 
Summary: CMPSCI 611: Advanced Algorithms
Micah Adler
Problem Set 4 Out: November 14, 2001
Due: November 21, 2001
1. Consider again the hockey league we saw in Problem Set 3. Let's say that we have reached a point in
the season where team T 1 has no games left, each of the other n 1 teams has 4n 8 games left, with
exactly 4 games against each of the other teams. Furthermore, let's assume that team T 1 has built up
a lead such that it will have more wins at the end of the season than any other team, provided that
no other team wins at least 3n of its remaining games. We want to derive a bound on the probability
that team T 1 has the most wins at the end of the season. We shall assume that in each game, both
teams have an equal probability of winning, and that the outcomes of all games are independent.
(a) Use Chebyshev's inequality to show that the probability that T 1 has the most wins at the end of
the season is at least 1=2.
(b) Derive the largest bound you can on the same probability using a Cherno bound. How do the
two bounds compare (for large n)?
2. Consider a parallel computer that consists of n processors and n memory modules. During a step of
computation, each processor chooses a memory module independently and uniformly at random, and
sends a memory request to that module.
(a) During a step of computation, what is the expected number of memory modules that do not receive
any requests?
