Description
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Graded Problems
1. What is the worst-case runtime performance of the procedure below?
c = 0
i = n
while i > 1 do
for j = 1 to i do
c = c + 1
end for
i = oor(i=2)
end while
return c
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Arrange these functions under the O notation using only = (equivalent) or (strict subset of):
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2log n
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23n
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nn log n
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log n
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n log n2
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nn2
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log(log(nn))
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E.g. for the function n, n + 1, n2, the answer should be
O(n + 1) = O(n) O(n2):
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Given functions f1; f2; g1; g2 such that f1(n) = O(g1(n)) and f2(n) = O(g2(n)). For each of the following statements, decide whether you think it is true or false and give a proof or counterexample.
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f1(n) f2(n) = O (g1(n) g2(n))
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f1(n) + f2(n) = O (max (g1(n); g2(n)))
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f1(n)2 = O g1(n)2
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log2 f1(n) = O (log2 g1(n))
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Given an undirected graph G with n nodes and m edges, design an O(m+ n) algorithm to detect whether G contains a cycle. Your algorithm should output a cycle if G contains one.
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Practice Problems
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Solve Kleinberg and Tardos, Chapter 2, Exercise 6.
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Solve Kleinberg and Tardos, Chapter 3, Exercise 6.
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