d) Implement your algorithm and show a test case of it
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- A numeric array of length N is given. We need to design a function that finds all positive numbers in the array. a) Describe approaches for solving optimal worst case and optimal average case performance, respectively.Algorithm A search using the heuristic h(n) = α for some fixed constant α > 0 is guaranteed to find an optimal solution Select one: True FalseExercise 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.
- We have N jobs and N workers to do these jobs. It is known at what cost each worker will do each job (as a positive numerical value). We want to assign jobs to workers in such a way that the total cost of completion of all jobs is minimal among other possible alternative assignments. For this problem, write the algorithm as pseudocode, whose input is a matrix representing worker/job costs, and the output is a list of tuples showing which work will be done by which worker, and that tries to reach the solution with GREEDY technique. Explain in what sense your algorithm exhibits greedy behavior. What is the time complexity of your algorithm? Interpret if your algorithm always produces the best (optimum) result for each instance of the problem.Write the formula for calculating the optimal cut of the text t of length n, where we have m cut points given in the field R.TIP: In the case of the first question, R = [3, 7, 17].C) Record the algorithm for calculating the optimal cut and estimate the time and complexity.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.
- 1. Let S = {1,2...n} be a set activities, where activity i € S has a start time si, a finish time f; and a profit p, asSociated with it. Design an dynamic programming algorithm for computing a set of compatible activities of maximum profit. Argue that the time complexity of you algorithm is O(n log(n)). (You must start with the correct definitions, and then write a recurrence relation.)A celebrity has planned to give his/her live performance in Mumbai. He/she needs to travel from Kolkata to Mumbai by their own vehicle due to the safety reasons. Create an algorithm to help them to reach the destination with optimal cost. The cost factor may be attributed by fuel consumption, time taken and/or expected speed on a particular route which are depicted in the below map. By using the information provided for each city. Use the following algorithm to solve the problem: a) Iterative Deepening A*Algorithm to An iterative solution to Towers of Hanoi.in: triplet S = s0, s1, s2 representing the current game stateout: triplet R = r0, r1, r2 representing the new game statelocal: pole indices a, b, z ∈ {0, 1, 2}; disc numbers g, h ∈ [2, n]; last(Q) = Q|Q|−1, if1 ≤ |Q|, otherwise, last(Q) = +∞
- We are to complete a set of n jobs using m identical machines. Each job has a known duration. The goal is to schedule the jobs on m machines so that the time to finish the last job is as small as possible. Once a job is scheduled on a machine, it can’t be stopped and restarted. Any job can run on any machine. A greedy algorithm to solve this problem is as follows: sort jobs in decreasing order of duration. Schedule jobs one by one, choosing the machine where it can start the earliest. a. For m = 3 and n = 6 where the durations are <9, 12, 3, 8, 6, 5>, what is the finish time of the last job using greedy schedule? b. Exhibit an input for which the greedy schedule is NOT optimal.Let's say there are n villages, {X1, . . . , Xn} on the country-road and we aim to build K < n restaurants to cover them. Each restaurant has to be built in a village, and we hope to minimize the average distance from each village to the closest restaurant. Please give an algorithm to compute the optimal way to place these K restaurants. The algorithm should run in O(k * n^2) time. Solutions with slightly higher time complexity also accepted.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.