For the primes p = 2, q = 3 and r = 7 and for n = 3, %3D provide a simulation of the algorithm to build an ordered list of integers of the form p°q°r© for non negative a, b and c with 1 < a + b + c < 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.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.
- AvgCompares(), a recursive function that calculates the average number of comparisons needed by a random search hit in a given BST (the internal path length of the tree divided by its size plus one), should be added to the BST. Create two implementations: a recursive method that adds a field to each node in the tree and takes linear space and constant time every query, and a method similar to size() that takes linear space and constant time per query.In Python, write a recursive implementation of Fibonacci without memoization. Include a timer to measure how long it takes. The sequence is defined by this recurrence: Fo = 0 F = 1 Fn = Fn-1+ Fn-2 The input should ask the user for the nth value in the sequence they want. Improvement: have your solution print all the values it computes along the way to the nth value in the sequence Bonus Question Improve the implementation above by using a memo dictionary (lecture notes slide 13)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)…
- Write in python programming language: 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. Let S be a sequence with ten distinct integers. Prove by…Write a program in JAVA Programming langaugue. Given an array of integers, in which each elements repeats twice except one. Your task is to find that element in O(n) Time complexity and O(1) Space Complexity.Given an unsorted array, A, of integers and an integer k, write a recursivejava code for rearranging the elements in A so that all elements less than or equal to k come before any elements larger than k. What is the running time of your algorithm on an array of n values.
- Please write a python program with explanation data structure& algorithm and time &space complexity Given a list of [personName, accessTime], for each person, return the earliest 1-hour interval in which he has swipe badges 3 or more times, e.g.[A, 0], [A, 5], [A, 59], [A, 60] ,[A, 61]return [A, 0], [A, 5], [A, 59], [A, 60] not [A, 5], [A, 59], [A, 60] ,[A, 61]Give a recursive algorithm for computing the greatest common divisor of two nonnegative integers a and b with a < b.Write Algorithm which computes all the orbits of a group, is a simple extension of Algorithm 1.Input" a set S = {s 1, s2 ..... Sm} of generators for a group G acting on f2;Output : the orbits orbit[l], orbit[2] .... of G on f~;