Alan M. Frieze: Difference between revisions

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-==Alan Frieze==
-'''Alan M. Frieze''' (born October 25, 1945 in [[London, England]]) is a [[professor]] in the Department of Mathematical Sciences at [[Carnegie Mellon University]], [[Pittsburgh]], [[United States]]. He graduated from the [[University of Oxford]] in 1966, and obtained his PhD from the [[University of London]] in 1975. His research interests lie in [[combinatorics]], [[discrete optimization]] and [[theoretical computer science]]. Currently, he focuses on the probabilistic aspects of these areas; in particular, the study of the asymptotic properties of [[random graphs]], the average case analysis of algorithms, and [[randomized algorithms]]. His recent work has included [[approximate counting]] and volume computation via [[random walks]]; finding edge disjoint paths in [[expander graphs]], and exploring [[anti-Ramsey theory]] and the stability of [[routing algorithms]].
-==Key contributions==
-Two key contributions made by Alan Frieze are:
-(1) - polynomial time algorithm for approximating the volume of [[convex bodies]]
-(2) - algorithmic version for [[Szemerédi Regularity Partition]]
-Both these algorithms wil be described briefly here.
-===Polynomial time algorithm for approximating the volume of convex bodies===
-The paper
-<ref name=JACM91>
-{{cite article
-|author= M.Dyer, A.Frieze and R.Kannan
-|title=A random polynomial-time algorithm for approximating the volume of convex bodies
-|journal= Journal of the ACM
-|volume = 38
-|number =1
-|pages =1--17
-|year= 1991
-|url=http://portal.acm.org/citation.cfm?id=102783}}
-</ref>
-is a joint work by [[Martin Dyer]], Alan Frieze and [[Ravindran Kannan]].
-The main result of the paper is a randomized algorithm for finding an <math>\epsilon</math> approximation to the volume of a convex body <math>K</math> in <math>n</math>-dimensional Euclidean space by assume the existence of a membership oracle. The algorithm takes time bounded by a polynomial in <math>n</math>, the dimension of <math>K</math> and <math>1/\epsilon</math>.
-The algorithm is a sophisticated usage of the so-called [[Markov Chain Monte Carlo]] (MCMC) method.
-The basic scheme of the algorithm is a nearly uniform sampling from within <math>K</math> by placing a grid consisting <math>n</math>-dimensional cubes and doing a random walk over these cubes. By using the theory of
-[http://en.wikipedia.org/wiki/Markov_chain_mixing_time rapidly mixing Markov chains], they show that it takes a polynomial time for the random walk to settle down to being a nearly uniform distribution.
-===Algorithmic version for Szemerédi Regularity Partition===
-This paper
-<ref>
-{{cite article
-|author= A.Frieze and R.Kannan
-|title=A Simple Algorithm for Constructing Szemere'di's Regularity Partition
-|journal=  Electr. J. Comb.
-|volume = 6
-|year= 1999
-|url=http://www.math.cmu.edu/~af1p/Texfiles/svreg.pdf}}
-</ref>
-is a combined work by Alan Frieze and [[Ravindran Kannan]]. They use 2 lemmas to derive the algorithmic version of the [[Szemerédi regularity lemma]] to find an <math>\epsilon</math>-regular partition.
-<br />'''Lemma 1:''' <br />Fix k and <math>\gamma</math> and let <math>G=(V,E)</math> be a graph with <math>n</math> vertices. Let <math>P</math> be an equitable partition of <math>V</math> in classes <math>V_0, V_1, \ldots ,V_k</math>. Assume <math>|V_1| > 4^{2k}</math> and <math>4^k >600 \gamma ^2</math>. Given proofs that more that <math>\gamma k^2</math> pairs <math>(V_r,V_s)</math> are not <math>\gamma</math>-regular, it is possible to find in O(n) time an equitable partition <math>P'</math> (which is a refinement of <math>P</math>) into <math>1+k4^k</math> classes, with an exceptional class of cardinality at most <math>|V_0|+n/4^k</math> and such that <math>ind(P')\geq ind(P) + \gamma^5/20</math>
-<br />'''Lemma 2:''' <br />Let <math>W</math> be a <math>R \times C</math> matrix with <math>|R|=p</math>, <math>|C|=q</math> and <math>\|W\|_\inf\leq1</math> and <math>\gamma</math> be a positive real.
-<br />(a) If there exist <math>S \subseteq R</math>, <math>T \subseteq C</math> such that <math>|S|\geq\gamma p</math>, <math>|T|\geq\gamma q</math> and <math>|W(S,T)|\geq\gamma |S||T|</math> then <math>\sigma_1(W)\geq\gamma^3\sqrt{pq}</math>
-<br />(b) If <math>\sigma_1(W)\geq\gamma\sqrt{pq}</math>, then there exist <math>S\subseteq R</math>, <math>T\subseteq C</math> such that <math>|S|\geq\gamma'p</math>, <math>|T|\geq\gamma'q</math> and <math>W(S,T)\geq\gamma'|S||T|</math> where <math>\gamma'=\gamma^3/108</math>. Furthermore <math>S</math>, <math>T</math> can be constructed in polynomial time.
-<br />These 2 lemmas are combined in the following algorithmic construction of the [[Szemerédi regularity lemma]].
-<br />'''[Step 1]''' Arbitrarily divide the vertices of <math>G</math> into an equitable partition <math>P_1</math> with classes <math>V_0,V_1,\ldots,V_b</math> where <math>|V_i|\lfloor n/b \rfloor</math> and hence <math>|V_0|<b</math>. denote <math>k_1=b</math>.
-<br />'''[Step 2]''' For every pair <math>(V_r,V_s)</math> of <math>P_i</math>, compute <math>\sigma_1(W_{r,s})</math>. If the pair <math>(V_r,V_s)</math> are not <math>\epsilon-</math>regular then by Lemma 2 we obtain a proof that they are not <math>\gamma=\epsilon^9/108-</math>regular.
-<br />'''[Step 3]''' If there are at most <math>\epsilon
-\left(
-  \begin{array}{c}
-    k_1\\
-\\
-  \end{array}
-\right)</math> pairs that produce proofs of non <math>\gamma-</math>regularity that halt. <math>P_i</math> is <math>\epsilon-</math>regular.
-<br />'''[Step 4]''' Apply Lemma 1 where <math>P=P_i</math>, <math>k=k_i</math>, <math>\gamma=\epsilon^9/108</math> and obtain <math>P'</math> with <math>1+k_i4^{k_i}</math> classes
-<br />'''[Step 5]''' Let <math>k_i+1 = k_i4^{k_i}</math>, <math>P_i+1=P'</math>, <math>i=i+1</math> and go to Step 2.
-==Awards and honors==
-In 1991, Dr. Frieze received the Fulkerson Prize in Discrete Mathematics (Jointly with Martin Dyer and Ravi Kannan for the paper "A random polynomial time algorithm for approximating the volume of convex bodies" in the Journal of the Association for Computing Machinery) awarded by the Americam Mathematical Society and the Mathematical Programming Society.
-In 1997 he was a Guggenheim Fellow
-In 2000, he received the IBM Faculty Partnership Award
-In 2006 he jointly received (with Michael Krivelevich) the Professor Pazy Memorial Research Award from the United States-Israel Binational Science Foundation
-==Personal==
-Dr. Alan Frieze is married to [[Carol Frieze]], Professor of [[Computer Science]] at Carnegie Mellon. They have two children, son Adam (currently teaching in England) and daughter Nancy, son-in-law Joe, and grandbabies Maisie and Molly. The family also includes Max, a [[Rottweiler]] mix.
-==References and external links==
-<references/>
-* [http://www.math.cmu.edu/~af1p/index.html Alan Frieze's web page]
-* [http://www.math.cmu.edu/~af1p/Texfiles/oldvolume.pdf Fulkerson prize-winning paper]
-* [http://www.cs.cmu.edu/~cfrieze/personal/index.html Carol Frieze's web page]
-* [http://www.informatik.uni-trier.de/~ley/db/indices/a-tree/f/Frieze:Alan_M=.html Alan Frieze's publications at DBLP]