endre szemerédi & tcs avi wigderson ias, princeton
TRANSCRIPT
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)
1
0
1
0
0
0
1
1
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 algorithm
Thm[PPST]: 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
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) separators
Thm[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-dimensional
Thm[Barak-Dvir-W-Yehudayoff’10]: If i Li covers a –fraction of P, then P is O(1/2)-dim.
Happy Birthday Endre !