MATH 1P67: Mathematics for Computer Science

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1. Let f(n) = 2n^3 −n^2 + 10n−7. Prove that f(n) is O(n^3). Show all work.

2. Prove by contradiction that 6^n is not O(2^2n).

3. Suppose that f(n) is O(g(n)). Does it follow that 2^f(n) is O(2^g(n))? Justify your answer.

4. Consider the following pseudocode for an algorithm called “Algorithm,” which reads

procedure Algorithm(a1, a2, …, an: integers)

x := a1

for i := 2 to n

x := x + ai

return x Suppose a list a1,a2,…,an of integers is given as an input.

a) Describe what Algorithm(a1, a2, …, an) returns as output.

b) How many times would the expression x := x + ai be executed?

c) Give a Θ estimate for the runtime of this algorithm.

5. [3 marks] Consider the following pseudocode for an algorithm called “Algorithm,” which reads

procedure Algorithm(a1, a2, …, an: integers)

set L as the empty list

sum := 0

i := 1

while i ≤ n

if ai > sum then append ai to L end if

sum := sum + ai

i := i + 1

return L Suppose a list a1,a2,…,an of integers is given as an input.

a) Describe what Algorithm(a1, a2, …, an) returns as output.

b) How many times would the expression i ≤ n be executed?

c) Give a Θ estimate for the runtime of this algorithm.


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