elyot talk
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<! Fall 2010 >


<eventitem date="20101026" 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 setpacking problem. Spencer, for seemingly no reason, introduces a time variable into his greedy algorithm and treats setpacking 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


nonzero 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 setpacking


problem. Spencer, for seemingly no reason, introduces a time variable into his


greedy algorithm and treats setpacking 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="20101019" time="04:30 PM" room="RCH 306" title="Machine learning vs human learning  will scientists become obsolete?">




<short><p><i>by Dr. Shai BenDavid</i>.




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