algorithms
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For relatively small values of n, algorithms with larger orders can be more efficient than algorithms with smaller orders. Use a graphing calculator or computer to answer this question.
For what values of n is an
[50 log2(n)]
operations? (Assume n is an integer such that n > 1.)
n?
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- A certain recursive algorithm takes an input list of n elements. Divides the list into Vn sub-lists, each with yn elements. Recursively solves each of these yn smaller sub- instances. Then spends an additional 0(n) time to combine the solutions of these sub- instances to obtain the solution of the main instance. As a base case, if the size of the input list is at most a specified positive constant, then the algorithm solves such a small instance directly in 0(1) time. a) Express the recurrence relation that governs T(n), the time complexity of this algorithm. b) Derive the solution to this recurrence relation: T(n) = 0(?). Mention which methods you used to derive your solution.In python, The Longest Subsequence Problem is a well-studied problem in Computer Science, where given a sequence of distinct positive integers, the goal is to output the longest subsequence whose elements appear from smallest to largest, or from largest to smallest. For example, consider the sequence S= [9,7,4,10,6,8,2,1,3,5]. The longest increasing subsequence of S has length three ([4,6,8] or [2,3,5]), and the longest decreasing subsequence of S has length five([9,7,4,2,1] or [9,7,6,2,1]). And if we have the sequence S = [531,339,298,247,246,195,104,73,52,31], then the length of the longest increasing subsequence is 1 and the length of the longest decreasing subsequence is 10. Question: Find a sequence with nine distinct integers for which the length of the longest increasing subsequence is 3, and the length of the longest decreasing subsequence is 3. Briefly explain how youconstructed your sequence.Consider a divide-and-conquer algorithm that calculates the sum of all elements in a set of n numbers by dividing the set into two sets of n/2 numbers each, finding the sum of each of the two subsets recursively, and then adding the result. What is the recurrence relation for the number of operations required for this algorithm? Answer is f(n) = 2 f(n/2) + 1. Please show why this is the case.
- The Longest Subsequence Problem is a well-studied problem in Computer Science, where given a sequence of distinct positive integers, the goal is to output the longest subsequence whose elements appear from smallest to largest, or from largest to smallest. For example, consider the sequence S = [9,7,4,10,6,8,2,1,3,5]. The longest increasing subsequence of S has length three ([4,6,8] or [2,3,5]), and the longest decreasing subsequence of S has length five([9,7,4,2,1] or [9,7,6,2,1]). And if we have the sequence S = [531,339,298,247,246,195,104,73,52,31], then the length of the longest increasing subsequence is 1 and the length of the longest decreasing subsequence is 10. Question: Let S be a sequence with ten distinct integers. Prove by Contradiction that there must exist an increasing subsequence of length 4 (or more) or a decreasing subsequence of length 4 (or more). Hint: for each integer k in the sequence you found in the first part, define the ordered pair (x(k), y(k)), where x(k)…Let C(n) = 1^3 + 2^3 + 3^3+…+n^3 be the sum of the first n cubes.a. Write a recursive algorithm to compute C(n).b. What is the recurrence relation for your algorithm?c. What is the time complexity of the algorithm you developed? Do not copy from online or Use AIA graph is a collection of vertices and edges G(V, E). A weighted graph has weights (numbers, etc.) on every edge. A multigraph can have more than one edges between any vertices. Explain why a person should use a weighted graph instead of a multigraph. Give examples. An adjacency matrix might be a better choice for speeding up a program, however, it consumes huge memory for large graphs. How this situation can be improved? What programming constructs better suit graph representation? Explain with example
- Write for the following problem a recursive algorithm whose worst-case timecomplexity is not worse than Θ(n ln n). Given a list of n distinct positiveintegers, partition the list into two sublists, each of size n/2, such that thedifference between the sums of the integers in the two sublists is maximized.You may assume that n is a multiple of 2.Consider the following algorithm and answer the following questions. procedure Algo1(a1, a2, . . . , an: integers) k := 0 for i := 1 to n if ai is even then k := i return k{k is the desired location(or 0 if there are no evens)} What does the given algorithm compute??? Execute the given algorithm and complete the following table by using the list 3, 5, 8, 10, 5, 2, 1, 7, 3, 13. i ai k 1 3 2 5 3 8 4 10 5 5 6 2 7 1 8 7 9 3 10 13Consider the problem of sorting elements in an array of n numbers. Design two algorithms to implement the quicksort as follows: Recursive algorithm, and Non-recursive algorithm Analyze the efficiency of your algorithms as follows: For the recursive algorithm, figure out the recurrence relation and use the Master Theorem to find out the time efficiency in Big-O notation. For the non-recursive algorithm, use counting method to count the number of comparisons in Big-O notation.
- Consider the definition h(0) = 1 h(n) h(n-1)+h(floor(n/2)) + 1 For example, h(2) = h(1) + h(1) + 1 = 3, h(3) = h(2) + h(1) + 1 = 5, etc. An obvious algorithm to compute h is a recursive one, based directly on the defining equations. Is that algorithm efficient? Why or why not? If not, how can the algorithm be improved? Justify your answer.Consider the recursive algorithm below for computing the sum of the first n cubes: S(n) = 13 + 23 + … + n3 ALGORITHM S(n) //n is a positive integer if (n==1) return 1 else return S(n – 1) + (n*n*n); Solve the recurrence relation in your answer for the previous question. The runtime efficiency of the algorithm is? A. O(n) B. O(n2) C. O(n3) D. None of the aboveUsing Java, give an algorithm for the following problem: given a list of n distinct positive integers, partition the list into two sublists, each of size n/2, such that the difference between the sums of the integers in the sublists is minimized. Determine the time complexityof the algorithm.you may assume that h is a multiple of 2