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 14.3, Problem 3E
Program Plan Intro
To describe an efficient
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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.
Given a set B of n planar points, construct an efficient algorithm to determine whether or notthere exist three points in S that are collinear. Hint: construct an algorithm that runs in o(n3) sequential time.
There are n people who want to carpool during m days. On day i, some subset si ofpeople want to carpool, and the driver di must be selected from si . Each person j hasa limited number of days fj they are willing to drive. Give an algorithm to find a driverassignment di ∈ si each day i such that no person j has to drive more than their limit fj. (The algorithm should output “no” if there is no such assignment.) Hint: Use networkflow.For example, for the following input with n = 3 and m = 3, the algorithm could assignTom to Day 1 and Day 2, and Mark to Day 3.
Person
Day 1
Day 2
Day 3
Limit
1 (Tom)
x
x
x
2
2 (Mark)
x
x
1
3 (Fred)
x
x
0
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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 disproofarrow_forwardThere are n people who want to carpool during m days. On day i, some subset ???? of people want to carpool, and the driver di must be selected from si . Each person j has a limited number of days fj they are willing to drive. Give an algorithm to find a driver assignment di ∈ si each day i such that no person j has to drive more than their limit fj. (The algorithm should output “no” if there is no such assignment.) Hint: Use network flow. For example, for the following input with n = 3 and m = 3, the algorithm could assign Tom to Day 1 and Day 2, and Mark to Day 3. Person Day 1 Day 2 Day 3 Driving Limit 1 (Tom) x x x 2 2 (Mark) x x 1 3 (Fred) x x 0arrow_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 We want to estimate its worst case running time using the big-Oh notation. Answer the following questions: a. Let x be a integer stored on n bits. How many bits will you need to store x/2? b. We note that if a ≥ b, then a mod b < a/2. Assume the values of the input integers a and b are encoded on n bits. How many bits will be used to store the values of a and b at the next iteration of the While loop? c. Deduce from this observation, the maximal number iterations of the While loop the algorithm will do.arrow_forward
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