Endre Szemer

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Endre Szemerédi TCS

• Avi Wigderson
• IAS, Princeton

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Happy Birthday Endre !
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Selection of omitted results
Babai-Hajnal-Szemerédi-Turan Lower bounds on
Branching Programs Ajtai-Iwaniec-Komlós-Pintz-Sz
emerédi   Explicit ?-biased set over Zm
Nisan-Szemerédi-W Undirected connectivity in
(log n)3/2 space Komlós-Ma-Szemerédi Matching
nuts and bolts in O(n log n) time .
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The dictionary problem Storage, retrieval, and
the power of universal hashing
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The Dictionary Problem
Store a set Uu1, u2, , un ? 0,1k (n ??
2k) using O(n) time space (each unit is k-bit
word). - Minimize of queries to determine if x
? U?
Classic log n Sort U and use a search tree.
u5 lt un lt lt
u7 QuestionYao Should tables be
sorted? ThmYao No! (for many k,n). Use
hashing! ThmFredman-Komlós-Szemerédi82
Never! 2 queries always suffice!
xltui
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h2k ? n universal hash h(x)axb(modn)
E?i ni2 O(n)
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Sorting networks The mamnoth of all expander
applications
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Sorting networks Ajtai-Komlós-Szemerédi
n inputs (real numbers), n outputs (sorted)
MIN MAX
Many sorting algorithms of O(n log n)
comparisons Several sorting networks of O(n log2
n) comparators
ThmAKS83 Explicit network with O(n log n)
comparators, and depth O(log n) Proof Extremely
sophisticated use analysis of expanders
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Monotone Threshold Formulae
n inputs (bits), n outputs (sorted)
1 0 1 0
0 0 1 1
AND OR
Threshold
Thm AKS83 Size O(n log n), depth O(log n)
network. CorAKS Monotone Majority formula of
size nO(1) (derandomizing a probabilistic
existence proof of Valiant) Open Find a simple
polynomial size Majority formula Open Prove size
lower bound gtgt n2 (best upper bound n5.3)
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Derandomization The mother of all randomness
extractors
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Derandomized error reduction CW,IZ
Prerror lt 1/3
Bxlt2n/3
Random bits kn nO(k)
ThmChernoff r1 r2. rk
independent
ThmAjtai-Komlós-Szemerédi87 r1 . rk random
path
then Prerror Prr1 r2. rk ?Bx
gt k/2 lt exp(-k)
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Derandomization of sampling via expander walks
G d-regular expander. f V(G) ? R, f(v)?1,
Ef0 Thm Chernoff r1 r2. rk independent
in V(G) Thm AKS,Gilman r1 r2. rk random
path in G then Pr?i f(ri) gt ?k lt
exp(-?2 k) f V(G) ? Md(R), f(v)?1, Ef0
Thm Ahlswede-Winter r1 r2. rk
independent Conjecture r1 r2. rk
random path then Pr ?? ?i f(ri) ?? gt
?k lt d exp(-?2 k)
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Black-box groups and computational group theory
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Black-box groups Babai-Szemerédi84
G a finite group (of permutations, matrices,
) Think of the elements as n-bit strings
(G?2n) Black-box BG representation of G is
BG
x y
x-1 xy
Membership problem Given g1, g2, , gd, h ?
G, does h ? ?g1, g2, , gd ? ? Standard proof
word (can be exponentially long!) e.g. m2n,
?g? Cm , hgm/2 ggggg.gggggggg Clever
proof SLP (Straight Line Program)
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Straight-line programs Babai-Szemerédi
An SLP for h ? ?S? with S g1, g2, , gd
is g1, g2, , gd , gd1, gd2, , gth
where for kgtd gkgi-1 or gkgigj (i,jltk). Let
SLPS(h) denote the smallest such t ThmBS
Membership ? NP For every G, every generators
?g1, g2,, gd ? G and every, h ? G, SLPS(h) lt
(log G)2 Open Is it tight, or perhaps O(log
G) possible? ThmBabai, Cooperman, Dixon
Random generation ? BPP
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Proof complexity Resolution of random formulae
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The Resolution proof system
A CNF over Boolean variables x1, x2, , xn is a
conjunction of clauses f C1? C2 ? ? Cm, with
every clause Ci of the form xi1 ? xi2 ??
xik Assume fFALSE. How can we prove it? A
resolution proof is a sequence of clauses
C1, C2, , Cm, Cm1, Cm2, , Ct? with (C?x,
D?x) ? C?D (Resolution Rule) Let Res(f) denote
the smallest such t ThmHaken85
Res(PHPn) gt exp (n) ThmChvátal-Szemerédi88
Res(f) gt exp(n) for almost all 3-CNFs f on
m20n clauses. Open Extend to the Frege proof
system.
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The Frege proof system
A CNF over Boolean variables x1, x2, , xn is a
conjunction of clauses f C1? C2 ? ? Cm
Assume fFALSE. How can we prove it? A Frege
proof is a sequence of formulae C1, C2, ,
Cm, Gm1, Gm2, , Gt? with (G, G?H) ? H
(Modus Ponens) Let Fre(f) denote the smallest
such t ThmBuss Fre(PHPn) poly(n) Open Is
there any f for which Fre(f) ? poly(n)
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Determinism vs. Non-determinism Separators and
segregators in k-page graphs
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Determinism vs. non-determinism in linear time
Paul-Pippenger-Szemerédi-Trotter

• Conj NP ? P ( NTIME(nO(1)) ? DTIME(nO(1)) )
• Conj SAT has no polynomial time algorithm
• ThmPPST SAT has no linear time algorithm
• Cor PPST NTIME(n) ? DTIME(n)
• Proof
• Block-respecting computation
• Simulation of alternating time.
• Diagonalization
• k-page graphs describe TM computation

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k-page graphs (k constant)

• Vertices on spine
• Planar per page
• k pages

ThmPPST k-page graphs have o(n) segregators
( Remove o(n) nodes. Each node has o(n)
descendents ) ConjGKS k-page graphs have o(n)
separators ThmBourgain k-page graphs can be
expanders!
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Point-Line configurations locally correctable
codes
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Point-Line configurations

• Pp1, p2, , pn points in Rn (or Cn).
• A line is special if it passes through 3 points.
• Li special lines through pi
• ThmSilvester-Gallai-Melchior40 If ?i, Li
  covers all of P, then P is 1-dimensional ( over
  C, 2-dim)
• ThmSzemerédi-Trotter83 If ?i, Li covers
• (1-?0)-fraction of P, then P is 1-dimensional
• ThmBarak-Dvir-W-Yehudayoff10 If ?i Li covers
  a ?fraction of P, then P is O(1/?2)-dim.

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Happy Birthday Endre !