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CS 373: Combinatorial Algorithms, Spring 2001
Homework 3 (due Thursday, March 8, 2001 at 11:59.99 p.m.)
Name:
Net ID:
Alias:
U
3
/
4
1
Name:
Net ID:
Alias:
U
3
/
4
1
Name:
Net ID:
Alias:
U
3
/
4
1
Starting with Homework 1, homeworks may be done in teams of up to three people. Each team
turns in just one solution, and every member of a team gets the same grade. Since 1unit graduate
students are required to solve problems that are worth extra credit for other students,
1unit grad
students may not be on the same team as 3/4unit grad students or undergraduates.
Neatly print your name(s), NetID(s), and the alias(es) you used for Homework 0 in the boxes above.
Please also tell us whether you are an undergraduate, 3/4unit grad student, or 1unit grad student
by circling U,
3
/
4
, or 1, respectively. Staple this sheet to the top of your homework.
Required Problems
1. Hashing:
A hash table of size
m
is used to store
n
items with
n
≤
m/
2. Open addressing is used for
collision resolution.
(a) Assuming uniform hashing, show that for
i
= 1
,
2
,... ,n
, the probability that the
i
th
insertion requires strictly more than
k
probes is at most 2
−
k
.
(b) Show that for
i
= 1
,
2
,... ,n
, the probability that the
i
th
insertion requires more than
2lg
n
probes is at most 1
/n
2
.
Let the random variable
X
i
denote the number of probes required by the
i
th
insertion.
You have shown in part (b) that Pr
{
X
i
>
2lg
n
} ≤
1
/n
2
. Let the random variable
X
=
max
1
≤
i
≤
n
X
i
denote the maximum number of probes required by any of the
n
insertions.
(c) Show that Pr
{
X >
2lg
n
} ≤
1
/n
.
(d) Show that the expected length of the longest probe sequence is
E
[
X
] =
O
(lg
n
).
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View Full DocumentCS 373
Homework 3 (due 3/8/2001)
Spring 2001
2. Reliable Network:
Suppose you are given a graph of a computer network
G
= (
V,E
) and a function
r
(
u,v
) that
gives a reliability value for every edge (
u,v
)
∈
E
such that 0
≤
r
(
u,v
)
≤
1. The reliability
value gives the probability that the network connection corresponding to that edge will
not
fail. Describe and analyze an algorithm to Fnd the most reliable path from a given source
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