Endre Szemerédi & TCS

Avi WigdersonIAS, Princeton

Happy Birthday Endre !

Selection of omitted results

[Babai-Hajnal-Szemerédi-Turan] Lower bounds on Branching Programs

[Ajtai-Iwaniec-Komlós-Pintz-Szemerédi]  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

……….

The dictionary problem

Storage, retrieval, and thepower of universal hashing

The Dictionary ProblemStore a set U={u1, u2, …, un} {0,1}k (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 < un < … < u7

Question[Yao] “Should tables be sorted?”Thm[Yao] No! (for many k,n). Use hashing!Thm[Fredman-Komlós-Szemerédi’82] Never!2 queries always suffice!

x<ui

hi

h1

h3

hn

ni2

n12

- Birthday paradox- Storage: O(n)- Search: 2 queries

hi:[2k] [ni2]h:[2k] [n]

universal hashh(x)=ax+b(modn)

E[i ni2 ] = O(n)

[2k]

h

n1

ni

n2

n3u2

u1

un

n

1

2

3

i

Sorting networks

The mamnoth of all expander applications

Sorting networks [Ajtai-Komlós-Szemerédi]

n inputs (real numbers), n outputs (sorted)

Many sorting algorithms of O(n log n) comparisonsSeveral sorting networks of O(n log2 n) comparatorsThm:[AKS’83] Explicit network with O(n log n)

comparators, and depth O(log n)Proof: Extremely sophisticated use & analysis of expanders

MIN

MAX

Monotone Threshold Formulaen inputs (bits), n outputs (sorted)

Thm: [AKS’83] Size O(n log n), depth O(log n) network.Cor[AKS]: Monotone Majority formula of size nO(1)

(derandomizing a probabilistic existence proof of Valiant)Open: Find a simple polynomial size Majority formulaOpen: Prove size lower bound >> n2 (best upper bound n5.3)

1010

0011

AND

OR Threshold

Derandomization

The mother of all randomness extractors

Derandomized error reduction [CW,IZ]

Algx

r

{0,1}n

random

strings

Thm[Chernoff] r1 r2…. rk independentThm[Ajtai-Komlós-Szemerédi’87] r1 …. rk random path

Algx

rk

Algx

r1

Majority

G explicit d-regularexpander graph Bx

Pr[error] < 1/3

then Pr[error] = Pr[|{r1 r2…. rk }Bx}| > k/2] < exp(-k)

|Bx|<2n/3

Random bits kn n+O(k)

Derandomization of sampling via expander walks

G d-regular expander.f: V(G) R, |f(v)|1, E[f]=0 Thm [Chernoff] r1 r2…. rk independent in V(G)Thm [AKS,Gilman] r1 r2…. rk random path in G then Pr[|i f(ri) | > k] < exp(-2 k)

f: V(G) Md(R), ||f(v)||1, E[f]=0 Thm [Ahlswede-Winter] r1 r2…. rk

independentConjecture: r1 r2…. rk random path then Pr[ i f(ri) > k] < d exp(-2 k)

Black-box groups

and computational group theory

Black-box groups [Babai-Szemerédi’84]

G a finite group (of permutations, matrices, …)Think of the elements as n-bit strings (|G|2n)Black-box BG representation of G is BGx

yx-1

xy

Membership problem: Given g1, g2, …, gd, h G,does h g1, g2, …, gd ?

Standard proof: word (can be exponentially long!) e.g. m=2n, g = Cm , h=gm/2 = ggggg…….ggggggggClever proof: SLP (Straight Line Program)

Straight-line programs [Babai-Szemerédi]

An SLP for h S with S = {g1, g2, …, gd } is g1, g2, …, gd , gd+1, gd+2, …, gt=h where for k>d gk=gi

-1 or gk=gigj (i,j<k).Let SLPS(h) denote the smallest such t

Thm[BS] Membership NPFor every G, every generators g1, g2,…, gd =Gand every, h G, SLPS(h) < (log |G|)2

Open: Is it tight, or perhaps O(log |G|) possible?

Thm[Babai, Cooperman, Dixon] Random generation BPP

Proof complexity

Resolution of random formulae

The Resolution proof systemA 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 f=FALSE. How can we prove it?A resolution proof is a sequence of clauses C1, C2, …, Cm, Cm+1, Cm+2, …, Ct= with (Cx, Dx) CD (Resolution Rule)Let Res(f) denote the smallest such tThm[Haken’85] Res(PHPn) > exp (n)Thm[Chvátal-Szemerédi’88] Res(f) > exp(n) for almost all 3-CNFs f on m=20n clauses.Open: Extend to the Frege proof system.

axioms

The Frege proof systemA CNF over Boolean variables {x1, x2, …, xn} is a conjunction of clauses f= C1 C2 … Cm Assume f=FALSE. How can we prove it?A Frege proof is a sequence of formulae C1, C2, …, Cm, Gm+1, Gm+2, …, Gt= with (G, GH) H (Modus Ponens)Let Fre(f) denote the smallest such t

Thm[Buss] Fre(PHPn) = poly(n)

Open: Is there any f for which Fre(f) poly(n)

axioms

Determinism vs.Non-determinism

Separators and segregators in k-page graphs

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 algorithmThm[PPST]: SAT has no linear time algorithmCor [PPST]: NTIME(n) DTIME(n) Proof:- Block-respecting computation- Simulation of alternating time.- Diagonalization- k-page graphs describe TM computation

k-page graphs (k constant)

n

1

2

3

Thm[PPST]: k-page graphs have o(n) segregators ( Remove o(n) nodes. Each node has o(n) descendents )Conj[GKS]: k-page graphs have o(n) separatorsThm[Bourgain]: k-page graphs can be expanders!

- Vertices on spine- Planar per page- k pages

Point-Line configurations

& locally correctable codes

Point-Line configurationsP={p1, p2, …, pn} points in Rn (or Cn).A line is special if it passes through ≥3 points.Li: special lines through pi

Thm[Silvester-Gallai-Melchior’40]: If i, Li covers all of P, then P is 1-dimensional ( over C, 2-dim)Thm[Szemerédi-Trotter’83]: If i, Li covers (1-0)-fraction of P, then P is 1-dimensionalThm[Barak-Dvir-W-Yehudayoff’10]: If i Li covers a –fraction of P, then P is O(1/2)-dim.

Happy Birthday Endre !