Suppose we assign n persons to n jobs. Let,C base ij, be the cost of assigning the i^th person to the j^th job. a) Write a greedy algorithm that finds an assignment that minimizes the total cost of assigning all n persons to all n jobs. b) Test your algorithm based on the following example and discuss if your algorithm provides an optimal solution to the problem
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- Given an array of integers, find the longest non-decreasing subsequences (the subsequence does not need to be consecutive). For example, A = [8,5,2,10,3,6,9,7] contains the longest subsequences [2,3,6,9] and [2,3,6,7].a. Formulate the recursive relation of the optimal solution (do not miss the base case); b. Design a bottom-up (iterative) algorithm to calculate the length of the longest non-decreasing subsequences (pseudo code); c. Analyze the complexity of your algorithm.K = 0, L = 18 Write and solve the following linear program using lingo, take screen shots of your model as well as the reports and the optimal solution. Clearly show the optimal solution.NB:K=the second digit of your student number;L=sum of the digits of your student number, For example if your student number is 17400159 thenK=7andL=1+7+4+0+0+1+5+9=27!!!! SAVE YOUR FILE BY YOUR STUDENT NUMBER!!!!minz=t∈T∑(AtYt+PtXt)+k∈K∑(HkUk+BkVk)s.t.Uk+Vk=50∀k∈KXt−CtYt<=0∀t∈Tk∈K∑Vk≥80t∈T∑Xt≥t∈T∑DtXt>=0∀t∈TYt∈{0,1}∀t∈TUk>=0∀k∈KVk>=0∀k∈KThe sets parameters and data are as follows: \[ \begin{array}{l} \mathrm{T}=\{1,2,3,4\} \\ \mathrm{K}=\{0,1,2,3,4\} \\ \mathrm{A}=\{5000,7000,8000,4000\} \\ \mathrm{D}=\{250,65,500,400\} \\ \mathrm{C}=\{500,900,700,800\} \\ \mathrm{P}=\{20, \mathrm{~L}, 25,20\} \\ \mathrm{H}=\{5,3,2, \mathrm{~K}, 9\} \\ \mathrm{B}=\{8,5,4,7,6\} \end{array} \]Implement the algorithm for an optimal parenthesization of a matrix chain product as dis-cussed in the class.Use the following recursive function as part of your program to print the outcome, assumethe matrixes are namedA1, A2, ..., An.PRINT-OPTIMAL-PARENS(s, i, j){if (i=j) thenprint “A”i else{print “(”PRINT-OPTIMAL-PARENS(s,i,s[i, j])PRINT-OPTIMAL-PARENS(s, s[i, j] + 1, j)print “)”} }a- Test your algorithm for the following cases:1. Find and print an optimal parenthesization of a matrix-chain product whose sequenceof dimensions is<5,10,3, X,12,5,50, Y,6>.2. Find and print an optimal parenthesization of a matrix-chain product whose sequenceof dimensions is<5,10,50,6, X,15,40,18, Y,30,15, Z,3,12,5>. 3. Find and print an optimal parenthesization of a matrix-chain product whose sequenceof dimensions is<50,6, X,15,40,18, Y,5,10,3,12,5, Z,40,10,30,5>. X=10 Y=20 Z=30
- 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.Roses 1 2 3 4 5 Profit $5 $15 $24 $30 $35 For each positive integer n, let f(n) be the maximum profit that Flora can make with n roses.For example, if n = 10, Flora can make numerous bouquet combinations, including two 5-rose bouquets (total profit of $70), and a 4-rose bouquet with three 2-rose bouquets (total profit of $75). Provide two different algorithms for calculating f(n): one using Recursion, and one using Dynamic Programming. Explain why both algorithms are guaranteed to return the correct value of f(n).If you can make an optimal solution for a problem by making optimal solutions for its subproblems, then the problem has the quality in question. a) Subproblems that overlap; b) best substructure; c) memory; d) greedy
- Using C++, code the following problem: In this problem, we are given a knapsack of fixed capacity C. We are also given a list of N objects, each having a weight W(I) and profit P(I). We can put any subset of the objects into the knapsack, as long as the total weight of our selection does not exceed C. The goal is to maximize the total profit, which is the sum of the profits of each object we put into the knapsack. Example: Input: C = 3 W: 1 2 3 P: 100 5 50 Selected Items: 1 2 Explanation: Possible selections that can be put in the knapsack are: 1 with profit 100 2 with profit 5 3 with profit 50 1 and 2 with profit 105 Note that in this problem, there is no need that the some of weights be exactly C. Hence, in this example, 1 is a valid solution but since the profit is lower than the maximum, is not the final answer. Input Format First line is C. Second line is the number of objects, N. The next N lines, each line contains two numbers: the first number is the weight of an…Given an array of intervals I[1, . . . , n] where I[i] = [si, fi], find the minimumnumber of intervals you need to remove to make the rest of the intervals non-overlapping. Precisely define the subproblem.Provide the recurrence equation.Describe the algorithm in pseudocode to compute the optimal value.Describe the algorithm in pseudocode to print out an optimal solution.There are n people who want to carpool during m days. On day i, some subset ???? of people want to carpool, and the driver di must be selected from si . Each person j has a limited number of days fj they are willing to drive. Give an algorithm to find a driver assignment di ∈ si each day i such that no person j has to drive more than their limit fj. (The algorithm should output “no” if there is no such assignment.) Hint: Use network flow. For example, for the following input with n = 3 and m = 3, the algorithm could assign Tom to Day 1 and Day 2, and Mark to Day 3. Person Day 1 Day 2 Day 3 Driving Limit 1 (Tom) x x x 2 2 (Mark) x x 1 3 (Fred) x x 0
- You are given an array of positive integers S[1, . . . , n]. You are initially positioned at the array’s first index 1, and each element S[i] represents your maximum length of jump at position i. For example, you can jump from position i to any position among i + 1, . . . , i + S[i]. Return true if you can reach the last index, or false otherwise. Precisely define the subproblem.Provide the recurrence equation.Describe the algorithm in pseudocode to compute the optimal value.Describe the algorithm in pseudocode to print out an optimal solution.You are given a collection of n bolts of different widths and n corresponding nuts. Youare allowed to try a nut and bolt together, from which you can determine whether thenut is larger than the bolt, smaller than the bolt, or matches the bolt exactly. However,there is no way to compare two nuts together or two bolts together. The problem isto match each bolt to each nut. Design an algorithm for this problem with Θ(n log n) average-case complexity (in terms of nut-bolt comparisons). Explain why your algorithm has Θ(n log n) average-case complexity. You may use any result given in class without explicitly solving recurrence equations. I know that one can use Quicksort to get nlogn but I just don't know how to implement it. Thank you so much.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.