Algorithm Design

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Chapter !3 Randomized Algorithms
n
Set fli=f(fll fl2"’’fli-l,~i)
Eadfor
Output fl
One could view this is as a type of "stream cipher with feedback." One
problem with this approach is that, if any bit a~ gets corrupted in transmission,
it will corrupt the computed value of ~1 for all ] > i.
We consider the following problem. A sender S wants to transmit the
R k. With each
one, he shares a different secret function f~. Thus he sends a different
encrypted message ~I0 to each receiver, so that a~0 decrypts to fl when
the above algorithm is run with the function f(0.
Unfortunately, the communication channels are very noisy, so each of
the n bits in each of the k transmissions is independently corrupted (i.e.,
flipped to its complement) with probability 1/4. Thus no single receiver
on his or her own is likely to be able to decrypt the message corrhctly.’
Show, however, that ff k is large enough as a function of n, then the k
receivers can jointly reconstruct the plain-text message in the following
way. They get together, and without revealing any of the ~0 or the f~0,
they interactively run an algorithm that will produce the correct fl with
probability at least 9/!0. (How large do you need k to be inyour algorithm?)
17. Consider the following simple model of gambling in the presence of bad
odds. At the beginning, your net profit is 0. You play for a sequence of n
rounds; and in each round, your net profit increases by 1 with probability
1/3, and decreases by 1 with probability 2/3.
Show that the expected number of steps in which your net profit is
positive can be upper-bounded by an absolute constant, independent of
the value of n.
18. In this problem, we will consider the following simple randomized algorithm
for the Vertex Cover Algorithm.
Start with S=~
While S is not a vertex cover,
Select au edge e not covered by S
Select one end of e at random (each end equally likely)
Add the selected node to $
Endwhile
Notes and Further Reading
We wi~ be interested in the expected cost of a vertex cover selected by
this algorithm.
(a) Is this algorithm a c-approximation algorithm for the Minimum
Weight Vertex Cover Problem for some constant c? Prove your answer.
Is this algorithm a c-approximation algorithm for the M~h~imum Cardinality
Vertex Cover Problem for some constant c? Prove your answer.
(Hint; For an edge, let Pe denote the probability that edge e is
selected as an uncovered edge in this algorithm. Can you express
the expected value of the solution in terms of these probabilities? To
bound the value of an optimal solution in terms of the Pe probabilities,
try to bound the sum of the probabilities for the edges incident to a
given vertex v, namely, Pe.)
e incident to u
Notes and Further Reading
The use of randomization in algorithms is an active research area; the books
by Motwani and Raghavan (1995) and Mitzenmacher and Upfal (2005) are
devoted to this topic. As the contents of this chapter make clear, the types
of probabilistic arguments used in the study of basic randomized algorithms
often have a discrete, combinatorial flavor; one can get background in this
style of probabilistic analysis from the book by Feller (1957).
The use of randomization for contention resolution is common in many
systems and networking applications. Ethernet-style shared communication
media, for example, use randomized backoff protocols to reduce the number
of collisions among different senders; see the book by Bertsekas and Gallager
(1992) for a discussion of this topic.
The randomized algorithm for the Minimum-Cut Problem described in the
text is due to Karger, and afier further optimizafions due to Karger and Stein
(1996), it has become one of the most efficient approaches to the minimum
cut problem. A number of further extensions and applications of the algorithm
appear in Karger’s (1995) Ph.D. thesis.
The approximation algorithm for MAX 3-SAT is due to Johnson (!974), in
a paper that contains a number of early approximation algorithms for NP-hard
problems. The surprising punch line to that section--that every instance of 3-
SAT has an assignment satisfying at least 7/8 of the clauses--is an example
of the probabilistic method, whereby a combinatorial structure with a desired
property is shown to exist simply by arguing that a random structure has
the property with positive probability. This has grown into a highly refined
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