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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Chapter 9.3, Problem 7E
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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)┤
Prove or disprove. f(n) is O(n).
Please show proof or disproof
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)┤
A. Prove or disprove. f(n) is O(n).
The Legendre Polynomials are a sequence of polynomials with applications in numerical analysis. They can be defined by the
following recurrence relation:
for any natural number n > 1.
Po(x) = 1,
P₁(x) = x,
Pn(x) = − ((2n − 1)x Pn-1(x) — (n − 1) Pn-2(x)),
n
Write a function P(n,x) that returns the value of the nth Legendre polynomial evaluated at the point x.
Hint: It may be helpful to define P(n,x) recursively.
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Introduction to Algorithms
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- Given an n-element array X of integers, Algorithm A executes an O(n) time computation for each even number in X and an O(log-n) time computation for each odd number in X. What are the best case and worst case for running time of algorithm C?arrow_forwardThe 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)…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_forwardGiven a list of n positive integers, show that there must two of these integers whose difference is divisible by n-1arrow_forward
- 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.arrow_forwardImplement a phi function that returns the count of coprime integers of a given positive integer n. Examples phi (1) 1 phi (3) 2 phi (8) → 4arrow_forwardGiven a set S of n planar points, construct an efficient algorithm to determine whether or not there exist three points in S that are collinear. Hint: While there are Θ(n3) triples of members of S, you should be able to construct an algorithm that runs in o(n3) sequential time.arrow_forward
- Given an n-element sequence of integers, an algorithm executes an O(n)-time computation for each even number in the sequence, and an O(logn)-time computation for each odd number in the sequence. What are the best-case and worst-case running times of this algorithm? Why? Show with proper notations.arrow_forwardFind the series which contains n terms of fibanacoi series in linear time complexity. Take the n from the user in Java.arrow_forwardThe algorithm of Euclid computes the greatest common divisor (GCD) of two integer numbers a and b. The following pseudo-code is the original version of this algorithm. Algorithm Euclid(a,b)Require: a, b ≥ 0Ensure: a = GCD(a, b) while b ̸= 0 do t ← b b ← a mod b a ← tend whilereturn a What is the maximum number of iterations of the While loop the algorithm will do?arrow_forward
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