CSCI 570 - HW 2
1 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 = floor(i/2)
end while
return c
2. Arrange these functions under the O notation using only = (equivalent)
or ⊂ (strict subset of):
(a) 2log n
(b) 23n
(c) n
n log n
(d) log n
(e) n log
n
2
(f) n
n
2
(g) log(log(n
n))
E.g. for the function n, n + 1, n
2
, the answer should be
O(n + 1) = O(n) ⊂ O(n
2
).
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.
(a) f1(n) · f2(n) = O (g1(n) · g2(n))
(b) f1(n) + f2(n) = O (max (g1(n), g2(n)))
(c) f1(n)
2 = O
g1(n)
2
(d) log2 f1(n) = O (log2
g1(n))
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.
