Seminar in Theoretical Computer Science
MIT 18.434 Spring 2016
Problem Set 1∗
Due: March 20, 11.59pm. Email the pdf to yuanzhou@mit.edu
There are 5 problems in this problem set. Each problem is worth 20 points.
Problem 1
Z
+∞
1
1
2
2
√ e−z /2 dz = 1, i.e. φ(z) = √ e−z /2 is indeed a probability density
2π
2π
−∞
function for standard Gaussian.
a) Prove that
b) Let X ∼ N (µ1 , σ12 ), Y ∼ N (µ2 , σ22 ) be two independent Gaussians. Let Z = aX + bY where
a, b ∈ R are two constants. Use Central Limit Theorem to prove that Z is also a Gaussian,
and Z ∼ N (aµ1 + bµ2 , a2 σ12 + b2 σ22 ).
Z z
c) Let Φ(z) =
φ(z)dz be the cumulative distribution function of a standard Gaussian, where
−∞
1
2
φ(z) = √ e−z /2 is the probability density function. Prove that for z > 1, Φ(−z) =
2π
Θ(φ(z)/z).
Problem 2
a) Let G be the complete graph Kn on n vertices. Determine the eigenvalues λ0 , λ1 , . . . , λn−1 of
G’s normalized Laplacian.
b) Same question for the complete graph with self-loops (which is an n-regular graph).
c) Let G be the “hypercube graph” with vertex set V = {−1, 1}d and an edge connecting two
strings ifQ
they differ on exactly one coordinate. For each set S ⊆ [d], define φS : V → R by
φS (u) = i∈S ui . Show that each φS is an eigenfunction for G’s Laplacian L, and determine
all eigenvalues of L.
d) Show two connected regular n-vertex graphs G1 and G2 , and prove that
p
ΦG1 = O(λ1 (LG1 )),
ΦG2 = Ω( λ1 (LG2 )),
and
ΦG1 → 0, ΦG2 → 0, as n → +∞,
∗
Part of Problem 2 and Problem 3 are borrowed from Ryan O’Donnell’s course http://www.cs.cmu.edu/
~odonnell/toolkit13/. Thanks Ryan for the permission!
Seminar in Theoretical Computer Science
MIT 18.434 Spring 2016
where LG1 is the normalized Laplacian of G1 and LG2 is the normalized Laplacian of G2 .
These two instances will demonstrate the asymptotic tightness of Cheeger’s inequality in
both directions.
Problem 3
a) Say that a pseudorandom generator G : F`2 → Fn2 is -almost k-wise independent if
1 1 − ≤ P [w · G(s) = 1] ≤ +
2 2 s∼F`2
2 2
for all nonzero w ∈ Fn2 with Hamming weight at most k. Explain how to get such a generator,
computable in poly(n/)-time, with ` = O(log k + log log(n) + log(1/)). (Note the extremely
small dependence on n!.)
b) A set U ⊆ {0, 1}n is called k-universal (for 1 ≤ k ≤ n) if for every 1 ≤ i1 < i2 < · · · < ik ≤ n
and every pattern b ∈ {0, 1}k , there is at least one string x ∈ U with xi1 xi2 . . . xik = b.
Show that for any k ≤ O(log n) we can deterministically construct a k-universal set U with
|U | = poly(2k , log n) in time poly(n).
Problem 4
There are n students playing the following game. When the game starts, each student has a hat
with a random color put on his/her head. Each hat is either red or green with equal probability,
while the colors of the n hats are independently assigned. Each student can see everyone else’s hat
but his own. Then every student is asked to guess the color of his/her own hat, while no information
exchange (including the guesses) between the students is allowed after the start of the game. The
group of n students collaboratively win the game only when all of them made the correct guess.
Please design and prove a strategy for the students to achieve the maximum possible winning
probability.
Problem 5
The n students are playing a game that is slightly different from the previous problem. Now let us
assume n = 2k − 1 where k ∈ Z+ . Similarly as in the previous problem, when the game starts, each
student has a hat with a random color put on his/her head. Each hat is either red or green with
equal probability, while the colors of the n hats are independently assigned. Each student can see
everyone else’s hat but his own. Then every student is asked to guess the color of his/er own hat.
In this game, however, each student has one more option other than guessing the color – he/she
can choose to give up. Again, no information exchange (including the guesses and whether to give
up) between the students is allowed after the start of the game. The students collaboratively win
the game when there is at least one student made the guess and no student made the wrong guess.
Please design and prove a strategy for the students to achieve the maximum possible winning
probability. (Hint: try n = 2k − 1 for some small constants k first and then try to generalize your
finding.)