Introduction to Algorithms
3rd Edition
ISBN: 9780262033848
Author: Thomas H. Cormen, Ronald L. Rivest, Charles E. Leiserson, Clifford Stein
Publisher: MIT Press
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Consider a function f: N → N that represents the amount of work done by some algorithm as follow:
f(n) = {(1 if n is oddn if n is even)┤
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Question: Design an algorithm for this problem with worst-case runtime of O(n).
Write an algorithm for a divide-and-conquer algorithm for finding values of both
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- 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.arrow_forwardA 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.arrow_forwardimplement Running time algorithm Careful(n) pre-cond: n is an integer. post-cond: Q(n) “Hi”s are printed for some odd function Qarrow_forward
- Given a sorted array A of n distinct integers, some of which may be negative, give an O(log(n)) algorithm to find an index i such that 1 ≤ i ≤ n and A[i] = i provided such an index exists. If there are many such indices, the algorithm can return any one of them.arrow_forwardIn 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.arrow_forwardShow Let f(.) be a computable, strictly monotonic function, that is, f(n+ 1) > f(n) for all n. Show B = {f(n) | n ∈ N} is recursive.arrow_forward
- Given a list of n positive integers, show that there must two of these integers whose difference is divisible by n-1arrow_forwardWrite an algorithm that inputs a number n followed by an array of n numbers and outputs the number of values of i, with i between 1 and n-1 such that ai < ai +1. {i| i is a natural number and 1 <= i <= n-1 and ai < ai +1} *The output is the cardinality of the set Can be written in pseudocodearrow_forwardGiven two sorted arrays A and B, design a linear (O(IA|+|B|)) time algorithm for computing the set C containing elements that are in A or B, but not in both. That is, C = (AU B) \ (AN B). You can assume that elements in A have different values and elements in B also have different values. Please state the steps of your algorithm clearly, prove that it is correct, and analyze its running time. Pls give the code in C++, or very clear steps of the algorithmarrow_forward
- 4 points Given an n-element array X of integers, Algorithm A executes an O(n3.4)-time computation for each even positive number in X, an Oin2.3-time computation for each odd positive number in X, and an O(n2.5)-time computation for each negative number in X. What are the best-case and worst-case running times of Algorithm A? Justify your answer. For the toolbar, press ALT+F10 (PC) or ALT+FN+F10 (Mac) BIUS Paragraph V Arial 10pt EVE 2 IXO QFS3Earrow_forwardThe reverse of a string x denoted by rev(x) and is defined by the following recursiverule: •rev(ε) = ε•rev(xa) = a.rev(x) Prove that if A is a regular set then so is the following set:rev(A) = {rev(x) : x ∈A}arrow_forwardComputer Science Let A be an array of n numbers. Design an O(n)-time algorithm to rearrange elements of A in a way that all negative numbers precede all positive numbers. Explain your algorithm in detail and show why it runs in O(n).arrow_forward
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