Exercise 4 20= 10+4+6 The rod-cutting problem consists of a rod of n units long that can be cut into integer-length pieces. The sale price of a piece i units long is Pi for i = 1,...,n. We want to apply dynamic programming to find the maximum total sale price of the rod. Let F(k) be the maximum price for a given rod of length k. 1. Give the recurrence on F(k) and its initial condition(s). 2. What are the time and space efficiencies of your algorithm? Now, consider the following instance of the rod-cutting problem: a rod of length n=5, and the following sale prices P1=2, P2=3, P3=7, P4=2 and P5=5.
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- Write a recurrence that would be used in dynamic programming for thefollowing problem: Given a rod of length n and an array A of prices thatcontain all prices of all the pieces smaller than n. Determine the maximumvalue obtained by cutting up the rod and selling the pieces. Note that youcould sell the rod at its original length without cutting it.Given A = {1,2,3} and B={u,v}, determine. a. A X B b. B X BConsider the same house rent prediction problem where you are supposed to predict price of a house based on just its area. Suppose you have n samples with their respective areas, x(1), x(2), ... , x(n), their true house rents y(1), y(2),..., y(n). Let's say, you train a linear regres- sor that predicts f(x()) = 00 + 01x(e). The parameters 6o and 0, are scalars and are learned by minimizing mean-squared-error loss with L2-regularization through gradient descent with a learning rate a and the regularization strength constant A. Answer the following questions. 1. Express the loss function(L) in terms of x), y@), n, 0, 01, A. 2. Compute L 3. Compute 4. Write update rules for 6, and O1
- A robot can move horizontally or vertically to any square in the same row or in the same column of a board. Find the number of the shortest paths by which a robot can move from one corner of a board to the diagonally opposite corner. The length of a path is measured by the number of squares it passes through, including the first and the least squares. Write the recurrence relation if you solve the problem by a dynamic programming algorithm.Considering the function f(x) = x – cos(x), what is the value of x7 after performing fixed point iteration. Assume an initial guess of 1. Use the equation form that will seem fit according to the choices provided.Group of answer choices 0.72210 0.71537 0.76396 0.72236More and more seafood is being farm-raised these days. A model (differentialequation) used for the rate of change for a fish population, P(t) in farmingponds is given by P'(t) = b(1-(P(t)/Pm)) - hP(t) where b is the birth rate, PM is the maximum number of fish the pond cansupport, and h is the rate the fish are harvested. Write a python code that implements the Forward Euler method to solve thedifferential equation Suppose that the carrying capacity PM = 20, 000 fish with a birth rateof 6% and a harvesting rate of h = 0%, use your Python code to findand plot the numerical solution for the first 400 days for different valuesof y0. Pick y0 < 20, 000, y0 > 20, 000. Don’t forget to label all your plotswith x and y axes label, titles and legends. Use a time step ∆t = 0.1
- Suppose that there are m students who want to take part in n projects. A student is allowed to join in a particular project only if the student is qualified for the project. Each project can only have a limited number of students, and each student can take part in at most one project. The goal is to maximize the number of students that are admitted to the projects. Show how to solve this problem by transforming it into a maximum flow problem. What is the running time of your algorithm?an Office consisting of m cabins enumerated from 1 to m. Each cabin is 1 meter long. Sadly, some cabins are broken and need to be repaired.You have an infinitely long repair tape. You want to cut some pieces from the tape and use them to cover all of the broken cabins. To be precise, a piece of tape of integer length t placed at some positions will cover segments 5,5+1-sit-1.You are allowed to cover non-broken cabins, it is also possible that some pieces of tape will overlap.Time is money, so you want to cut at most k continuouspieces of tape to cover all the broken cabins. What is theminimum total length of these pieces?Input FormatThe first line contains three integers n,m and k(1sns10°, namsloº, Isksn) - the number of broken cabins, the length of the stick and the maximum number of pieces you can useThe second line contains n integers bl,b2,bn (Isbism) - the positions of the broken cabins. These integers are given in increasing order, that is, blOutput Format:Print the minimum total…Consider the problem of making change for n cents using the fewest number of coins. Assume that we live in a country where coins come in k dierent denominations c1, c2, . . . , ck, such that the coin values are positive integers, k ≥ 1, and c1 = 1, i.e., there are pennies, so there is a solution for every value of n. For example, in case of the US coins, k = 4, c1 = 1, c2 = 5, c3 = 10, c4 = 25, i.e., there are pennies, nickels, dimes, and quarters. To give optimal change in the US for n cents, it is sufficient to pick as many quarters as possible, then as many dimes as possible, then as many nickels as possible, and nally give the rest in pennies. Design a bottom-up (non-recursive) O(nk)-time algorithm that makes change for any set of k different coin denominations. Write down the pseudocode and analyze its running time. Argue why your choice of the array and the order in which you ll in the values is the correct one.
- Consider the problem of making change for n cents using the fewest number of coins. Assume that we live in a country where coins come in k dierent denominations c1, c2, . . . , ck, such that the coin values are positive integers, k ≥ 1, and c1 = 1, i.e., there are pennies, so there is a solution for every value of n. For example, in case of the US coins, k = 4, c1 = 1, c2 = 5, c3 = 10, c4 = 25, i.e., there are pennies, nickels, dimes, and quarters. To give optimal change in the US for n cents, it is sufficient to pick as many quarters as possible, then as many dimes as possible, then as many nickels as possible, and nally give the rest in pennies. Prove that the coin changing problem exhibits optimal substructure. Design a recursive backtracking (brute-force) algorithm that returns the minimum number of coins needed to make change for n cents for any set of k different coin denominations. Write down the pseudocode and prove that your algorithm is correct.Consider the problem of making change for n cents using the fewest number of coins. Assume that we live in a country where coins come in k dierent denominations c1, c2, . . . , ck, such that the coin values are positive integers, k ≥ 1, and c1 = 1, i.e., there are pennies, so there is a solution for every value of n. For example, in case of the US coins, k = 4, c1 = 1, c2 = 5, c3 = 10, c4 = 25, i.e., there are pennies, nickels, dimes, and quarters. To give optimal change in the US for n cents, it is sufficient to pick as many quarters as possible, then as many dimes as possible, then as many nickels as possible, and nally give the rest in pennies. Design a bottom-up (non-recursive) O(nk)-time algorithm that makes change for any set of k different coin denominations. Write down the pseudocode and analyze its running time. Argue why your choice of the array and the order in which you fill in the values is the correct one. Notice how it is a lot easier to analyze the running time of…please try to simulate the probability of rolling a Die with Sample Space* S={1,2,3,4,5,6} and the probability of each sample point has a 1/6 chance of occurring, i.e., you need to verify that your simulation converges to 1/6 when you select one point of sample space. When X is a random variable for sample point of rolling a Die, Pr(X<=4)=2/3. Please verify this result by simulation. Please let me know how to make an Excel file as stated above.