PowerPoint Presentation

[Virtual Presenter] Welcome everyone to this presentation. As we start Chapter 3, we will explore how we can make use of graph algorithms and dynamic programming to more effectively solve recursive problems. Let's take a look..

[Audio] Dynamic Programming is a powerful tool for solving a wide range of complex problems. It works by breaking down the problem into smaller subproblems and finding the optimal solutions for each one, with the answers then being combined to form a single solution for the entire problem. This technique reduces complexity and makes the problem easier to solve, by using a memorization technique, typically via a memory table, to store the solutions of the subproblems for future reference. For Dynamic Programming to be useful, the problem must have two main properties: optimal substructure, meaning that the optimal solution can be constructed from the optimal solutions of its subproblems, and overlapping subproblems, meaning that the problem can be divided into smaller subproblems..

[Audio] Dynamic programming can be used to analyze the Fibonacci sequence in order to determine its patterns and behaviors. This knowledge can be used to calculate stock prices, create animations, and obtain deeper insights into the sequence and its potential uses..

[Audio] In order to find the fifth member of the Fibonacci sequence, the fourth and third sequence members must be located. This task serves as a subproblem to the main objective. A table with corresponding values from the sequence can be used, such as the third being 3, fourth being 4, and so forth. A recursive formula, f sub n equals f sub n minus 1 plus f sub n minus 2, can then be used to fill in the missing data in the table and locate the fifth member of the Fibonacci sequence..

[Audio] This slide presents the implementation of a Dynamic Programming algorithm to solve the Fibonacci Sequence problem. As seen in the table, fib(5) can be solved by finding the values of f(4) and f(3) first. This is done through a recursive code which calculates the value of f(n) by adding the two preceding values, f(n-1) and f(n-2). The code is a recursive approach, which means it will keep repeating until the base value of the Fibonacci Sequence is reached. The result is a fast, efficient and reliable solution to the Fibonacci problem..

[Audio] Dynamic programming is an effective method to solve intricate issues. It permits us to divide problems into substeps which can be responded to productively. Through memorization, the solutions of each subproblem can be kept in a storing table to be used later on, thereby giving the opportunity to retrieve solutions quickly when needed. A common example is the 1D array which is employed to store the Fibonacci sequence, or a 2D array to keep an adjacency matrix. With this strategy, time is saved and it becomes a substantial asset when dealing with intricate problems..

[Audio] The Fibonacci sequence is a mathematical structure composed of a set of numbers following a specific rule. To find the nth value, the table presents the values for n = 0, 1, 2, 3, 4 and 5. To determine f(5), the simple recursive equation can be used, meaning that the nth value is equal to the sum of the two previous values. In this case, f(5) equals f(4) + f(3), which is 4 + 3 = 7. This logic is applied to all the numbers in the sequence until the value of f(0) is reached, which is 3. Therefore, f(1) is 3, f(2) is equal to 7, etc. Solving the values in this manner allows us to reach f(11), which equals 29..

[Audio] Dynamic programming is an algorithm utilized to address a variety of optimization questions. This table displays its usefulness. It can be seen that for each 'Nth' the 'value' is the sum of the prior two 'values'. This algorithm shows how to enhance the productivity of a particular issue. It is an excellent resource for expeditiously solving intricate problems in our daily lives..

[Audio] We will discuss Fibonacci series, a type of graph algorithm that can be solved using Dynamic Programming. It is a well-known sequence of numbers with a recursive formula of adding the two previous numbers in the sequence to get the sum of the next number. The code in this example directly calculates the desired Fibonacci number in linear time complexity using Dynamic Programming, providing an efficient solution..

[Audio] We are examining a Space Optimized method to calculate Fibonacci series. This uses a bottom-up dynamic programming approach which has a constant space and time complexity. It involves a loop that runs for up to the input number, and calculates the Fibonacci series for each. The result is returned from the function, making it a highly efficient method for calculating Fibonacci series..

[Audio] We are discussing the Graph Algorithm - Dynamic Programming, and more specifically the problem of Matrix Chain Multiplication. This algorithm is used to find the most efficient way to multiply a sequence of matrices together. The focus here is to decide the sequence of the matrix multiplications and not to perform the multiplications. It has the potential to revolutionize 3D rendering, network analysis, and many other fields. It is an extremely powerful tool that, when used correctly, can cause a major shift in the way things are done..

For Example:.

[Audio] Matrix chain multiplication is an algorithm commonly used to solve dynamic programming problems. It consists of taking an NxN matrix and using a set of partial solutions to compute the cost of the best chain possible in terms of computing time and effort. This algorithm allows us to determine the optimal solution for problems with a large number of possible chain combinations, making it indispensable for many real-world and industrial problems..

Problem: Matrix Chain Multiplication. Basic Matrix.

Problem: Matrix Chain Multiplication.

[Audio] We will look at matrix chain multiplication and its dynamic programming approach. Need to consider the number of multiplication operations and the optimal path for the solution. Cost table and parentheses table can provide helpful information to help determine best approach. To arrive at desired solution, want to obtain last element diagonally..

[Audio] Our 3rd chapter focuses on the Graph Algorithm – Dynamic Programming. We start by multiplying the same matrix to slot in 0 as the first diagonal elements. To calculate the second diagonal, we multiply the elements with the first one and add the cost on the left, the cost below it and the product of the two dimensions, which yields 120. This step forms a key part of the four-part Graph Algorithm – Dynamic Programming. Moving on, let's see what's next..

Problem: Matrix Chain Multiplication. A 1 . A 2 . A 3 . A 4 5x4 4x6 6x2 2x7.

[Audio] For this third chapter on Graph Algorithm, we will learn how to multiply the elements in the third diagonal, beginning with the first element. To calculate the cost, we add up three elements: the cost on the left of the (1,3) element, the cost below (1,3), and the product of the left and the below dimensions. The lowest cost is then found. For instance, the second element in row 1 and the second element in column 3 results in a cost of 180, whereas the first element in row 1 and the first element in column 3 results in a cost of 88..

[Audio] Dynamic Programming algorithm is a powerful tool for improving the performance of programming problems. In this slide, an example of its application is presented. The first element of the row and column is calculated by adding the element in the previous row and the element in the same column, and then multiplying the result with the row number, column number and the provided value. For the second element, the same principle is applied but with the element in the previous row and the third column. This approach yields two different outcomes..

[Audio] To determine the lowest cost, the procedure of adding the cost on the left of the (1,4) element, the cost below the (1,4) element and the product of the dimensions of the left and the below must be repeated for the 2nd and 3rd elements in row 1 and the respective elements in column 4. The cost for the 1st element was calculated as 244, for the 2nd as 414, and for the 3rd as 158..

Problem: Matrix Chain Multiplication. A 1 . A 2 . A 3 . A 4 5x4 4x6 6x2 2x7.

[Audio] In this chapter, we explore a graph algorithm that employs dynamic programming. Our goal is to find the optimum number of matrix multiplications. To accomplish this, we begin by selecting the last element from the parentheses table. We then divide the matrix, eliminating any redundant parentheses. If there are more than two matrices present, we keep dividing the matrix until none remain. Once done, the last element chosen is the optimum number of matrix multiplications..

[Audio] For the Matrix Chain Multiplication Problem of Chapter 3, we need to create two memory tables: a cost table and a parentheses table. As an example, we'll use a table of 5 matrices, with the respective dimensions of 5x6, 6x8, 8x10, 10x4, and 4x7. We can then solve this problem step by step..

[Audio] A 5x5 table is looked at with regard to the matrix chain multiplication problem. A value of 0 is assigned to the table elements in row 1, column 1 and the remaining subsequent elements at the bottom right, since when n = m, only one matrix is multiplied, thus no multiplication is needed. The value in the first row, last element is the final answer, which is the least amount of operations possible for the matrix chain multiplication problem..

[Audio] We have already looked at the data in the memory table from (1,1) to the bottom right concerning the matrix chain multiplication problem. Now, focusing on the next digonal row, let's take a closer look at the cost and parentheses of the elements (1,2), (2,3), (3,4), and (4,5)..

[Audio] This slide illustrates a table of data pertaining to the matrix chain multiplication problem. It displays the dimensions of each matrix, which enables us to find the cost and parentheses of the chain incrementally. We can calculate the cost of a matrix by adding together the costs of the matrix to the left and below it, plus the product of their dimensions. For example, the cost of matrix M12 in this example is 240, as marked in the parentheses memory table (1,2). The same process holds true for any matrix displayed in the example. This data shows that matrix chain multiplication is an important issue requiring the application of dynamic programming..

[Audio] Matrix chain multiplication is an efficient method for computing the product of a sequence of matrices. It provides the ability to compute the product recursively for any given sequence and to reduce the number of scalar multiplications required. We can observe an example of this problem via a step by step explanation. This table includes the costs per chain matrix; cost increases with the number of matrices. Eventually, we are left with a cost table and parentheses table which illustrate the optimal cost for each matrix chain and the optimal combination of parentheses..

[Audio] Matrix chain multiplication is a problem that can be solved using dynamic programming. We used a table of five matrices with dimensions of 5x6, 6x8, 8x10, 10x4, and 4x7 respectively to calculate the optimal number of operations required to multiply these four matrices. Applying the dynamic programming algorithm, we found that the optimal cost is 320 and the optimal parentheses order is 1, 3, 5, meaning matrices must be multiplied in the order A1 x A3 x A5. To illustrate the solution, we provided a step-by-step explanation. Dynamic programming is an effective technique to find the optimal cost and parenthesis order for solving matrix chain multiplication problems..

[Audio] We can observe the Matrix Chain Multiplication Problem, step-by-step, in this slide. The cost table indicated that the minimum cost of multiplying the matrices is 544, and the parentheses table specified the optimal way of multiplying these matrices with the minimum cost. This has the potential to provide efficient solutions to a range of real-world problems..

[Audio] This slide looks at the Matrix Chain Multiplication Problem and how it can be solved using Dynamic Programming. We can see a table containing the dimensions of the chain of matrices. To determine the cost of (1,4), we examine the costs in row one and column four and find the lowest cost among them. The cost turns out to be 640. We then perform the same process for (2,5) which works out to the product of the dimensions of the chain, 5x6 multiplied by 6x10 and 10x4, which gives us 5x10 for the cost. Dynamic Programming can be used to easily solve the Matrix Chain Multiplication Problem..

[Audio] Matrix Chain Multiplication is a problem that can be solved using Dynamic Programming. Cost table and parentheses table are calculated with the given data. Cost table provides the minimum cost of computing each matrix, and parentheses table shows which matrices should be grouped together for multiplication. Thus, the problem can be solved with minimum cost and time..

[Audio] We explore the 'Matrix Chain Multiplication Problem' in this slide. To calculate the number of least operations for this problem, we compare the cost of three subproblems; the cost left to the final answer (row 1, column 5), the cost below the final answer (row 1, column 3), and the cost of the final answer (row 1, column 5). Using the following data: 5x6 6x8 8x10 10x4 4x7, we are able to calculate the final answer which is the number of least operations for this problem..

[Audio] We have solved the Matrix Chain Multiplication Problem using dynamic programming. The cost table reveals that the lowest cost for multiplying matrices with sizes A1 A2 A3 A4 A5 is 772. The parentheses table provides a way to understand which matrices should be multiplied, thus finding the optimal order of matrices with minimal cost..

[Audio] Without greetings, without beginning with Today, and without thanks: In this slide, we will be looking at the Matrix Chain Multiplication Problem and step by step explanation. To find the best parenthesizing of the sequence A1 A2 A3 A4 A5 that resulted in 772 cost operations, we use the parentheses table. We can observe from the table that the value of the first row and last element is 4. We divide the sequence after A4 into two partitions, adding a bracket into the two partitions before eliminating any redundant brackets. This process is repeated until a single bracket has no more than two matrices. We can see in our example that the multiplication sequence before divide is (A1 A2 A3 A4 A5) and after the divide is ((A1 A2 A3 A4) (A5)), and after eliminating the redundant bracket is ((A1 A2 A3 A4)A5)..

[Audio] Matrix Chain Multiplication Problem is a method of optimizing sequence of matrix multiplication. To obtain the optimal result, Dynamic Programming technique is used which involves creating a table with the values of the matrices that must be multiplied. Through step by step example, this process can be applied to determine the optimal sequence of matrix multiplication..

[Audio] The Matrix Chain Multiplication Problem is a powerful technique to solve complex issues. To start, parenthesize A1. When there are more than two elements, parenthesize A2, A3 and A4. To proceed, look at row 2, beginning with A2. The fourth element of this row ending in A4 has a value of two, which indicates the sequence should be divided after A2. Create a table with the given data and then eliminate unneeded brackets to finish. By using this process, the sequence can be parenthesized so that only two elements are in a single bracket..

[Audio] The Matrix Chain Multiplication Problem is an algorithm that finds the most efficient way to multiply a chain of matrices. In this case, A1, A2, A3, A4 and A5 are the matrices and the best sequence is to first multiply A3 with A4, followed by A2, A1 and A5, resulting in only 772 operations needed to complete the problem. This problem can be used to quickly and efficiently determine the optimal solution for similar problems..

[Audio] The complexity of the matrix chain multiplication can be determined by the number of operations required to generate the product sequence. In order to calculate this, a dynamic programming approach is used. This involves creating a cost table which is filled in a top-down manner. Each row is filled before continuing on to the next. To calculate the cost of each element, all elements in the same row and column are checked which has a time complexity of O(n). An example cost table, with the given data, is shown below: one equals zero, two equals two hundred and forty, three equals four hundred and eighty, four equals three hundred and twenty, and five equals two hundred and eighty..

[Audio] The complexity analysis of the Matrix Chain Multiplication has a time complexity of o(n^2); as shown in the table, the cost for a chain of length 5 is 280, as is the case for chain length of up to n. Therefore, the resulting time complexity of this chain is o(n^2)..

[Audio] We will be examining the outcomes of our investigation into the Matrix Chain Multiplication Problem through dynamic programming. Our investigation revealed the time complexity to be o(n^3) and the space complexity to be o(n^2). We can observe this from the data in our table which represents the cost of the different computations. Examining the entire Matrix Chain requires analyzing multiple rows, thus having a complexity of o(n^3). In terms of space complexity, we need to utilize a two-dimensional array for our memory table, which has a complexity of o(n^2)..

[Audio] Today, we will be looking at a code example for Matrix Chain Multiplication, a graph algorithm in the field of Dynamic Programming. This code creates a matrix to store optimal solutions to subproblems, making efficient use of memory. It then transfers those solutions to the original problem, providing an optimal solution. Let's dive right into it!.

[Audio] Dynamic Programming is a popular technique in Graph Algorithms and the slide on Graph Algorithms - Dynamic Programming shows a code snippet for matrix chain multiplication. It includes global variable declarations, function declarations, and functions for matrix multiplication and ordering. Matrix chain multiplication is useful for optimization as it enables us to find the most efficient way to multiply a sequence of matrices. Analysis of this code helps to better understand the concept of Dynamic Programming and its application in Graph Algorithms..

[Audio] Given the data in the table, our C code for matrix chain multiplication has been implemented successfully. Looping through the length and calculating the cost determines the minimum amount of operations needed for the matrix chain multiplication. Furthermore, printing out the cost of each element in the table and the multiplication sequence is possible. In this case, the minimum number of multiplication is zero. By utilizing dynamic programming, it is ensured that the minimum amount of operations needed for the matrix chain multiplication is determined, and that the cost of each element and the multiplication sequence can be printed out clearly. These results demonstrate that our C code for matrix chain multiplication is accurate and complete..

[Audio] In this slide, we will be learning about how to write the code for matrix chain multiplication. We will begin by reading in the number of elements and their respective dimensions. Afterwards, we will set all matrix elements, such as m11, m22, m33 etc., to 0, since the cost of multiplying with one matrix is 0. We will then have three tables of data: the first table contains a row of the matrices as well as a row of 0s; the second table contains the matrix size with the i numbers ranging from 1 to 5; and the third table contains the matrices with their respective dimensions. Lastly, we will write the dimensions of the matrix into the program..

[Audio] This slide looks at the C code for matrix chain multiplication. A table is presented which shows the various lengths of the chain, as well as the optimal parenthesization and the minimum cost. The code uses equations to calculate the cost for each length, selecting the parenthesization that produces the least cost. Thus, the code is able to find the optimal parenthesization and minimum cost for each length of the chain, as seen in the table. This is the purpose of the C code discussed in this slide..

[Audio] We need to understand the underlying concept of dynamic programming to solve the matrix chain multiplication problem. We must find the most efficient way to multiply a chain of matrices and use suitable data structures to represent the matrices, store the calculated results in a cost matrix, and use loops to perform the matrix multiplication. The code will print the cost matrix, which can then be used to find the most efficient order for the multiplication..

[Audio] The goal of this algorithm is to find the best method of multiplying a given sequence of matrices. To do this, the algorithm employs a divide-and-conquer approach. By breaking the problem into smaller sub-problems, the algorithm can achieve an optimal solution. The C code on the slide is an example of a matrix chain multiplication algorithm, which uses dynamic programming to split up the sequence of matrices into smaller sub-problems and then rebuilds the solution. This technique benefits from the reoccurring sub-problems which leads to faster solutions and a more effective utilization of memory. By adhering to the instructions in the code, the algorithm can determine the optimal way to multiply the sequence of matrices that is displayed on the slide..

[Audio] In the C code for Matrix Chain Multiplication, iteration 2 is running with a value of i and j being 0 and 3 respectively. This leads to the program executing the else statement and printing a bracket. Following that, the recursive function gets called and the value 0 and 0 get passed to that function. This results in two brackets being printed on the screen. Overall, it can be stated that the C code for Matrix Chain Multiplication is an effective way to solve the data given in the table..