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 3.1, Problem 6E
Program Plan Intro
To prove that the running time complexity of an
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2. For a problem we have come up with three algorithms: A, B, and C. Running time
of Algorithm A is O(n¹000), Algorithm B runs in 0(2¹) and Algorithm C runs in
O(n!). How do these algorithms compare in terms of speed, for large input?
Explain why.
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?
Find the Worst case time Complexity of the following recursive functions
T(n) = T(n-1)+n -1, T(1) = 0
Chapter 3 Solutions
Introduction to Algorithms
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- For a given problem with inputs of size n, algorithms running time,one of the algorithm is o(n), one o(nlogn) and one o(n2). Some measured running times of these algorithm are given below: Identify which algorithm is which and explain the observed running Times which algorithm would you select for different value of n?arrow_forwardGiven 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_forward1. If I prove that an algorithm takes Θ(n2) worst-case time, is it possible that it takes O(n) on some inputs? 2. If I prove that an algorithm takes Θ(n2) worst-case time, is it possible that it takes O(n) on all inputs?arrow_forward
- Find the Worst case time Complexity of the following recursive functions T(n)=T(n/3)+2T(n/3)+narrow_forwardGiven an algorithm with runtime: T(n) = log(2") + 3n +1 Formally prove whether T(n) is O(n)arrow_forwardWhich of the following best describes the useful criterion for comparing the efficiency of algorithms? None of the above Memory Time Both Time and Memoryarrow_forward
- 2. Algorithm A has a running time described by the recurrence T(n) = 7T(n/2) + n². A competing algorithm B has a running time described by the recurrence T(n) = aT (n/4) + n². What is the largest integer value for a such that B is asymptotically faster than A? Explain your answer.arrow_forwardFind time complexity T(n).arrow_forwardShow that f (n) is O(g(n)) if and only if g(n) is Ω( f (n)).arrow_forward
- Let the sequence (n) be recursively defined by x1 = √2 and Xn+1 = √√2+xn, n≥ 1. Show that (n) converges and evaluate its limit.arrow_forwardAl and Bob are arguing about their algorithms. Al claims his O(nlogn)-time method is always faster than Bob's O(n²)-time method. To settle the issue, they perform a set of experiments. To Al's dismay, they find that if n 100 is the O(nlogn)-time one better. Explain how this is possible.arrow_forwardDetermine φ (m), for m=12,15, 26, according to the definition: Check for each positive integer n smaller m whether gcd(n,m) = 1. (You do not have to apply Euclid’s algorithm.)arrow_forward
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