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Homework 1 Solution
Assigned Problems
Exercises (Do not hand in) Chapter 1, Problems 1-3, 5. Chapter 2, Problems 3-5, 7.
Following are the problems to be handed in, 25 points each.
(Resident Matching, 2-page limit { your solutions should t on two sides of 1 page).
The situation is the following. There were m teams at Google, each with a certain number of available positions for hiring interns. There were n students who want internships at Google this summer, each interested in joining one of the teams. Each team had a ranking of the students in order of preference, and each student had a ranking of the teams in order of preference. We will assume that there were more students who want an internship at Google than there were slots available in the m teams.
The interest, naturally, was in nding a way of assigning each student to at most one team at Google, in such a way that all available positions in all teams were lled. (Since we are assuming a surplus of potential interns, there would be some students who do not get assigned to any team.) We say that an assignment of students to Google teams is stable if neither of the following situations arises.
The rst type of instability that can occur is that there is a team t, and there are students s and s0, so that
{ s is matched with t, and
{ s0 is assigned to no team, and { t favors s0 over s.
The second type of instability that can occur is that there are teams t and t0 and students s and s0 so that
{ s is matched with t, and { s0 is matched with t0, and
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{ t favors s0 over s, and { s0 favors t over t0.
So we basically have the Stable Matching Problem, except that, one, teams generally want more than one intern, and, two, there is a surplus of students who want internships at Google. Show that there is always a stable assignment of students to Google teams, and give an algorithm to nd one.
Please give a clear description of your algorithm. Don’t forget to prove its correctness and analyze its time and space complexity.
(Time complexity, 2-page limit { your solutions should t on two sides of 1 page). Part (a) has 15 points and part (b) has 10 points. The top of your solution for part (a) should have the functions in order by their letter, with no spaces, commas, etc. between them. (For example, abc). If you do not include this you will automatically lose 75% of the credit. (Functions that are equivalent should be in alphabetical order)
Rank the following functions by increasing order of growth, that is, nd an arrangement g1; ::: of the functions satisfying g1(n) = O(g2(n)); g2(n) = O(g3(n)); :::. Break the functions into equivalence classes so that f and g are in the same class if and only if f(n) = (g(n)). Note that log( ) is the base 2 logarithm, logb( ) is the base b logarithm, ln( ) is the natural logarithm, and logc(n) denotes (log(n))c (for example, log2(n) = log(n) log(n)).
a: ln(ln n)
f: n2
n
X
k: 3i
i=1
p: 2log4 n
n
(i + 1)
b:
n
1
i
14 log3 n
Xi
n log n
c:
d:
2
g:
i=1
2
h:
log(n!)
=5
i:
3n
X
l: 2log2(n)
m:
2log n
n:
n!
q:
p
r:
log(n2)
s:
4log n
n
e: log2(n)
j: nlog 7
o: n
t: (54 )n
For each of the following statements, decide whether it is always true, never true, or some-times true for asymptotically nonnegative functions f and g. If it is always true or never true, give a proof. If it is sometimes true, give one example for which it is true, and one for which it is false.
f(n) + g(n) = (max(f(n); g(n)))
f(n) = !(g(n)) and f(n) = O(g(n))
Either f(n) = O(g(n)) or f(n) = (g(n)) or both.
(Induction, 2-page limit { your solutions should t on two sides of 1 page). Part (a) has 10 points and part (b) has 15 points.)
(a) (Uniform shu ing) Let A[1; ; n] be an array of integers. A uniform shu e of A is a set of n random elements from A (without replacement), such that the probability of selecting any such set is the same. Consider the following algorithm to generate a uniform random shu e:
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UniformShuffle(A)
1 for i n downto 1
do j random integer such that 1 j i
exchange A[i] and A[j] 4 return A
Prove that the algorithm indeed generates a uniform random shu e of A. What is the running time of the algorithm, given that generating random integer takes time O(1)? Hint: Start by thinking of what a uniform shu e means in terms of probability.
Point out the error in the following proof by induction.
Claim: Given any set of b buses, all buses lead to the same destination.
Proof: We proceed by induction on the number of buses, b.
Base case: If b = 1, then there is only one bus in the set, and so all buses in the set lead to the same destination.
Induction step: For k 1, we assume that the claim holds for b = k and prove that it is true for b = k + 1. Take any set B of b + 1 buses. To show that all buses lead to the same destination, we take the following approach. Remove one bus from this set to obtain the set B1 with just b buses. By the induction hypothesis, all the buses in B1 lead to the same destination. Now go back to the original set and remove a di erent bus to obtain a the set B2. By the same argument, all the buses in B2 lead to the same destination. Therefore all the buses in B = B1 [ B2 must lead to the same destination, and the proof is complete.
(Divide and Conquer, 2-page limit { your solutions should t on two sides of 1 page).
After dating for several years, Jack and Anthony have nally decided to move in together. As part of this process, each of them wants to bring his n alphabetically sorted books over to the new place. Due to some weird reason, they want to nd out who owns the median book of the joint book collection, which has 2n books. In this joint book collection, the median would be the n-th book among the union of the 2n alphabetically sorted books.
Because their original book collections are already sorted, they manage to nd out who owns the median in (log n). They did not have to reorder the joint book collection, but rather it was enough for them to just query individual values from their original book collections. What algorithm did they use? Prove that this algorithm is correct. Find the recurrence relation and show that it resolves to (log n).
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