1

Why is it useful to walk randomly?

László Lovász

Mathematical InstituteEötvös Loránd University

October 2012

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Random walk on a graph

October 2012

Graph G=(V,E)

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Random walk on a graph

October 2012

t(v): probability of being

at node v after t steps

1

( )

1( ) ( )

deg( )t t

j N i

i jj

1deg( ) deg( )( ) ( )

2 2t ti i

i i i im m

Stationary distribution

4

Hitting time

H(s,t) = hitting time from s to t

= expected # of steps, starting at s,

before hitting t

k(s,t) = commute time between s and t

= H(s,t) + H(t,s)

October 2012

( )

1( )

ha

( ).deg(

rmonic in

)

:

j N i

f i f ji

f i V

( )

1( )

ha

( )deg(

s pole i

)

n :

j N i

f i V

f i f ji

Every nonconstant function has at least 2 poles.

Harmonic functions

Every function defined on SV (S) has a unique extension harmonic on V \ S.

G=(V,E) graph,

f: V

October 2012 5

S

2

3

1 f(v)= E(f(Zv))

Zv: (random) point where

random walk from v hits Sv

0v

1

f(v)= P(random walk from

v hits t before s)

s t

Harmonic functions and random walks

October 2012 6

0v

1f(v)=electrical potentials t

Harmonic functions and electrical networks

October 2012 7

f(v) = position of nodes

0 1

Harmonic functions and rubber bands

October 2012 8

2( , ) 2 ( , )

force acting on

mu v mR u v

u

Commute time and resistance

October 2012 9

effective resistence between u and v

( )

11 ( , )

d )( )

eg(,

i N s

H i ts

H s t

( )

deg( ) ( ( , ) ( , )) 0i N s

s H i t H s t

Distance from s to t = H(s,t).

t

weight=degree

strength=1

Hitting time and rubber bands

October 2012 10

11

1{

7

1{

5

}1

}12

7

} 1

7

9

3

5

Hitting time and rubber bands

October 2012 11

12

Random maze

October 2012

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Random maze

October 2012

14October 2012

We obtain every mazewith the same probability!

Random maze

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Random spanning tree

October 2012

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- card shuffling

- statistics

- simulation

- counting

- numerical integration

- optimization

- …

Sampling: a general algorithmic task

October 2012

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{ : ( ) ( , )}L x y xA y

polynomial time algorithm

certificate

October 2012

L: a „language” (a family of graphs, numbers,...)

Sampling: a general algorithmic task

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{ : ( ) ( , )}L x y xA y

Find: - a certificate

Given: x

- an optimal certificate

- the number of certificates

- a random certificate

(uniform, or given distribution)

October 2012

L: a „language” (a family of graphs, numbers,...)

Sampling: a general algorithmic task

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One general method for sampling: Random walks

(+rejection sampling, lifting,…)

Construct regular graph with node set V

Want: sample uniformly from V

Simulate (run) random walk for T steps

Output the final node ????????????

mixing timeOctober 2012

Sampling by random walk

Given: convex body K n

Want: volume of K

Not possible in polynomial time, even if an errorof nn/10 is allowed.

Elekes, Bárány, Füredi

Volume computation

October 2012 20

Dyer-Frieze-Kannan 1989

But if we allow randomization:

There is a polynomial time randomized algorithmthat computes the volume of a convex body

with high probability with arbitrarily small relative error

Volume computation

October 2012 21

B

K

Why not just....

***

*

*

*

*

*

**

* *

**

*

*

*

*

S

| |vol( ) vol( )

| |

S KK B

S

Need exponential size S

to get nonzero!

Volume computation by plain Monte-Carlo

October 2012 22

i iK K B

0B1B

2B

mB

1 10

1 2 0

vol( ) vol( ) vol( )vol( ) vol( )

vol( ) vol( ) vol( )m m

m m

K K KK K

K K K

mK K

0 0K B

1/1 2 n

i iB B

Volume computation by multiphase Monte-Carlo

October 2012 23

1vol( )1 2

vol( )i

i

K

K

Can use Monte-Carlo!

But...Now we have to generate random points from Ki+1.

Need sampling to computethe volume

Volume computation by multiphase Monte-Carlo

October 2012 24

Do sufficiently longrandom walk on centersof cubes in K

Construct sufficiently dense lattice

Pick random point p from little cube

If p is outside K, abort;else return p

Dyer-Frieze-Kannan 1989

Sampling by random walk on lattice

October 2012 25

Sampling by ball walk

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Sampling by hit-and-run walk

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steplength can be large!

Sampling by reflecting walk

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- Stepsize

- Where to start the walk?

- How long to walk?

- How close will be the returned point to random?

Issues with all these walks

October 2012 29

bottleneck

1S 2S1 1 2( ) ( , ' )PS x S x S

1 21

1 2

( , ' )( )

( ) ( ' )

P

P P

x S x SS

x S x S

isoperimetric quantity

inf { ( ) : }S S K

: uniform random point inx K

' : one step fromx x

x

'x

Conductance

October 2012 30

Dyer-Frieze-Kannan 1989 ** 27 * 32 23( ) (( ) )O nO n O n

Polynomial time!

Cost of volume computation

(number of oracle calls)Amortized cost

of sample point

Cost ofsample point

Time bounds

October 2012 31

Dyer-Frieze-Kannan 1989

Lovász-Simonovits 1990

Applegate-Kannan 1990

Lovász 1991

Dyer-Frieze 1991

Lovász-Simonovits 1992,93

Kannan-Lovász-Simonovits 1997

** 27 * 32 23( ) (( ) )O nO n O n

Lovász 1999

** 16 * 41 13( ) (( ) )O nO n O n** 10 * 87(( ) ( ))O n O nO n** 10 * 87(( ) ( ))O n O nO n

* 8 ** 6 7( ) ( )( )O nO n O n* 7 ** 5 6( ) ( )( )O nO n O n* 5 ** 3 4( ) ( )( )O nO n O n

Kannan-Lovász 1999

Lovász-Vempala 2002 * 3( )O n

Lovász-Vempala 2003 * 4( )O n

* 3( )O n

* 3( )O n

Time bounds

October 2012 32

- The Slicing Conjecture

- Reflecting walk

Possibilities for further improvement

October 2012 33

Reflecting random walk in K

v

u

steplength h large

How fast does this mix?

Stationary distribution: uniform

Chain is time-reversible

(e.g. exponentially distributedwith expectation = diam(K))

October 2012 34

Smallest bisecting surface

F H

Smallest bisecting hyperplane

1 1vol ( ) vol ( )n nH F

??

The Slicing Conjecture

October 2012 35