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

a) Prove that

∫ +∞

−∞

1√2πe−z

2/2dz = 1, i.e. φ(z) =1√2πe−z

2/2 is indeed a probability density

function for standard Gaussian.

b) Let X ∼ N (µ1, σ21), Y ∼ N (µ2, σ

22) 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, a

2σ21 + b2σ22).

c) Let Φ(z) =

∫ z

−∞φ(z)dz be the cumulative distribution function of a standard Gaussian, where

φ(z) =1√2πe−z

2/2 is the probability density function. Prove that for z > 1, Φ(−z) =

Θ(φ(z)/z).

Problem 2

a) Let G be the complete graph Kn on n vertices. Determine the eigenvalues λ0, λ1, . . . , λn−1 ofG’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, 1d and an edge connecting twostrings if they differ on exactly one coordinate. For each set S ⊆ [d], define φS : V → R byφS(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

ΦG1 = O(λ1(LG1)), ΦG2 = Ω(√λ1(LG2)),

andΦG1 → 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 inboth directions.

Problem 3

a) Say that a pseudorandom generator G : F`2 → Fn

2 is ε-almost k-wise independent if

1

2− ε

2≤ P

s∼F`2

[w ·G(s) = 1] ≤ 1

2+ε

2

for all nonzero w ∈ Fn2 with Hamming weight at most k. Explain how to get such a generator,

computable in poly(n/ε)-time, with ` = O(log k+ log log(n) + log(1/ε)). (Note the extremelysmall dependence on n!.)

b) A set U ⊆ 0, 1n is called k-universal (for 1 ≤ k ≤ n) if for every 1 ≤ i1 < i2 < · · · < ik ≤ nand every pattern b ∈ 0, 1k, there is at least one string x ∈ U with xi1xi2 . . . 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 hatwith 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 hatbut his own. Then every student is asked to guess the color of his/her own hat, while no informationexchange (including the guesses) between the students is allowed after the start of the game. Thegroup 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 winningprobability.

Problem 5

The n students are playing a game that is slightly different from the previous problem. Now let usassume n = 2k−1 where k ∈ Z+. Similarly as in the previous problem, when the game starts, eachstudent has a hat with a random color put on his/her head. Each hat is either red or green withequal probability, while the colors of the n hats are independently assigned. Each student can seeeveryone 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/shecan choose to give up. Again, no information exchange (including the guesses and whether to giveup) between the students is allowed after the start of the game. The students collaboratively winthe 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 winningprobability. (Hint: try n = 2k − 1 for some small constants k first and then try to generalize yourfinding.)