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elyot talk

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Edgar Bering 2010-10-25 13:21:43 -04:00
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<eventdefs>
<!-- Fall 2010 -->
<eventitem date="2010-10-26" time="04:30 PM" room="MC4040" title="Analysis of randomized algorithms via the probabilistic method">
<short><p>In this talk, we will give a few examples that illustrate the basic method and show how it can be used to prove the existence of objects with desirable combinatorial properties as well as produce them in expected polynomial time via randomized algorithms. Our main goal will be to present a very slick proof from 1995 due to Spencer on the performance of a randomized greedy algorithm for a set-packing problem. Spencer, for seemingly no reason, introduces a time variable into his greedy algorithm and treats set-packing as a Poisson process. Then, like magic, he is able to show that his greedy algorithm is very likely to produce a good result using basic properties of expected value.
</p></short>
<abstract><p>The probabilistic method is an extremely powerful tool in combinatorics that can be
used to prove many surprising results. The idea is the following: to prove that an
object with a certain property exists, we define a distribution of possible objects
and use show that, among objects in the distribution, the property holds with
non-zero probability. The key is that by using the tools and techniques of
probability theory, we can vastly simplify proofs that would otherwise require very
complicated combinatorial arguments.
</p><p>As a technique, the probabilistic method developed rapidly during the latter half of
the 20th century due to the efforts of mathematicians like Paul Erdős and increasing
interest in the role of randomness in theoretical computer science. In essence, the
probabilistic method allows us to determine how good a randomized algorithm's output
is likely to be. Possibly applications range from graph property testing to
computational geometry, circuit complexity theory, game theory, and even statistical
physics.
</p><p>In this talk, we will give a few examples that illustrate the basic method and show
how it can be used to prove the existence of objects with desirable combinatorial
properties as well as produce them in expected polynomial time via randomized
algorithms. Our main goal will be to present a very slick proof from 1995 due to
Spencer on the performance of a randomized greedy algorithm for a set-packing
problem. Spencer, for seemingly no reason, introduces a time variable into his
greedy algorithm and treats set-packing as a Poisson process. Then, like magic,
he is able to show that his greedy algorithm is very likely to produce a good
result using basic properties of expected value.
</p><p>Properties of Poisson and Binomial distributions will be applied, but I'll remind
everyone of the needed background for the benefit of those who might be a bit rusty.
Stat 230 will be more than enough. Big O notation will be used, but not excessively.
</p></abstract>
</eventitem>
<eventitem date="2010-10-19" time="04:30 PM" room="RCH 306" title="Machine learning vs human learning - will scientists become obsolete?">
<short><p><i>by Dr. Shai Ben-David</i>.