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Random Sorting Networks. Omer Angel. Alexander Holroyd. Dan Romik. Balint Virag. math.PR/0609538. Adv. in Math. 2007. Thanks to: Nathanael Berestycki, Alex Gamburd, Alan Hammond, Pawel Hitczenko, Martin Kassabov, Rick Kenyon, Scott Sheffield, David Wilson, Doron Zeilberger. 1 2 3

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slide1

Random Sorting Networks

Omer Angel

Alexander Holroyd

Dan Romik

BalintVirag

math.PR/0609538

Adv. in Math. 2007

Thanks to: Nathanael Berestycki,

Alex Gamburd, Alan Hammond,

Pawel Hitczenko, Martin Kassabov,

Rick Kenyon, Scott Sheffield,

David Wilson, Doron Zeilberger

slide2

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slide8

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slide9

2

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slide10

2

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To get from 1Ln to nL1

requires

N:=

nearest-neighbour swaps

slide11

E.g. n=4:

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A Sorting Network =

any route from 1Ln to nL1

in exactly

N:=

nearest-neighbour swaps

slide12

Theorem (Stanley 1984).

# of n-particle sorting networks =

Uniform Sorting Network (USN):

choose an n-particle sorting network

uniformly at random.

E.g. n=3:

slide13

2

1

3

4

2

3

1

4

2

3

4

1

3

2

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1

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2

1

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slide14

swap

locations

2

1

3

4

2

3

1

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1

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1

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2

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slide15

swap

locations

2

1

3

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2

3

1

4

2

3

4

1

3

2

4

1

3

4

2

1

4

3

2

1

1

2

3

4

particle

trajectory

slide16

2

3

4

1

swap

locations

2

1

3

4

2

3

1

4

2

3

4

1

3

2

4

1

3

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2

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2

1

1

2

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4

particle

trajectory

permutation matrix

(at half-time)

1 2 3 4

9 efficient simulation algorithm for USN...

slide19

swap

locations

2

1

3

4

2

3

1

4

2

3

4

1

3

2

4

1

3

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2

1

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2

1

1

2

3

4

s1

=

1

s2

=

2

s3

=

3

s4

=

1

s5

=

2

s6

=

1

slide20

Theorem:For USN:

1. Sequence of swap locations

(s1,...,sN) is stationary

2. Scaled first swap location

3. Scaled swap process

8 n

as n!1

as n!1

(Note: not true for all sorting networks,

e.g. bubble sort)

slide21

In progress:

Process of first k swaps

in positions cn...cn+k

! random limit

as n!1

not depending on c2(0,1)

slide22

Proof of stationarity:

2

1

3

4

2

3

1

4

2

3

4

1

3

2

4

1

3

4

2

1

4

3

2

1

1

2

3

4

slide23

Proof of stationarity:

2

1

3

4

2

3

1

4

2

3

4

1

3

2

4

1

3

4

2

1

4

3

2

1

slide24

Proof of stationarity:

1

2

3

4

1

3

2

4

1

3

4

2

3

1

4

2

3

4

1

2

4

3

1

2

slide25

Proof of stationarity:

1

2

3

4

1

3

2

4

1

3

4

2

3

1

4

2

3

4

1

2

4

3

1

2

slide26

Proof of stationarity:

1

2

3

4

1

3

2

4

1

3

4

2

3

1

4

2

3

4

1

2

4

3

1

2

4

3

2

1

slide27

So for USN:

Proof of stationarity:

1

2

3

4

1

3

2

4

1

3

4

2

3

1

4

2

3

4

1

2

4

3

1

2

4

3

2

1

(s1,...,sN) a (s2,...,sN,n-s1) is a bijection

from {sorting networks} to itself.

slide29

Scaled trajectory of particle i:

Ti:[0,1]![-1,1]

1

i

-1

1

0

slide30

Conjecture:

trajectories ! random Sine curves:

as n!1

(random)

Theorem:

scaled trajectories have subsequential

limits which are a.s. Hölder(½).

as n!1

slide32

projection of surface

measure on S2½R3 to R2

(unique circularly symmetric measure

with uniform linear projections;

on x2+y2<1 )

scaled permutation

matrix at time tN

Arch.

meas.

Conjecture:

scaled permutation

matrix at time N/2

Archimedes

measure

slide33

(1-½p3-e)n

Theorem:

scaled permutation matrix at time tN

is supported within a certain octagon w.h.p.

as n!1

slide34

Some Proofs:

  • Key tools:
  • Bijection (Edelman-Greene 1987)
  • {sorting networks} $ {standard staircase
  • Young tableaux}
  • New result for
  • profile of random staircase Young tableau
  • (deduced from Pittel-Romik 2006)
slide35

Staircase Young diagram:

(E.g. n=5)

n-1

N cells

slide36

Standard staircase Young tableau:

1

2

4

8

3

5

6

7

10

9

Fill with 1,L,N so each row/col increasing

slide37

Edelman-Greene algorithm:

1

2

4

8

3

5

6

7

10

9

1. Remove largest entry

slide38

Edelman-Greene algorithm:

1

2

4

8

3

5

6

7

9

1. Remove largest entry

slide39

Edelman-Greene algorithm:

1

2

4

8

3

5

6

7

9

2. Replace with larger of neighbours "Ã

slide40

Edelman-Greene algorithm:

1

2

4

8

3

5

6

7

9

2. Replace with larger of neighbours "Ã

slide41

Edelman-Greene algorithm:

1

2

4

8

5

6

3

7

9

2. Replace with larger of neighbours "Ã

...repeat

slide42

Edelman-Greene algorithm:

2

4

8

1

5

6

3

7

9

2. Replace with larger of neighbours "Ã

...repeat

slide43

Edelman-Greene algorithm:

0

2

4

8

1

5

6

3

7

9

3. Add 0 in top corner

slide44

Edelman-Greene algorithm:

1

3

5

9

2

6

7

4

8

10

4. Increment

slide45

Edelman-Greene algorithm:

1

3

5

9

2

6

7

4

8

5. Repeat everything...

slide46

Edelman-Greene algorithm:

1

3

5

9

2

6

7

8

4

5. Repeat everything...

slide47

Edelman-Greene algorithm:

1

3

5

9

6

7

2

8

4

5. Repeat everything...

slide48

Edelman-Greene algorithm:

3

5

9

6

7

1

2

8

4

5. Repeat everything...

slide49

Edelman-Greene algorithm:

0

3

5

9

6

7

1

2

8

4

5. Repeat everything...

slide50

Edelman-Greene algorithm:

1

4

6

10

7

8

2

3

9

5

5. Repeat everything...

slide51

Edelman-Greene algorithm:

1

4

6

7

8

2

3

9

5

5. Repeat everything...

slide52

Edelman-Greene algorithm:

1

2

5

7

8

9

3

4

10

6

5. Repeat everything...

slide53

Edelman-Greene algorithm:

1

2

5

7

8

9

3

4

6

5. Repeat everything...

slide56

Edelman-Greene Theorem:

After N steps,

45o

slide57

Edelman-Greene Theorem:

After N steps,

get swap process

of a sorting network!

1 1 1 1 4 4 4 4 5 5 5

2 2 4 4 1 1 5 5 4 4 4

3 4 2 2 2 5 1 2 2 2 3

4 3 3 5 5 2 2 1 1 3 2

5 5 5 3 3 3 3 3 3 1 1

slide58

Edelman-Greene Theorem:

After N steps,

get swap process

of a sorting network

And this is a bijection!

And can explicitly describe inverse!

slide59

Corollary (AHRV): For uniform random

staircase tableau, limiting shape is half

of this.

(Proof uses Greene-Nijenhuis-Wilf

Hook Walk)

Theorem (Pittel-Romik): For a uniform random

n x n square tableau, 9 limiting shape

with contours:

slide60

come from entries >(1-e)N in tableau:

bn

µ

an

entries exiting in [an,bn]

µ

# ¼ area under contour ¼ semicircle

Proof of LLN (swap process ) semic. x Leb.)

Swaps in space-time window [an,bn]x[0,eN]

slide61

# entries <k in 1st row

¸ longest % subseq. of swaps

by time k

¸ furthest any particle moves up

by time k

So can bound this using

limit shape.

Proof of octagon and Holder bounds

Inverse Edelman-Greene bijection

(¼ RSK algorithm) )

slide62

Why do we believe the conjectures?

Thepermutahedron:

embedding of Cayley graph

(Sn, n.n. swaps) in Rn :

sas-1=(s-1(1),...,s-1(n))2Rn

n=4:

embeds

in an

(n-2)-sphere

n=5

slide63

Theorem: If a non-random sorting network

  • lies close to some great circle on the
  • permutahedron, then:
  • Trajectories ¼ Sine curves
  • 2. Half-time permutation ¼ Archimedes
  • measure
  • 3. Swap process ¼ semicircle x Lebesgue
slide64

Theorem: If a non-random sorting network

  • lies close to some great circle on the
  • permutahedron, then:
  • Trajectories ¼ Sine curves
  • 2. Half-time permutation ¼ Archimedes
  • measure
  • 3. Swap process ¼ semicircle x Lebesgue

Simulations suggest

O(pn) for USN!

o(n) in | |1

slide65

Proof of Theorem:

close to great circle )

¼ Sine trajectories (up to a time change)

,¼ rotating disc

projections uniform )¼ Archimedes

swap rate uniform ) rotation uniform

) no time change

calculation ) semicircle law

slide66

Stretchable sorting network:

4

1

(, rotating disc)

2

3

2

3

1

4

USN not stretchable w.h.p. as n!1

Main conj. ) - USN “¼ stretchable” w.h.p.

- sub-network of k random

items is stretchable w.h.p.

slide67

1

...

n/2

n/2+1

...

n

n

...

n/2+1

n/2

...

1

USN

USN

N.B. Not every sorting network lies close to

a great circle! E.g. typical network through

n/2

...

1

n

...

n/2+1

(But this permutation is very unlikely).