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CS 225 - Pseudorandomness: Problem Set 3, Summaries of Data Structures and Algorithms

Cornell CollegeData Structures and Algorithms

Your problem set solutions must be typed (in e.g. LATEX) and submitted electronically to ... Problem 4.10 (Unbalanced Vertex Expanders and Data Structures).

Typology: Summaries

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CS 225 - Pseudorandomness Prof. Salil Vadhan
Problem Set 3
Harvard SEAS - Fall 2016 Due: Fri. Oct. 14, 2016 (5pm sharp)
Your problem set solutions must be typed (in e.g. L
A
T
E
X) and submitted electronically to
[email protected]. You are allowed 12 late days for the semester, of which at most 5
can be used on any individual problem set. (1 late day = 24 hours exactly). Please name your file
ps3-lastname.*.
The problem sets may require a lot of thought, so be sure to start them early. You are encouraged
to discuss the course material and the homework problems with each other in small groups (2-3
people). Identify your collaborators on your submission. Discussion of homework problems may
include brainstorming and verbally walking through possible solutions, but should not include one
person telling the others how to solve the problem. In addition, each person must write up their
solutions independently, and these write-ups should not be checked against each other or passed
around.
Strive for clarity and conciseness in your solutions, emphasizing the main ideas over low-level
details. Do not despair if you cannot solve all the problems! Difficult problems are included to
stimulate your thinking and for your enjoyment, not to overwork you. *ed problems are extra
credit.
Problem 4.2 (More Combinatorial Consequences of Spectral Expansion)
Let Gbe a graph on Nvertices with spectral expansion γ= 1 −λ. Prove that:
1. The independence number α(G) is at most (λ/(1 + λ))N, where α(G) is defined to be the
size of the largest independent set, i.e. subset Sof vertices s.t. there are no edges with both
endpoints in S.
2. The chromatic number χ(G) is at least (1 + λ)/λ, where χ(G) is defined to be the smallest
number of colors for which the vertices of Gcan be colored s.t. all pairs of adjacent vertices
have different colors.
3. The diameter of Gis O(log1/λ N).
Recall that computing α(G) and χ(G) exactly are NP-complete problems. However, the above
shows that for expanders, nontrivial bounds on these quantities can be computed in polynomial
time.
Problem 4.6 (Error Reduction For Free)
Show that if a problem has a BPP algorithm with constant error probability, then it has a BPP
algorithm with error probability 1/n that uses exactly the same number of random bits.
1
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Download CS 225 - Pseudorandomness: Problem Set 3 and more Summaries Data Structures and Algorithms in PDF only on Docsity!

CS 225 - Pseudorandomness Prof. Salil Vadhan

Problem Set 3

Harvard SEAS - Fall 2016 Due: Fri. Oct. 14, 2016 (5pm sharp)

Your problem set solutions must be typed (in e.g. LATEX) and submitted electronically to [email protected]. You are allowed 12 late days for the semester, of which at most 5 can be used on any individual problem set. (1 late day = 24 hours exactly). Please name your file ps3-lastname.*. The problem sets may require a lot of thought, so be sure to start them early. You are encouraged to discuss the course material and the homework problems with each other in small groups (2- people). Identify your collaborators on your submission. Discussion of homework problems may include brainstorming and verbally walking through possible solutions, but should not include one person telling the others how to solve the problem. In addition, each person must write up their solutions independently, and these write-ups should not be checked against each other or passed around. Strive for clarity and conciseness in your solutions, emphasizing the main ideas over low-level details. Do not despair if you cannot solve all the problems! Difficult problems are included to stimulate your thinking and for your enjoyment, not to overwork you. *ed problems are extra credit.

Problem 4.2 (More Combinatorial Consequences of Spectral Expansion)

Let G be a graph on N vertices with spectral expansion γ = 1 − λ. Prove that:

  1. The independence number α(G) is at most (λ/(1 + λ))N , where α(G) is defined to be the size of the largest independent set, i.e. subset S of vertices s.t. there are no edges with both endpoints in S.
  2. The chromatic number χ(G) is at least (1 + λ)/λ, where χ(G) is defined to be the smallest number of colors for which the vertices of G can be colored s.t. all pairs of adjacent vertices have different colors.
  3. The diameter of G is O(log 1 /λ N ).

Recall that computing α(G) and χ(G) exactly are NP-complete problems. However, the above shows that for expanders, nontrivial bounds on these quantities can be computed in polynomial time.

Problem 4.6 (Error Reduction For Free)

Show that if a problem has a BPP algorithm with constant error probability, then it has a BPP algorithm with error probability 1/n that uses exactly the same number of random bits.

Problem 4.9 (The Replacement Product)

Given a D 1 -regular graph G 1 on N 1 vertices and a D 2 -regular graph G 2 on D 1 vertices, consider the following graph G 1 ©r^ G 2 on vertex set [N 1 ] × [D 1 ]: vertex (u, i) is connected to (v, j) iff (a) u = v and (i, j) is an edge in G 2 , or (b) v is the i’th neighbor of u in G 1 and u is the j’th neighbor of v. That is, we “replace” each vertex v in G 1 with a copy of G 2 , associating each edge incident to v with one vertex of G 2.

  1. Prove that there is a function g such that if G 1 has spectral expansion γ 1 > 0 and G 2 has spectral expansion γ 2 > 0 (and both graphs are undirected), then G 1 ©r^ G 2 has spectral expansion g(γ 1 , γ 2 , D 2 ) > 0. (Hint: Note that (G 1 ©r^ G 2 )^3 has G 1 ©z^ G 2 as a subgraph.)
  2. Show how to convert an explicit construction of constant-degree (spectral) expanders into an explicit construction of degree 3 (spectral) expanders.
  3. Without using Theorem 4.14, prove an analogue of Part 1 for edge expansion. That is, there is a function h such that if G 1 is an (N 1 / 2 , ε 1 ) edge expander and G 2 is a (D 1 / 2 , ε 2 ) edge expander, then G 1 ©r^ G 2 is a (N 1 D 1 / 2 , h(ε 1 , ε 2 , D 2 )) edge expander, where h(ε 1 , ε 2 , D 2 ) > 0 if ε 1 , ε 2 > 0. (Hint: given any set S of vertices of G 1 ©r^ G 2 , partition S into the clouds that are more than “half-full” and those that are not.)
  4. Prove that the functions g(γ 1 , γ 2 , D 2 ) and h(ε 1 , ε 2 , D 2 ) must depend on D 2 , by showing that G 1 ©r^ G 2 cannot be a (N 1 D 1 / 2 , ε) edge expander if ε > 1 /(D 2 + 1) and N 1 ≥ 2.

Problem 4.10 (Unbalanced Vertex Expanders and Data Structures)

Consider a (K, (1 − ε)D) bipartite vertex expander G with N left vertices, M right vertices, and left degree D.

  1. For a set S of left vertices, a y ∈ N (S) is called a unique neighbor of S if y is incident to exactly one edge from S. Prove that every left-set S of size at most K has at least (1− 2 ε)D|S| unique neighbors.
  2. For a set S of size at most K/2, prove that at most |S|/2 vertices outside S have at least δD neighbors in N (S), for δ = O(ε).

Now we’ll see a beautiful application of such expanders to data structures. Suppose we want to store a small subset S of a large universe [N ] such that we can test membership in S by probing just 1 bit of our data structure. A trivial way to achieve this is to store the characteristic vector of S, but this requires N bits of storage. The hashing-based data structures mentioned in Section 3.5. only require storing O(|S|) words, each of O(log N ) bits, but testing membership requires reading an entire word (rather than just one bit.) Our data structure will consist of M bits, which we think of as a { 0 , 1 }-assignment to the right vertices of our expander. This assignment will have the following property.

Property Π: For all left vertices x, all but a δ = O(ε) fraction of the neighbors of x are assigned the value χS (x) (where χS (x) = 1 iff x ∈ S).