Foundations of Data Science

Exercise 4.7 Let f (n) be a function that is asymptotically less than n. Some such functions
are 1/n, a constant d, log n or n 1 3 . Show that
( ) n
1 + f(n)
n ≃ e f(n)(1±o(1)) .
for large n. That is
Exercise 4.8
[( ) n ]
ln 1 + f(n)
n
lim
n→∞ f (n)
= 1.
1. In the limit as n goes to infinity, how does ( 1 − n) 1 n ln n
behave.
(
2. What is lim n+1
) n?
n→∞ n
Exercise 4.9 Consider a random permutation of the integers 1 to n. The integer i is
said to be a fixed point of the permutation if i is the integer in the i th position of the
permutation. Use indicator variables to determine the expected number of fixed points in
a random permutation.
Exercise 4.10 Generate a graph G ( n, d n)
with n = 1000 and d=2, 3, and 6. Count the
number of triangles in each graph. Try the experiment with n=100.
Exercise 4.11 What is the expected number of squares (4-cycles) in G ( n, d n)
? What is
the expected number of 4-cliques in G ( n, d n)
?
Exercise 4.12 Carry out an argument, similar to the one used for triangles, to show that
p = 1 is a threshold for the existence of a 4-clique. A 4-clique consists of four vertices
n 2/3
with all ( 4
2)
edges present.
Exercise 4.13 What is the expected number of simple paths of length 3, log n, √ n, and
n − 1 in G(n, d )? A simple path is a path where no vertex appears twice as in a cycle.
n
The expected number of simple paths of a given length being infinite does not imply that a
graph selected at random has such a path.
Exercise 4.14 Let x be an integer chosen uniformly at random from {1, 2, . . . , n}. Count
the number of distinct prime factors of n. The exercise is to show that the number of prime
factors almost surely is Θ(ln ln n). Let p stand for a prime number between 2 and n.
1. For each fixed prime p, let I p be the indicator function of the event that p divides x.
Show that E(I p ) = 1 p + O ( 1
n
)
.
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