Alan M. Frieze (born 25 October 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 optimisation 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 randomised 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.
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 lemma
Both these algorithms will be described briefly here.
The paper[1] is a joint work by Martin Dyer, Alan Frieze and Ravindran Kannan.
The main result of the paper is a randomised algorithm for finding an
\epsilon
K
n
n
K
1/\epsilon
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
K
This paper[2] is a combined work by Alan Frieze and Ravindran Kannan. They use two lemmas to derive the algorithmic version of the Szemerédi regularity lemma to find an
\epsilon
Lemma 1:
Fix k and
\gamma
G=(V,E)
n
P
V
V0,V1,\ldots,Vk
|V1|>42k
4k>600\gamma2
\gammak2
(Vr,Vs)
\gamma
P'
P
1+k4k
k | |
|V | |
0|+n/4 |
\operatorname{ind}(P')\geq\operatorname{ind}(P)+\gamma5/20
Lemma 2:
Let
W
R x C
|R|=p
|C|=q
\|W\|inf\leq1
\gamma
S\subseteqR
T\subseteqC
|S|\geq\gammap
|T|\geq\gammaq
|W(S,T)|\geq\gamma|S||T|
3\sqrt{pq} | |
\sigma | |
1(W)\geq\gamma |
\sigma1(W)\geq\gamma\sqrt{pq}
S\subseteqR
T\subseteqC
|S|\geq\gamma'p
|T|\geq\gamma'q
W(S,T)\geq\gamma'|S||T|
\gamma'=\gamma3/108
S
T
These two lemmas are combined in the following algorithmic construction of the Szemerédi regularity lemma.
[Step 1] Arbitrarily divide the vertices of
G
P1
V0,V1,\ldots,Vb
|Vi|\lfloorn/b\rfloor
|V0|<b
k1=b
(Vr,Vs)
Pi
\sigma1(Wr,s)
(Vr,Vs)
\epsilon-
\gamma=\epsilon9/108-
\epsilon \left(\begin{array}{c} k1\\ 2\\ \end{array} \right)
\gamma-
Pi
\epsilon-
P=Pi
k=ki
\gamma=\epsilon9/108
P'
ki | |
1+k | |
i4 |
ki+1=
ki | |
k | |
i4 |
Pi+1=P'
i=i+1
Frieze is married to Carol Frieze, who directs two outreach efforts for the computer science department at Carnegie Mellon University.