---
title: "Alan M. Frieze"
chunk: 2/2
source: "https://en.wikipedia.org/wiki/Alan_M._Frieze"
category: "reference"
tags: "science, encyclopedia"
date_saved: "2026-05-05T17:17:20.488241+00:00"
instance: "kb-cron"
---

(b) If 
  
    
      
        
          σ
          
            1
          
        
        (
        W
        )
        ≥
        γ
        
          
            p
            q
          
        
      
    
    {\displaystyle \sigma _{1}(W)\geq \gamma {\sqrt {pq}}}
  
, then there exist 
  
    
      
        S
        ⊆
        R
      
    
    {\displaystyle S\subseteq R}
  
, 
  
    
      
        T
        ⊆
        C
      
    
    {\displaystyle T\subseteq C}
  
 such that 
  
    
      
        
          |
        
        S
        
          |
        
        ≥
        
          γ
          ′
        
        p
      
    
    {\displaystyle |S|\geq \gamma 'p}
  
, 
  
    
      
        
          |
        
        T
        
          |
        
        ≥
        
          γ
          ′
        
        q
      
    
    {\displaystyle |T|\geq \gamma 'q}
  
 and 
  
    
      
        W
        (
        S
        ,
        T
        )
        ≥
        
          γ
          ′
        
        
          |
        
        S
        
          |
        
        
          |
        
        T
        
          |
        
      
    
    {\displaystyle W(S,T)\geq \gamma '|S||T|}
  
 where 
  
    
      
        
          γ
          ′
        
        =
        
          γ
          
            3
          
        
        
          /
        
        108
      
    
    {\displaystyle \gamma '=\gamma ^{3}/108}
  
. Furthermore, 
  
    
      
        S
      
    
    {\displaystyle S}
  
, 
  
    
      
        T
      
    
    {\displaystyle T}
  
 can be constructed in polynomial time.
These two lemmas are combined in the following algorithmic construction of the Szemerédi regularity lemma.
[Step 1] Arbitrarily divide the vertices of 
  
    
      
        G
      
    
    {\displaystyle G}
  
 into an equitable partition 
  
    
      
        
          P
          
            1
          
        
      
    
    {\displaystyle P_{1}}
  
 with classes 
  
    
      
        
          V
          
            0
          
        
        ,
        
          V
          
            1
          
        
        ,
        …
        ,
        
          V
          
            b
          
        
      
    
    {\displaystyle V_{0},V_{1},\ldots ,V_{b}}
  
 where 
  
    
      
        
          |
        
        
          V
          
            i
          
        
        
          |
        
        ⌊
        n
        
          /
        
        b
        ⌋
      
    
    {\displaystyle |V_{i}|\lfloor n/b\rfloor }
  
 and hence 
  
    
      
        
          |
        
        
          V
          
            0
          
        
        
          |
        
        <
        b
      
    
    {\displaystyle |V_{0}|<b}
  
. denote 
  
    
      
        
          k
          
            1
          
        
        =
        b
      
    
    {\displaystyle k_{1}=b}
  
.
[Step 2] For every pair 
  
    
      
        (
        
          V
          
            r
          
        
        ,
        
          V
          
            s
          
        
        )
      
    
    {\displaystyle (V_{r},V_{s})}
  
 of 
  
    
      
        
          P
          
            i
          
        
      
    
    {\displaystyle P_{i}}
  
, compute 
  
    
      
        
          σ
          
            1
          
        
        (
        
          W
          
            r
            ,
            s
          
        
        )
      
    
    {\displaystyle \sigma _{1}(W_{r,s})}
  
. If the pair 
  
    
      
        (
        
          V
          
            r
          
        
        ,
        
          V
          
            s
          
        
        )
      
    
    {\displaystyle (V_{r},V_{s})}
  
 are not 
  
    
      
        ϵ
        −
      
    
    {\displaystyle \epsilon -}
  
regular then by Lemma 2 we obtain a proof that they are not 
  
    
      
        γ
        =
        
          ϵ
          
            9
          
        
        
          /
        
        108
        −
      
    
    {\displaystyle \gamma =\epsilon ^{9}/108-}
  
regular.
[Step 3] If there are at most 
  
    
      
        ϵ
        
          (
          
            
              
                
                  
                    k
                    
                      1
                    
                  
                
              
              
                
                  2
                
              
            
          
          )
        
      
    
    {\displaystyle \epsilon \left({\begin{array}{c}k_{1}\\2\\\end{array}}\right)}
  
 pairs that produce proofs of non 
  
    
      
        γ
        −
      
    
    {\displaystyle \gamma -}
  
regularity that halt. 
  
    
      
        
          P
          
            i
          
        
      
    
    {\displaystyle P_{i}}
  
 is 
  
    
      
        ϵ
        −
      
    
    {\displaystyle \epsilon -}
  
regular.
[Step 4] Apply Lemma 1 where 
  
    
      
        P
        =
        
          P
          
            i
          
        
      
    
    {\displaystyle P=P_{i}}
  
, 
  
    
      
        k
        =
        
          k
          
            i
          
        
      
    
    {\displaystyle k=k_{i}}
  
, 
  
    
      
        γ
        =
        
          ϵ
          
            9
          
        
        
          /
        
        108
      
    
    {\displaystyle \gamma =\epsilon ^{9}/108}
  
 and obtain 
  
    
      
        
          P
          ′
        
      
    
    {\displaystyle P'}
  
 with 
  
    
      
        1
        +
        
          k
          
            i
          
        
        
          4
          
            
              k
              
                i
              
            
          
        
      
    
    {\displaystyle 1+k_{i}4^{k_{i}}}
  
 classes
[Step 5] Let 
  
    
      
        
          k
          
            i
          
        
        +
        1
        =
        
          k
          
            i
          
        
        
          4
          
            
              k
              
                i
              
            
          
        
      
    
    {\displaystyle k_{i}+1=k_{i}4^{k_{i}}}
  
, 
  
    
      
        
          P
          
            i
          
        
        +
        1
        =
        
          P
          ′
        
      
    
    {\displaystyle P_{i}+1=P'}
  
, 
  
    
      
        i
        =
        i
        +
        1
      
    
    {\displaystyle i=i+1}
  
 and go to Step 2.

== Awards and honours ==
In 1991, Frieze received (jointly with Martin Dyer and Ravi Kannan) the Fulkerson Prize in Discrete Mathematics awarded by the American Mathematical Society and the Mathematical Programming Society. The award was for the paper "A random polynomial time algorithm for approximating the volume of convex bodies" in the Journal of the ACM).
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.
In 2011 he was selected as a SIAM Fellow.
In 2012 he was selected as an AMS Fellow.
In 2014 he gave a plenary talk at the International Congress of Mathematicians in Seoul, South Korea.
In 2015 he was selected as a Simons Fellow.
In 2017 he was promoted to University professor.
In 2022 he became the Orion Hoch, S 1952 Professor.

== Personal life ==
Frieze is married to Carol Frieze, who directs two outreach efforts for the computer science department at Carnegie Mellon University.

== References ==

== External links ==
Alan Frieze's web page
Fulkerson prize-winning paper
Alan Frieze's publications at DBLP
Certain self-archived works are available here