Prove the theorem: The Hungarian Algorithm finds a maximum weight mathcing and a minimum cost cover
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Prove the theorem: The Hungarian
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- 4. The following algorithm for approximating roots combines the bisection method with Newton's method and ensures the convergence even in the case where Newton's method fail to converge. Choose an interval [a, b] such that f(a)f(b) ≤ 0. Compute initial approximation xo = (a+b)/2 of the root using the bisection method. For i=1,2,... Compute x₁ using Newton's method and X₁-1. If x₁ [a, b], set x₁ = (a + b)/2 (from bisection). Check for convergence. If f(a) f(xi) ≤0 set b = xi, else set a = xį. (a) Implement this algorithm in a PYTHON function with the following specifications: def findzero (a, b, tol, maxit, f, df) # Input: # a, b = The endpoints of the interval # tol = The required tolerance # maxit = Maximum number of iterations # f, df = The function and its derivative # Output: # star #niter # ierr # # # approximation of root = number of iterations for convergence 0, the method converged 1, df was zero or undefined 2, maximum number of iterations has been reached return xstar, niter,…Consider the two versions of Quicksort that are based on the following: (i) Median selection algorithm using the linear-time select method; (ii) The randomized pivot selection algorithm Which of the two algorithms has a faster worst-case asymptotic behavior? Which of the two algorithms has a faster average-case performance? Provide empirical evidence of your answer by writing the corresponding programs for each and conducting experimental evaluation. Your experimental results should provide evidence of your answer. For this you need to run an adequate number of experiments on a variety of array/list sizes (use large randomly generated lists of integers).38. Suppose that, in a divide-and-conquer algorithm, we always divide an instance of size n of a problem into n subinstances of size n/3, and the dividing and combining steps take linear time. Write a recurrence equation for the running time T(n), and solve this recurrence equation for T(n). Show your solution in order notation.
- Computer Science C++ please, use Monte Carlo integration to calculate the volume of a d-dimensional hypersphere of radius r = 1. (Note that for d=1, 2, and 3, the common names for d-dimensional volume are length, area, and volume, respectively.) Print out each volume and narrow the answer down to 4 digits with 99% confidence. How far can you push d for this method?5. You are given a set of n positive numbers A = {a₁,..., an} and a positive integer t. Design a dynamic programming algorithm running in O(nt) time that decides whether there exists a subset A' CA such that Σ x = t. Note that each element of A can be xЄA' used at most once. Is the run-time of your algorithm polynomial with respect to the size of the input?Part 3 You are part of a team responsible for running a successful video streaming service with millions of views daily. For marketing and research reasons, you have been asked to implement an algorithm that efficiently finds the k most viewed videos daily. You should expect k<Given a matrix of size N x M where N is the number of rows and M is the number of columns, write an algorithm to find the shortest path from the top-left cell to the bottom-right cell that passes through all the cells with a prime value and avoids cells with composite values. The algorithm should have a time complexity of O(NM log(max(N,M))).Below is a list of functions that commonly appear in complexity analyzes as a function of the size n of the problem. Sort the functions in ascending order of growth rate, that is, the slowest growing one is 1, and so on. Note that the functions are described in the notation O(.), which represents the cost of the algorithm as a function of the predominant term of the cost expression.Subject: Theory of Computation Design a brute-force algorithm for solving this problem (below) and demonstrate that this algorithm's time complexity grows exponentially with input length. Adding 2 binary numbers, w is the length of the input which is the number of digits in the numbers. Suppose the input has the same number of digitsGiven an array of unsorted integers, find the maximum product of two integers in an array. For example, if the arc = [10, 8, -1, 7, 14] then the maximum product would be 140, as output of (10 x 14). To solve this problem, the following idea can be used: Design a brute-force algorithm to solve this problem using a Pseudocode and find its complexity - Another solution would be "Sort the array first, and then to get the product of its highest two integers". Is this idea better that the first one? Explain why? Is there any extreme case that should be considered while designing your algorithm? Justify your answer. Design your own idea to solve this problem in O(n) complexity, where n is the size of the array. [note: your idea should be explained in simple English, and then transformed into algorithm (pseudocode)].You are required to start from the first room (I) and collect the all item in the maze before arriving at the last room (O) using the least cost. Please execute the Breadth-first Search algorithm in Python to solve the problem with the assumption the cost that is required to move between rooms is constant and always the same.
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