Description

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 = floor(i/2)
end while return c
2. Arrange these functions under the O notation using only = (equivalent) or ⊂ (strict subset of):
(a) 2logn
(b) 23n
(c) nnlogn

(g) log(log(nn))
E.g. for the function n, n + 1, n2, the answer should be
O(n + 1) = O(n) ⊂ O(n2).
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3. 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.

4. 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.
2 Practice Problems
1. Solve Kleinberg and Tardos, Chapter 2, Exercise 6.
2. Solve Kleinberg and Tardos, Chapter 3, Exercise 6.
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