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 24.3, Problem 9E
Program Plan Intro
To modify the Dijkstra’s
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Given is a strictly increasing function, f(x). Strictly increasing meaning: f(x)< f(x+1). (Refer to the example graph of functions for a visualization.)
Now, define an algorithm that finds the smallest positive integer, n, at which the function, f(n), becomes positive.
The things left to do is to:
Describe the algorithm you came up with and make it O(log n).
Analyze the Print Shortest Path algorithm (Algorithm 3.5) and show that it
has a linear-time complexity.
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.
Chapter 24 Solutions
Introduction to Algorithms
Ch. 24.1 - Prob. 1ECh. 24.1 - Prob. 2ECh. 24.1 - Prob. 3ECh. 24.1 - Prob. 4ECh. 24.1 - Prob. 5ECh. 24.1 - Prob. 6ECh. 24.2 - Prob. 1ECh. 24.2 - Prob. 2ECh. 24.2 - Prob. 3ECh. 24.2 - Prob. 4E
Ch. 24.3 - Prob. 1ECh. 24.3 - Prob. 2ECh. 24.3 - Prob. 3ECh. 24.3 - Prob. 4ECh. 24.3 - Prob. 5ECh. 24.3 - Prob. 6ECh. 24.3 - Prob. 7ECh. 24.3 - Prob. 8ECh. 24.3 - Prob. 9ECh. 24.3 - Prob. 10ECh. 24.4 - Prob. 1ECh. 24.4 - Prob. 2ECh. 24.4 - Prob. 3ECh. 24.4 - Prob. 4ECh. 24.4 - Prob. 5ECh. 24.4 - Prob. 6ECh. 24.4 - Prob. 7ECh. 24.4 - Prob. 8ECh. 24.4 - Prob. 9ECh. 24.4 - Prob. 10ECh. 24.4 - Prob. 11ECh. 24.4 - Prob. 12ECh. 24.5 - Prob. 1ECh. 24.5 - Prob. 2ECh. 24.5 - Prob. 3ECh. 24.5 - Prob. 4ECh. 24.5 - Prob. 5ECh. 24.5 - Prob. 6ECh. 24.5 - Prob. 7ECh. 24.5 - Prob. 8ECh. 24 - Prob. 1PCh. 24 - Prob. 2PCh. 24 - Prob. 3PCh. 24 - Prob. 4PCh. 24 - Prob. 5PCh. 24 - Prob. 6P
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- Given 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_forwardLet P2(x) be the least squares interpolating polynomial for f(x) := sin(πx) on the interval [0,1] (with weight function w(x) = 1). Determine nodes (x0,x1,x2) for the second-order Lagrange interpolating polynomial Pˆ2(x) so that P2 = Pˆ2. You are welcome to proceed theoretically or numerically using Python.arrow_forwardGiven a linked list L storing n integers, present an algorithm (either in words or in a pseudocode) that decides whether L contains any 0 or not. The output of your algorithm should be either Yes or No. What is the running time of your algorithm in the worst-case, using O notation?arrow_forward
- with n=6 and A=(3,5,4,1,3,2). Draw the corresponding walkthrough as shown in P.158arrow_forward2. 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_forwardConsider 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 disproofarrow_forward
- I needed the algorithm for G whose length is even. You can read it in the last line of the question. I don't know where the divisible by 5 part came from. Please solve the question again.arrow_forwardConsider 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).arrow_forwardAlgorithm X has a growth rate that is proportional to n ∗ n ∗ n . What is the function that represents the growth rate of algorithm X? What is the order of algorithm X?arrow_forward
- Suppose that, in a divide-and-conquer algorithm, we always divide an instance of size n of a problem into 10 subinstances of size , and the dividing and combining steps take a time in O(n²). Write a recurrence equation for the running time T(n), and solve the equation for T(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_forwardApply algorithm 4.3.3 in P.158 with n=6 and A=(3,5,4,1,3,2). Draw the corresponding walkthrough as shown in P.158.arrow_forward
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