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 35.4, Problem 1E
Program Plan Intro
Toprove that both variable and its negation yields a randomized 8/7 − approximation
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If five integers are chosen from the set {1, 2, 3, 4, 5, 6, 7, 8}, must there be at least two integers with the property that the larger minus the smaller is 2? Write an answer that would convince a good but skeptical fellow student who has learned the statement of the pigeonhole principle but not seen an application like this one. Describe the pigeons, the pigeonholes, and how the pigeons get to the pigeonholes.
Write an expression for the decomposition of selection bias in each of the following cases.
Which is the "worst"?
(a)
(b)
(c)
Sx = 0
X = [0, 1], Six = [0, 0.75], Sox= [0.25, 1], f(x) = 1
X = 2¹
Let pn(x) be the probability of selling the house to the highest bidder when there are n people, and you adopt the Look-Then-Leap algorithm by rejecting the first x people. For all positive integers x and n with x < n, the probability is equal to p(n(x))= x/n (1/x + 1/(x+1) + 1/(x+2) + … + 1/(n-1))
If n = 100, use the formula above to determine the integer x that maximizes the probability n = 100 that p100(x). For this optimal value of x, calculate the probability p100(x).
Briefly discuss the significance of this result, explaining why the Optimal Stopping algorithm produces a result whose probability is far more than 1/n = 1/100 = 1%.
Chapter 35 Solutions
Introduction to Algorithms
Ch. 35.1 - Prob. 1ECh. 35.1 - Prob. 2ECh. 35.1 - Prob. 3ECh. 35.1 - Prob. 4ECh. 35.1 - Prob. 5ECh. 35.2 - Prob. 1ECh. 35.2 - Prob. 2ECh. 35.2 - Prob. 3ECh. 35.2 - Prob. 4ECh. 35.2 - Prob. 5E
Ch. 35.3 - Prob. 1ECh. 35.3 - Prob. 2ECh. 35.3 - Prob. 3ECh. 35.3 - Prob. 4ECh. 35.3 - Prob. 5ECh. 35.4 - Prob. 1ECh. 35.4 - Prob. 2ECh. 35.4 - Prob. 3ECh. 35.4 - Prob. 4ECh. 35.5 - Prob. 1ECh. 35.5 - Prob. 2ECh. 35.5 - Prob. 3ECh. 35.5 - Prob. 4ECh. 35.5 - Prob. 5ECh. 35 - Prob. 1PCh. 35 - Prob. 2PCh. 35 - Prob. 3PCh. 35 - Prob. 4PCh. 35 - Prob. 5PCh. 35 - Prob. 6PCh. 35 - Prob. 7P
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