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An example: diophantine approximation and continued fractions Givenfind rational approximation such that and continued fraction expansion
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Classical mathematicsand new challenges
László Lovász Microsoft Research
One Microsoft Way, Redmond, WA 98052 [email protected]
Theorems and Algorithms
Algorithmic vs. structural mathematics
Geometric constructions
Euclidean algorithm
Newton’s method
Gaussian elimination
ancient and classical algorithms
An example: diophantine approximation and continued fractions
Given , find rational approximation /p q
such that | / | /p q q and 1/ .q
m
n
| | | |m n p q p
0a 0 ?a
10 0
1 1a
a a
01
1a
a
continued fraction expansion1/q
30’s: Mathematical notion of algorithms
Church, Turing, Post
recursive functions, Λ-calculus, Turing-machines
Church, Gödel
algorithmic and logical undecidability
A mini-history of algorithms
50’s, 60’s: Computers the significance of running time
simple and complex problems
sortingsearchingarithmetic …
Travelling Salesmanmatchingnetwork flowsfactoring …
late 60’s-80’s: Complexity theory
P=NP?
Time, space, information complexity
Polynomial hierarchy
Nondeterminism, good characteriztion, completeness
Randomization, parallelism
Classification of many real-life problems into P vs. NP-complete
90’s: Increasing sophistication upper and lower bounds on complexity
algorithms negative results
factoringvolume computationsemidefinite optimization
topologyalgebraic geometrycoding theory
Higlights of the 90’s:Approximation algorithms
positive and negative results
Probabilistic algorithms
Markov chains, high concentration, nibble methods, phase transitions
Pseudorandom number generators
from art to science: theory and constructions
Approximation algorithms:The Max Cut Problem
maximize
NP-hard
…Approximations?
Easy with 50% error Erdős ~’65
Polynomial with 12% error Goemans-Williamson ’93
???
Arora-Lund-Motwani-Sudan-Szegedy ’92Hastad
NP-hard with 6% error
(Interactive proof systems, PCP)
(semidefinite optimization)
Randomized algorithms (making coin flips):
Algorithms and probability
Algorithms with stochastic input:
difficult to analyze
even more difficult to analyze
important applications (primality testing, integration, optimization, volume computation, simulation)
even more important applications
Difficulty: after a few iterations, complicated functions of the original random variables arise.
Strong concentration (Talagrand)
Laws of Large Numbers: sums of independent random variables is strongly concentratedGeneral strong concentration: very general “smooth” functions of independent random variables are strongly concentrated
Nibble, martingales, rapidly mixing Markov chains,…
New methods in probability:
Example
1 2 33, , ,. ( ).. Ga Fa qa Want: such that:
- any 3 linearly independent
- every vector is a linear combination of 2
Few vectors
O(q)?
(was open for 30 years)
Every finite projective plane of order qhas a complete arc of size q polylog(q).
Kim-Vu
Second idea: choose 1 2 3, , ,...a a a at random
?????
Solution: Rödl nibble + strong concentration results
First idea: use algebraic construction (conics,…)
gives only about q
Driving forces for the next decade
New areas of applications
The study of very large structures
More tools from classical areas in mathematics
New areas of application: interaction between discrete and continuousBiology: genetic code population dynamics protein folding
Physics: elementary particles, quarks, etc. (Feynman graphs) statistical mechanics (graph theory, discrete probability)
Economics: indivisibilities (integer programming, game theory)
Computing: algorithms, complexity, databases, networks, VLSI, ...
Very large structures
-genetic code
-brain
-animal
-ecosystem
-economy
-society
How to model them?
non-constant but stablepartly random
-internet
-VLSI
-databases
Very large structures: how to model them?
Graph minors Robertson, Seymour, Thomas
If a graph does not contain a given minor,then it is essentially a 1-dimensional structure of essentially 2-dimensional pieces.
up to a bounded number of additional nodes
tree-decomposition
embedable in a fixed surface
except for “fringes” of bounded depth
The nodes of graph can be partitioned into a bounded number of essentially equal parts so that almost all bipartite graphs between 2 partsare essentially random(with different densities).
with k2 exceptions
Very large structures: how to model them?Regularity Lemma Szeméredi 74
given >0 and k>1, # of parts is between k and f(k, )
difference at most 1
for subsets X,Y of the two parts,# of edges between X and Y
is p|X||Y| n2
How to model them?
How to handle themalgorithmically?
heuristics/approximation algorithms
-internet
-VLSI
-databases
-genetic code -brain
-animal
-ecosystem
-economy
-society
A complexity theory of linear time?
Very large structures
linear time algorithms
sublinear time algorithms (sampling)
Example: Volume computation
nK Given: , convex
Want: volume of K
by a membership oracle;2(0,1) (0, )B K B n
with relative error ε
Not possible in polynomial time, even if ε=ncn. Elekes, Bárány, Füredi
Possible in randomized polynomial time,for arbitrarily small ε. Dyer, Frieze, Kannan
in n
More and more tools from classical math
Complexity:For self-reducible problems,counting sampling (Jerrum-Valiant-Vazirani)
Enough to samplefrom convex bodies
*
**
*
**
* *
*
( ) | |( ) | |
vol K K Svol B S
must be exponentialin n
Complexity:For self-reducible problems,counting sampling (Jerrum-Valiant-Vazirani)
Enough to samplefrom convex bodies
0K B
1/1 2 nK B K 2/
2 2 nK B K
0
1
vol( )vol( )
KK
by sampling
1
2
vol( )vol( )
KK
by sampling
…
Complexity:For self-reducible problems,counting sampling (Jerrum-Valiant-Vazirani)
Enough to samplefrom convex bodies
Algorithmic results:Use rapidly mixing Markov chains (Broder; Jerrum-Sinclair)
Enough to estimate the mixing rate of random walk on lattice in K
Complexity:For self-reducible problems,counting sampling (Jerrum-Valiant-Vazirani)
Enough to samplefrom convex bodies
Algorithmic results:Use rapidly mixing Markov chains (Broder; Jerrum-Sinclair)
Enough to estimate the mixing rate of random walk on lattice in K
Graph theory (expanders):use conductance toestimate eigenvalue gapAlon, Jerrum-Sinclair
Probability:use eigenvalue gap
K’K”
F
1vol ( ) vol( )vol( ') vol( ")
n F KK K
Complexity:For self-reducible problems,counting sampling (Jerrum-Valiant-Vazirani)
Enough to samplefrom convex bodies
Algorithmic results:Use rapidly mixing Markov chains (Broder; Jerrum-Sinclair)
Enough to estimate the mixing rate of random walk on lattice in K
Graph theory (expanders):use conductance toestimate eigenvalue gapAlon, Jerrum-Sinclair
Enough to proveisoperimetric inequalityfor subsets of K
Differential geometry: Isoperimetric inequality
DyerFriezeKannan1989
* 27( )O n
Probability:use eigenvalue gap
Differential equations:bounds on Poincaré constantPaine-Weinberger
bisection method,improvedisoperimetric inequalityLL-Simonovits 1990
* 16( )O nLog-concave functions: reduction to integration
Applegate-Kannan 1992* 10( )O n
Convex geometry: Ball walkLL 1992
* 10( )O n
Statistics: Better error handlingDyer-Frieze 1993
* 8( )O n
Optimization: Better prepocessingLL-Simonovits 1995
* 7( )O n
achieving isotropic positionKannan-LL-Simonovits 1998
* 5( )O nFunctional analysis:isotropic position ofconvex bodies
Geometry:projective (Hilbert)distance
affine invariant isoperimetric inequalityanalysis of hit-and-run walkLL 1999
* 5( )O n
Differential equations:log-Sobolev inequality
elimination of “start penalty” for lattice walkFrieze-Kannan 1999
log-Cheeger inequality elimination of “start penalty” for ball walkKannan-LL 1999
* 5( )O n
Scientific computing:non-reversible chainsmix better; liftingDiaconis-Holmes-NealFeng-LL-Pak
walk with inertiaAspnes-Kannan-LL
* 3( )??O n
Linear algebra : eigenvalues semidefinite optimization higher incidence matrices homology theory
More and more tools from classical math
Geometry : geometric representations convexity
Analysis: generating functions Fourier analysis, quantum computing
Number theory: cryptography
Topology, group theory, algebraic geometry,special functions, differential equations,…