Submitting:
 All homeworks should be turned in electronically to myCourses.
 Paper submissions may be scanned in if legible.
 If you take a photo of your paper solution, please take the photo outside or close to a bright window, for legibility.
Due dates:
 Homeworks are due by 10pm on the due date (Thu).
 No late submissions are accepted.
 If there are extenuating circumstances, contact the instructor as soon as possible to make arrangements.
References:
 If you find a resource helpful (from internet search, blog post, Stack Overflow, online courses, our lecture notes, etc), be sure to cite it. Citation format doesn’t matter, but make it easy for the TA to find the source: e.g., a clickable link.
 Code / pseudocode should be your own work. If you find code online, understand how it works, then go write up your solution on your own.
 The basic machine model was discussed in Lecture 1.
Don’t use library functions which obviate the bulk of
a question. E.g., if you’re asked to implement a sort, don’t use
Python’s
sorted()
.
HW1 (20pts) (solutions)

(4pts) Demonstrate insertion sort on the following array of integers. Show as much work as you can.
[ 35, 50, 44, 61, 17, 75, 23, 9 ]

(4pts) Prove from the definition of Theta: ` n^3 (2 + sin(n pi / 8)) + n^2 in Theta( n^3 ) `
 Consider the (pseudo)code given at the end of this question.
 a. (3pts) Derive and simplify its asymptotic complexity T(n), in Θ notation, as a function of n.
 b. (1pt) Is this function polynomial time, exponential time, or other?

c. (1pts) Rewrite this function to run in O(1) time. You can use pseudocode, Python, whatever you like.
def fib(n): if (n <= 0): return 0 if (n == 1): return 1 return fib(n2) + fib(n1)
 Consider a group of n people.

a. (2pts) If each person needs to shake hands with every other person in the group exactly once, how many hand shakes will there be total? Compute exactly, as a function of n.

b. (1pt) Express (a) in Θ asymptotic notation. Is this linear, quadratic, exponential, or other?

c. (3pt) In 2011 Facebook users had on average 190 friends. For each user in a group of n users, say you want to find that user’s oldest Facebook friend; i.e., the person with which that user has been Facebook friends for the longest time.
The input would be a list of n user IDs. The output should be a list of n user IDs, where the ith user in the output is the oldest friend for the ith user in the input.
Discuss how you might design such a function, and what its asymptotic complexity would be, as a function of n. Is it linear, quadratic, exponential, or other?

d. (1pt) What’s different about the situations in parts (b) and (c) that explains their different asymptotic complexities?