproblems. It summarizes dynamic programming, provides examples for the Fibonacci numbers, binomial coefficients, and knapsack problems, and analyzes the time and space complexity of algorithms developed using dynamic programming approaches.
The document summarizes optimization techniques for training deep neural networks, including gradient descent, mini-batch gradient descent, and Newton's method. While Newton's method converges the fastest, it is rarely used due to high computational cost and instability. Limited memory BFGS provides an approximate Newton method that is more efficient and stable by using a low-dimensional approximation of the Hessian matrix inverse. Stochastic gradient descent converges more slowly than other methods but is widely used in practice due to its ability to train large networks on massive datasets.
Dynamic programming is an algorithm design paradigm that can be applied to problems exhibiting optimal substructure and overlapping subproblems. It works by breaking down a problem into subproblems and storing the results of already solved subproblems, rather than recomputing them multiple times. This allows for an efficient bottom-up approach. Examples where dynamic programming can be applied include the matrix chain multiplication problem, the 0-1 knapsack problem, and finding the longest common subsequence between two strings.
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.
Dynamic programming can be used to solve optimization problems involving overlapping subproblems, such as finding the most valuable subset of items that fit in a knapsack. The knapsack problem is solved by considering all possible subsets incrementally, storing the optimal values in a table. Warshall's and Floyd's algorithms also use dynamic programming to find the transitive closure and shortest paths in graphs by iteratively building up the solution from smaller subsets. Optimal binary search trees can also be constructed using dynamic programming by considering optimal substructures.
Distributed Resilient Interval Observers for Bounded-Error LTI Systems Subjec...Mohammad Khajenejad
This is my presentation slides at the American Control Conference (ACC), at May 2023, in San Diego. This was one of the talks in an invited session on "Resiliency and Privacy Throughout Networked Cyber-Physical Systems", that we co-organized and co-chaired during ACC 2023. I this invited sessions, great scholars presented their work on the intersection of resilient estimation, control and learning, as well as designing privacy-preserving mechanisms in multi-agent cyber-physical systems.
In my presentation, I discussed our progress on designing distributed interval-valued input and state observers for multi-agent LTI systems that are subject to bounded noise, as well as adversarial unknown inputs on both sensors and actuators. We introduced the notion of min-max consensus as a counterpart to average consensus in bounded-error settings. Moreover, we developed structural conditions for the stability of the proposed observers for different classes of dynamics and networks.
The document describes several algorithms that use dynamic programming techniques. It discusses the coin changing problem, computing binomial coefficients, Floyd's algorithm for finding all-pairs shortest paths, optimal binary search trees, the knapsack problem, and multistage graphs. For each problem, it provides the key recurrence relation used to build the dynamic programming solution in a bottom-up manner, often using a table to store intermediate results. It also analyzes the time and space complexity of the different approaches.
This document discusses algorithms for rendering lines in raster graphics. It begins by introducing common line primitives in OpenGL and reviewing basic line drawing math. It then describes the Digital Differential Analyzer (DDA) line algorithm and its limitations. The document introduces Bresenham's midpoint line algorithm as a faster alternative that uses integer arithmetic. It explains how Bresenham's algorithm works by tracking the sign of a decision variable to select the next pixel along the line. The document concludes by generalizing Bresenham's algorithm and discussing optimizations.
I am Jayson L. I am a Signals and Systems Homework Expert at matlabassignmentexperts.com. I hold a Master's in Matlab, from the University of Sheffield. I have been helping students with their homework for the past 7 years. I solve homework related to Signals and Systems.
Visit matlabassignmentexperts.com or email info@matlabassignmentexperts.com.
You can also call on +1 678 648 4277 for any assistance with Signals and Systems homework.
Embedding and np-Complete Problems for 3-Equitable GraphsWaqas Tariq
We present here some important results in connection with 3-equitable graphs. We prove that any graph G can be embedded as an induced subgraph of a 3-equitable graph. We have also discussed some properties which are invariant under embedding. This work rules out any possibility of obtaining a forbidden subgraph characterization for 3-equitable graphs.
This document discusses the longest common subsequence problem and provides an example of how it can be solved using dynamic programming. It begins by defining the problem of finding the longest subsequence that is common to two input sequences. It then shows that this problem exhibits optimal substructure and can be solved recursively. However, a recursive solution is inefficient due to redundant subproblem computations. Instead, it presents an algorithm that uses dynamic programming to compute the length of the longest common subsequence in O(mn) time by filling out a 2D table in a bottom-up manner and returning the value at the last index. It also describes how to construct the actual longest common subsequence by tracing back through the table.
The document provides contact information for Python Homework Help, including their website, email address, and phone number. It then presents sample code defining two Python classes - A and B (which inherits from A) - and evaluates various expressions involving objects of these classes. Next, it defines Student and StudentBody classes to represent university students and evaluates methods on a sample StudentBody object. Finally, it presents additional examples involving state machine definitions and difference equations.
Arithmetic progressions - Poblem based Arithmetic progressionsLet's Tute
Arithmetic progressions - problem based Arithmetic progressions.
Lets tute is an online learning centre. We provide quality education for all learners and 24/7 academic guidance through E-tutoring.
Our Mission- Our aspiration is to be a renowned unpaid school on Web-World.
Contact Us -
Website - www.letstute.com
YouTube - www.youtube.com/letstute
This document provides solutions to quizzes from the textbook "Probability and Stochastic Processes" by Roy D. Yates and David J. Goodman. The solutions summarize the key concepts and formulas tested in each quiz question. MATLAB code is also provided to simulate some of the probabilistic experiments described in the textbook. Errors found in any quiz solutions will be corrected and posted online.
The document discusses dynamic programming and its application to solving the longest common subsequence (LCS) problem. It presents the LCS algorithm, which uses dynamic programming to find the length and sequence of the longest subsequence common to two strings X and Y in O(mn) time, where m and n are the lengths of X and Y, respectively. It provides an example running the LCS algorithm on strings X="ABCB" and Y="BDCAB" to determine their longest common subsequence is "BCB".
The document discusses using dynamic programming to solve optimization problems like finding the longest increasing subsequence in a sequence, cutting a rod into pieces for maximum profit, and finding the shortest path in a directed acyclic graph. It provides examples and explanations of how to model these problems as dynamic programming problems and efficiently solve them using techniques like memoization and bottom-up computation.
Lyapunov-type inequalities for a fractional q, -difference equation involvin...IJMREMJournal
The document summarizes a research paper that presents new Lyapunov-type inequalities for a fractional boundary value problem involving a fractional difference equation with a p-Laplacian operator. The paper obtains necessary conditions for the existence of nontrivial solutions to the equation. It also presents some applications to eigenvalue problems. Key concepts from fractional calculus such as fractional derivatives and integrals are reviewed. Lemmas establishing uniqueness of solutions to related problems are also presented.
Successive Differentiation is the process of differentiating a given function successively times and the results of such differentiation are called successive derivatives. The higher order differential coefficients are of utmost importance in scientific and engineering applications.
Solutions manual for logic and computer design fundamentals 5th edition by ma...Beckham000
Solutions Manual for Logic and Computer Design Fundamentals 5th Edition by Mano IBSN 9780133760637
Download at:
http://downloadlink.org/p/solutions-manual-for-logic-and-computer-design-fundamentals-5th-edition-by-mano-ibsn-9780133760637/
logic and computer design fundamentals 5th edition solutions
logic and computer design fundamentals 5th edition chegg
logic and computer design fundamentals 3rd edition pdf
logic and computer design fundamentals 2nd edition pdf
digital logic and computer design
logic and computer design fundamentals ppt
logic and computer design fundamentals companion website
logic and computer design fundamentals table of contents
The document discusses backtracking algorithms. It begins by defining backtracking as a methodical way to try different sequences of decisions to solve a problem until a solution is found. It then provides examples of backtracking for finding a maze path and coloring a map. The key aspects of a backtracking algorithm are that it uses depth-first search to recursively explore choices, pruning paths that do not lead to solutions.
The document discusses various backtracking algorithms and problems. It begins with an overview of backtracking as a general algorithm design technique for problems that involve traversing decision trees and exploring partial solutions. It then provides examples of specific problems that can be solved using backtracking, including the N-Queens problem, map coloring problem, and Hamiltonian circuits problem. It also discusses common terminology and concepts in backtracking algorithms like state space trees, pruning nonpromising nodes, and backtracking when partial solutions are determined to not lead to complete solutions.
The document discusses the divide and conquer algorithm design strategy. It begins by explaining the general concept of divide and conquer, which involves splitting a problem into subproblems, solving the subproblems, and combining the solutions. It then provides pseudocode for a generic divide and conquer algorithm. Finally, it gives examples of divide and conquer algorithms like quicksort, binary search, and matrix multiplication.
This document discusses randomized data structures and algorithms. It begins by motivating randomized data structures as a way to transform average case runtimes into expected runtimes that are not dependent on specific inputs. It then provides examples of randomized data structures like treaps and randomized skip lists that provide efficient operations like insertion, deletion, and search in expected logarithmic time. It also discusses how randomization can be applied in algorithms like primality testing.
This document discusses randomized data structures and algorithms. It begins by motivating randomized data structures by noting that some data structures like binary search trees have average case performance but worst case inputs. Randomizing the data structure removes dependency on inputs and provides expected case performance. The document then discusses treaps and randomized skip lists as examples of randomized data structures that provide efficient expected case performance for operations like insertion, deletion, and search. It also covers topics like randomized number generation, primality testing, and how randomization can transform average case runtimes into expected case runtimes.
Skip lists are a data structure for implementing dictionaries. They consist of multiple sorted lists, with the top list containing all elements and lower lists being subsequences. Searching works by dropping down lists until finding the target element or determining it is absent. Insertion and deletion use a randomized algorithm adding/removing elements from the appropriate lists. Analysis shows the expected space is O(n) and search, insertion and deletion times are O(log n), with these bounds also holding with high probability. Skip lists provide fast, simple dictionary implementation in practice.
This document summarizes a talk on dynamic graph algorithms. It begins with an introduction to dynamic graph algorithms, which involve maintaining a graph structure and answering queries efficiently as the graph undergoes a sequence of edge insertions and deletions. It then discusses several examples of fully dynamic algorithms for problems like connectivity, minimum spanning trees, and graph spanners. A key data structure introduced is the Euler tour tree, which represents a dynamic tree as a one-dimensional structure to support efficient updates and queries. The document concludes by outlining a fully dynamic randomized algorithm for maintaining connectivity under edge updates with polylogarithmic update time, using a hierarchical approach with multiple levels of edge partitions and ET trees.
This document discusses divide-and-conquer algorithms and their time complexities. It begins with examples of finding the maximum of a set and binary search. It then presents the general steps of a divide-and-conquer algorithm and analyzes time complexity. Several algorithms are discussed including quicksort, merge sort, 2D maxima finding, closest pair problem, convex hull problem, and matrix multiplication. Strategies like divide, conquer, and merge are used to solve problems recursively in fewer comparisons than brute force methods. Many algorithms have a time complexity of O(n log n).
This document discusses the merge sort algorithm for sorting a sequence of numbers. It begins by introducing the divide and conquer approach, which merge sort uses. It then provides an example of how merge sort works, dividing the sequence into halves, sorting the halves recursively, and then merging the sorted halves together. The document proceeds to provide pseudocode for the merge sort and merge algorithms. It analyzes the running time of merge sort using recursion trees, determining that it runs in O(n log n) time. Finally, it covers techniques for solving recurrence relations that arise in algorithms like divide and conquer approaches.
This document discusses divide-and-conquer algorithms and their time complexities. It begins with examples of finding the maximum of a set and binary search. It then presents the general steps of a divide-and-conquer algorithm and analyzes time complexity. Several algorithms are discussed including quicksort, merge sort, 2D maxima finding, closest pair problem, convex hull problem, and matrix multiplication. Strategies like divide, conquer, and merge are used to solve problems recursively in fewer comparisons than brute force methods. Many algorithms have a time complexity of O(n log n).
The document discusses greedy algorithms and provides examples of problems that can be solved using greedy techniques. It introduces the coin changing problem and activity selection problem. For activity selection, it demonstrates that a greedy approach of always selecting the activity with the earliest finish time results in an optimal solution. It provides pseudo-code for a greedy algorithm and proves that the greedy solution is optimal for the activity selection problem by showing there is always an optimal solution that makes the greedy choice and combining the greedy choice with the optimal solution to the remaining subproblem yields an optimal solution to the original problem.
The document discusses various optimization problems that can be solved using the greedy method. It begins by explaining that the greedy method involves making locally optimal choices at each step that combine to produce a globally optimal solution. Several examples are then provided to illustrate problems that can and cannot be solved with the greedy method. These include shortest path problems, minimum spanning trees, activity-on-edge networks, and Huffman coding. Specific greedy algorithms like Kruskal's algorithm, Prim's algorithm, and Dijkstra's algorithm are also covered. The document concludes by noting that the greedy method can only be applied to solve a small number of optimization problems.
This document discusses greedy algorithms and provides examples of their use. It begins by defining characteristics of greedy algorithms, such as making locally optimal choices that reduce a problem into smaller subproblems. The document then covers designing greedy algorithms, proving their optimality, and analyzing examples like the fractional knapsack problem and minimum spanning tree algorithms. Specific greedy algorithms covered in more depth include Kruskal's and Prim's minimum spanning tree algorithms and Huffman coding.
The document outlines various data structures and algorithms for implementing dictionaries and hash tables, including:
- Separate chaining, which handles collisions by storing elements that hash to the same value in a linked list. Find, insert, and delete take average time of O(1).
- Open addressing techniques like linear probing and quadratic probing, which handle collisions by probing to alternate locations until an empty slot is found. These have faster search but slower inserts and deletes.
- Double hashing, which uses a second hash function to determine probe distances when collisions occur, reducing clustering compared to linear probing.
This document discusses hashing and hash tables. It begins by introducing hash tables and describing how hashing works by mapping keys to array indices using a hash function to allow for fast insertion, deletion and search operations in O(1) average time. However, hash tables do not support ordering of elements efficiently. The document then discusses issues with hash functions such as collisions when different keys map to the same index. It describes techniques for collision resolution including separate chaining, where each index points to a linked list, and open addressing techniques like linear probing, quadratic probing and double hashing that resolve collisions by probing alternate indices in the array.
This document summarizes key points about extendible hashing and discusses its use for a spelling dictionary case study. Extendible hashing is a hashing technique that optimizes disk accesses for huge datasets by storing hash buckets in disk blocks. It uses a directory to hash to the correct bucket. The document explains how to insert keys, split buckets, and rehash the table. It also discusses solutions for the spelling dictionary case study, comparing storage and time efficiency of sorted arrays, open hashing, and closed hashing with linear probing.
This document discusses extendible hashing, which is a hashing technique for dynamic files that allows efficient insertion and deletion of records. It works by using a directory to map hash values to buckets, and dynamically expanding the directory size and number of buckets as needed to accommodate new records. When a bucket overflows, it is split into two buckets, and the directory is expanded to distinguish them. The directory size can also be contracted when buckets can be combined due to deletions. Alternative approaches like dynamic hashing and linear hashing that address the same problem of dynamic files are also overviewed.
The document discusses searching data structures like binary search trees and linked lists. It provides pseudocode for iterative searching algorithms on both ordered and unordered linked lists. The algorithms traverse the list by iterating through each node until the target value is found or the end is reached. For ordered lists, searching can stop early if the current node value is greater than the target. Tracing examples are provided to demonstrate searching for a value in sample linked lists.
1) Tree data structures involve nodes that can have zero or more child nodes and at most one parent node. Binary trees restrict nodes to having zero, one, or two children.
2) Binary search trees have the property that all left descendants of a node are less than the node's value and all right descendants are greater. This property allows efficient searches, inserts, and deletes that take O(log n) time on average.
3) Trees can become unbalanced over many insertions and deletions, affecting performance of operations. Various self-balancing binary search tree data structures use tree rotations to maintain balance.
This document discusses binary search trees (BSTs) and their use for dynamic sets. It covers BST operations like search, insert, find minimum/maximum, and successor/predecessor. It also discusses how BSTs can be used to sort in O(n log n) time by inserting elements in order and performing an inorder traversal, similar to quicksort. Maintaining a height of O(log n) for BSTs is discussed as an area for future improvement.
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.
Walmart Business+ and Spark Good for Nonprofits.pdfTechSoup
"Learn about all the ways Walmart supports nonprofit organizations.
You will hear from Liz Willett, the Head of Nonprofits, and hear about what Walmart is doing to help nonprofits, including Walmart Business and Spark Good. Walmart Business+ is a new offer for nonprofits that offers discounts and also streamlines nonprofits order and expense tracking, saving time and money.
The webinar may also give some examples on how nonprofits can best leverage Walmart Business+.
The event will cover the following::
Walmart Business + (https://business.walmart.com/plus) is a new shopping experience for nonprofits, schools, and local business customers that connects an exclusive online shopping experience to stores. Benefits include free delivery and shipping, a 'Spend Analytics” feature, special discounts, deals and tax-exempt shopping.
Special TechSoup offer for a free 180 days membership, and up to $150 in discounts on eligible orders.
Spark Good (walmart.com/sparkgood) is a charitable platform that enables nonprofits to receive donations directly from customers and associates.
Answers about how you can do more with Walmart!"
How to Make a Field Mandatory in Odoo 17Celine George
In Odoo, making a field required can be done through both Python code and XML views. When you set the required attribute to True in Python code, it makes the field required across all views where it's used. Conversely, when you set the required attribute in XML views, it makes the field required only in the context of that particular view.
This document provides an overview of wound healing, its functions, stages, mechanisms, factors affecting it, and complications.
A wound is a break in the integrity of the skin or tissues, which may be associated with disruption of the structure and function.
Healing is the body’s response to injury in an attempt to restore normal structure and functions.
Healing can occur in two ways: Regeneration and Repair
There are 4 phases of wound healing: hemostasis, inflammation, proliferation, and remodeling. This document also describes the mechanism of wound healing. Factors that affect healing include infection, uncontrolled diabetes, poor nutrition, age, anemia, the presence of foreign bodies, etc.
Complications of wound healing like infection, hyperpigmentation of scar, contractures, and keloid formation.
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 ..
আমাদের সবার জন্য খুব খুব গুরুত্বপূর্ণ একটি বই ..বিসিএস, ব্যাংক, ইউনিভার্সিটি ভর্তি ও যে কোন প্রতিযোগিতা মূলক পরীক্ষার জন্য এর খুব ইম্পরট্যান্ট একটি বিষয় ...তাছাড়া বাংলাদেশের সাম্প্রতিক যে কোন ডাটা বা তথ্য এই বইতে পাবেন ...
তাই একজন নাগরিক হিসাবে এই তথ্য গুলো আপনার জানা প্রয়োজন ...।
বিসিএস ও ব্যাংক এর লিখিত পরীক্ষা ...+এছাড়া মাধ্যমিক ও উচ্চমাধ্যমিকের স্টুডেন্টদের জন্য অনেক কাজে আসবে ...
Communicating effectively and consistently with students can help them feel at ease during their learning experience and provide the instructor with a communication trail to track the course's progress. This workshop will take you through constructing an engaging course container to facilitate effective communication.
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.
32. Example: keyExample: key A B C DA B C D
probability 0.1 0.2 0.4 0.3probability 0.1 0.2 0.4 0.3
The tables below are filled diagonal by diagonal: the left one is filledThe tables below are filled diagonal by diagonal: the left one is filled
using the recurrenceusing the recurrence
CC[[i,ji,j] =] = min {min {CC[[ii,,kk-1] +-1] + CC[[kk+1,+1,jj]} + ∑]} + ∑ pps ,s , CC[[i,ii,i] =] = ppii ;;
the right one, for trees’ roots, recordsthe right one, for trees’ roots, records kk’s values giving the minima’s values giving the minima
00 11 22 33 44
11 00 .1.1 .4.4 1.11.1 1.71.7
22 00 .2.2 .8.8 1.41.4
33 00 .4.4 1.01.0
44 00 .3.3
55 00
00 11 22 33 44
11 11 22 33 33
22 22 33 33
33 33 33
44 44
55
ii ≤≤ kk ≤≤ jj ss == ii
jj
optimal BSToptimal BST
B
A
C
D
ii
jj
ii
jj