Homework #0
This homework will not be handed in or graded.
1. There is a class of
n
≥
1 students. Student
A
is a
celebrity
if every student knows
A
,
and
A
does not know any other student. Student
B
is an
observer
if
B
knows every
student, but no other student knows
B
. You are allowed to query any student
i
as to
whether they know some other student
j
.
•
Design an algorithm which identifies which students are
celebrities
and which
students are
observers
, using as few queries as possible. Prove the correctness of
your algorithm.
•
Is it possible for a student to be both a
celebrity
and
observer
simultaneously?
Give a formal proof of your answer.
2. A prime number is an integer
p >
1 which is divisible only by itself and 1. Prove that
there are an infinite number of primes.
3. Order the following functions in order from smallest asymptotic running time to great
est. Additionally, identify all pairs of functions
i
and
j
where
f
i
(
n
) =
θ
(
f
j
(
n
)). Assume
all logarithms have base2 unless otherwise specified. Justify all answers.
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 Spring '06
 Shamsian
 Algorithms, Graph Theory, Directed acyclic graph, following recurrence relations, strongly connected graph

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