## Theory of Combinatorial Algorithms

Prof. Emo Welzl and Prof. Bernd Gärtner

# Mittagsseminar (in cooperation with M. Ghaffari, A. Steger and B. Sudakov)

 Mittagsseminar Talk Information

Date and Time: Thursday, October 04, 2012, 12:15 pm

Duration: 30 minutes

Location: CAB G51

Speaker: Timon Hertli

## Using Integrals to Compute Probabilities on Permutations

Suppose you want to compute the probability that in a random permutation some element x comes before element y and z, or before element c and d. One way to compute this is to list all subpermutations on the involved elements, but this becomes quickly infeasible.

There is a nice technique to compute such probabilities. It was first used to analyze the famous PPSZ algorithm for k-SAT by Paturi, Pudlák, Saks and Zane. I used it also in my semester thesis for a similar topic.

We can make permutations continuous as follows: For each element x, we choose a real number u.a.r. in [0,1] (the "place" of x); the permutation is then obtained by ordering the elements in ascending order. By symmetry, we again obtain a (uniformly) random permutation. The places of the elements are independent, unlike the positions in a permutation. Using this independence, probabilities like the above can then be computed using simple integrals.

In the talk I will first introduce the technique and will give some examples what to do with it; among them how it is used in the analysis of the PPSZ algorithm. I think this technique is nice to know, even if it is quite simple and, as far as I know, not used outside of SAT yet.

Previous talks by year:   2018  2017  2016  2015  2014  2013  2012  2011  2010  2009  2008  2007  2006  2005  2004  2003  2002  2001  2000  1999  1998  1997  1996

Information for students and suggested topics for student talks