Foundations of Data Science

2. The random variable of interest, y = ∑ p
I p , is the number of prime divisors of x
picked at random. Show that the variance of y is O(ln ln n). For this, assume the
known
∑
result that the number of primes p between 2 and n is O(n/ ln n) and that
1
= ln ln n. To bound the variance of y, think of what E(I p pI q ) is for p ≠ q, both
p
primes.
3. Use (1) and (2) to prove that the number of prime factors is almost surely θ(ln ln n).
Exercise 4.15 Suppose one hides a clique of size k in a random graph G ( n, 1 2)
. I.e.,
in the random graph, choose some subset S of k vertices and put in the missing edges to
make S a clique. Presented with the modified graph, find S. The larger S is, the easier it
should be to find. In fact, if k is more than c √ n ln n, then with high probability the clique
leaves a telltale sign identifying S as the k vertices of largest degree. Prove this statement
by appealing to Theorem 4.1. It remains a puzzling open problem to do this when k is
smaller, say, O(n 1/3 ).
Exercise 4.16 The clique problem in a graph is to find the maximal size clique. This
problem is known to be NP-hard and so a polynomial time algorithm is thought unlikely.
We can ask the corresponding question about random graphs. For example, in G ( )
n, 1 2
there almost surely is a clique of size (2 − ε) log n for any ε > 0. But it is not known how
to find one in polynomial time.
1. Show that in G(n, 1 2 ), there almost surely are no cliques of size ≥ 2 log 2 n.
2. Use the second moment method to show that in G(n, 1 ), almost surely there are
2
cliques of size (2 − ε) log 2 n.
3. Show that for any ε > 0, a clique of size (2 − ε) log n can be found in G ( n, 1 2)
in
time n O(ln n) if one exists.
4. Give an O (n 2 ) algorithm that finds a clique of size Ω (log n) in G(n, 1 ) with high
2
probability. Hint: use a greedy algorithm. Apply your algorithm to G ( 1000, 2) 1 .
What size clique do you find?
5. An independent set in a graph is a set of vertices such that no two of them are
connected by an edge. Give a polynomial time algorithm for finding an independent
set in G ( n, 1 2)
of size Ω (log n) with high probability.
Exercise 4.17 Suppose H is a fixed graph on cn vertices with 1 4 c2 (log n) 2 edges. Show
that if c ≥ 2, with high probability, H does not occur as a subgraph of G(n, 1/4).
Exercise 4.18 Given two instances, G 1 and G 2 of G(n, 1 ), what is the largest vertexinduced
subgraph common to both G 1 and G 2
2
?
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