cs161___Homework_1.pdf - CS 161 (Stanford, Winter 2022)...

CS 161 (Stanford, Winter 2022) Homework 1
Style guide and expectations: Please see the “Homework” part of the “Resources” sec-
tion on the webpage for guidance on what we are looking for in homework solutions. We
will grade according to these standards. You should cite all sources you used outside of
the course material.
What we expect: Make sure to look at the “ We are expecting ” blocks below each
problem to see what we will be grading for in each problem!
Exercises. The following questions are exercises. We suggest you do these on your own.
As with any homework question, though, you may ask the course staff for help.
1 Exercise: Course policies
(1 pt.) Have you thoroughly read the course policies on the webpage?
Yes
2 Exercise: Big-Oh Basics
(2 pt.) Which of the following functions are O ( n
2
) ?
No explanation is required, but you might want to prove your answer to yourself to convince
yourself that you are correct.
1

f
5
( n ) = sin( n ) + 5
f
7
( n ) = (100
3
)
4
3 Exercise: Big-Oh Definitions
Terry the Terrific Tiger wants to prove, from the definition of big-Oh, that f ( n ) = O ( g ( n )) ,
where f ( n ) = 6 n and g ( n ) = n 2 − 3 n .
3.1 “ c = 1 ”
(2 pt.) In the definition of big-Oh, Terry really wants to take c = 1 . Give a proof that
f ( n ) = O ( g ( n )) in which " c " is chosen to be 1.
3.2 “ n
0
= 4 ”
(2 pt.) Terry has changed their mind. Now, they don’t care what c is, but would like to take
n
0
= 4 . Give a different proof that f ( n ) = O ( g ( n )) in which " n
0
" is chosen to be 4.
3.1:
We want to prove that f ( n ) = O ( g ( n )) , meaning that we have to prove, from the definition
of big-Oh, that ∃ c, n
0
> 0 s.t ∀ n ≥ n
0
, f ( n ) ≤ c · g ( n ) where c is chosen to be 1 .
Let n
0
= 9 . We will prove that f ( n ) ≤ c · g ( n ) holds for all n ≥ n
0
, where c = 1 . We can
simplify the inequality:
f ( n ) ≤ 1 · g ( n ) (1)
6 n ≤ n
2
− 3 n (2)
0 ≤ n
2
− 9 n (3)
Solving gives us n ≥ 9 , so we can see that for all n ≥ 9 , f ( n ) ≤ 1 · g ( n ) holds. Therefore,
we have proven that f ( n ) ≤ 1 · g ( n ) holds for all n ≥ n
0
where n
0
= 9 , which proves that
f ( n ) = O ( g ( n )) . ■
3.2:
We want to prove that f ( n ) = O ( g ( n )) , meaning that we have to prove, from the definition
of big-Oh, that ∃ c, n
0
> 0 s.t ∀ n ≥ n
0
, f ( n ) ≤ c · g ( n ) where n
0
is chosen to be 4 .