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Multi-Embedding and Path Approximation of Metric Spaces Yair Bartal and Manor Mendel [email protected]

The Hebrew University

Multi-Embedding and Path Approximation of Metric Spaces – p.1/4




 

is



in

: a “host" space.






for any






  



 

 








is called non-contractive if 



An embedding of








: a finite metric space.



 



Metric Embedding









  

 

 

!"  #

 

  

 

The distortion of non-contractive embedding is

Multi-Embedding and Path Approximation of Metric Spaces – p.2/4

Algorithmic Paradigm $




for metrics in

.

*

$

.

/

 

is “easy".













 %0

- , .



 %



,

Algorithm for metric : On input : Embeds . Apply on

with

  



*

, an algorithm to solve 

Suppose

+ 

Suppose

+ - , .

*

: a class of metric spaces for which

$

)

'(

&

%

&

 %



$

: An algorithmic minimization problem. : Instance of . Contains a metric . : a feasible solution of . is a linear combination of the distances.

. Multi-Embedding and Path Approximation of Metric Spaces – p.3/4

)

'(



%0  

76

5

%0  

- ,

1 )



'(

76

5



% 

)

43/2



%




 %



'(

76

5

%0  

1) 

432

%  0

- ,

 *










 %0

, and

%

76

5

1 )

'(

'(




,

1) 

'(

Prop.: Suppose for any

/3

432





, and

432



%

,

)

'(

Then, for any



)

 %0

Proof. Let '(

Algorithmic Paradigm ,

,

. Then

Multi-Embedding and Path Approximation of Metric Spaces – p.4/4

>

?=: ;

>

@=: ;

Ultrametrics.

#



8 

>

>A

>

0

! =?: ;

B;

>A

?=: ;

0

@=: ;

9

8

Caveat: The -point distorcycle has tion when embedded into a tree metric.

Example of an ultrametric Multi-Embedding and Path Approximation of Metric Spaces – p.5/4

E/






*



E 

, where

. .

Theorem [B] Any

8J ( K (J K

(J K

I

8

-point metric has a probabilistic embedding into ultrametrics with distortion Furthermore, the distribution can be sampled efficiently.

8 

E / E

F

G

,

is non-contractive. H

E 





*

E   E E/

E

is an embedding into a metric

 0

Def. [AKPW, B] A probabilistic embedding



Probabilistic Embedding

.

It has many algorithmic applications. . .

Multi-Embedding and Path Approximation of Metric Spaces – p.6/4

Paradigm for Prob’ Embedding for metric

:



,

A randomized Algorithm

*



E  

.




0



N(

.

/

 

'(

  

432

%0 


 %0

1 )

*  %  0

,

 

) N(

% 

23

/

'(

 Q

, )

'(

%


OP

 %

M



probabilistic, ,








E

0

, and

M

.

- ,

1 )



+



*

'(

Prop.: Suppose that

and Then,






on

 %0

Apply

- , L

Sample an embedding

E

:

 %

On input

Multi-Embedding and Path Approximation of Metric Spaces – p.7/4

Applications of Prob’ Embedding Probabilistic embedding has found many applications for approximation algorithms, online algorithms, and distributed algorithms. Examples: Group Steiner tree [Garg, Konjevod, Ravi]. Metrical task systems [Bartal, Blum, Burch, Tomkins] [Fiat, M] Metric labeling [Kleinberg, Tardos] Clustering [Bartal, Charikar, Raz] ...

Multi-Embedding and Path Approximation of Metric Spaces – p.8/4

In This Talk Stronger notion

special metrics

Probabilistic Embedding

Improved embedding

Weaker Notion

A weaker notion of embedding. Useful for some algorithmic applications. Sometimes has a better “distortion". Multi-Embedding and Path Approximation of Metric Spaces – p.9/4

Motivation Weaker notions of embeddings may be of interest when: 1. There are algorithmic problems for which they make for feasible reductions. Examples: group Steiner problem, metrical task systems. 2. They provide reduced “overhead" (distortion), at least for some interesting metrics. Examples: expanders, low diameter graphs. 3. The constructions are much simpler than those for probabilistic approximations. 4. They are entertaining.

Multi-Embedding and Path Approximation of Metric Spaces – p.10/4

Multi Embedding 







V

UT

 SR











maps a point into a subset . The inverse mapping, , is a function.

 

 








 

 

V
 T

T 





V




M





The non-contraction property:

M H \ 8 

( ]^J

N

8 

We require

W

8

 Y

Y

[ZY



XY  

8 

W

The blow-up:

Multi-Embedding and Path Approximation of Metric Spaces – p.11/4

Path Approximation 

_ 
Ea fE

V

a

. For any path p

There exists path p’

E





e

Its length:

sequence of points in

_ 





V ab

a

c d

a

a`

“Path":



E 0 E d



M H

e

/3

e

_ 

E

a

_ 0 

a

E 0  

V

UT





a`

_  0

+



E d

E

_

M

a`

/



A multi-embedding is called -path approximation if , s.t. and

Multi-Embedding and Path Approximation of Metric Spaces – p.12/4

Metrics of Expander graphs g e h

Constant degree expanders are badly embeddable in . D/ 2

D/ 2

D

D

D

D



k

8

j



I



k

H

I

i



j



8

i

Prop. : -vertex unweighted graph, maximal degree , diameter . Then has path-approx’ by a tree with blow-up of .

j

j

i

38



I

Proof. There are only paths of length in . Put them all in one metric space with pairwise distance of .

Multi-Embedding and Path Approximation of Metric Spaces – p.13/4

l q

n

Rm


 p

.

r

Rq

R

l

,

. , satisfying

is connected. v

2.

p

1.

and a collection of

o






Instance: A metric space subsets (“groups") of points Feasible sol’: A graph



Group Steiner Problem (GSP)



w

x

y

p 





t

s '(

Minimize:

. z

u

,

m 

DsM

3.

.

8 

mY

(J K

8 

8J ( K (J K

(J b K

mY

poly-time

I J( K Y

Using probab’-approx’: GSP on

-point metric space has

8

I J( K Y

8

Thm. [GKR]: GSP on -point tree metrics has poly-time approx’ alg’.

approx’ alg’. Multi-Embedding and Path Approximation of Metric Spaces – p.14/4

Reduction via Path Approx’ Given *

m

:

m 


 m0 



. .

T { m0

implies that

is a feasible solution for




m

The definition of

. p

p

Return

p

V

- ,

.




for GSP instance

.

\

,

Z  s

path-approx’ of

s



X {

Apply

p0

Let

 m0

2

/

We construct: An -approx’ alg’

/

*

,







2

- ,

, an -approx’ alg’ to solve GSP for metrics in

. Multi-Embedding and Path Approximation of Metric Spaces – p.15/4

&




76

5

 2

Let

& 2

'(



n /

'( /

_  

. 

_



in

2

 '( 2

'(



p 

_ 0 

path approx’ of

p0 

be an

/

&

_

/

'(



n

p 

.

an Euler tour of .

 '(

.

'(

Let

_ 0

 &

Let Claim. Proof.

m

Analysis of the Reduction

Multi-Embedding and Path Approximation of Metric Spaces – p.16/4

GSP on Expanders D/ 2

D/ 2

D



mY

has

I J( K Y

Prop’. GSP on metrics of the type: approximation algorithms.

D

D

D

Proof. Two cases: j

The optimal solution is inside one -path: It is an interval in that path, and therefore easy to find.



j

The optimal solution spans more than one path: Its cost is dominated by the inter-distances between paths. ), therefore It is an These distances are all equal ( instance of the Hitting Set Problem. Multi-Embedding and Path Approximation of Metric Spaces – p.17/4

GSP on Expanders 

mY

I J( K Y

Corollary. GSP on constant degree expander graphs has approx’ alg’. poly-time H

I



This is almost optimal, since expanders contains large distortion from equilateral space. subset with

.

8

to improve the approximation factor below

mY J( b K

(J K Y

Perspective: using probabilistic embedding, it’s unclear how

Multi-Embedding and Path Approximation of Metric Spaces – p.18/4

€ }ˆ‡† ‚ƒ€}~ Š‹‰ „ „ … B ^…B ; ;

DC# > DC# >

Multi-embedding into Ultrametrics

 (

K (J K

Œ

\



(J K ŒJ


8

K (J K

( JX K 8J (

‘ 

I

8 ŽV

8

Œ



|

Def’. the aspect ratio of metric space: Thm. Any -point metric space with a.r. , has with path-approx’ at multi-embedding into UM of size most

Remarks: Œ

9

Œ

The dependence on is much better than in probab’-embedding, for which it is .

on . /

, and

9 J( K Œ

8 

(J K

There are lower bounds of

9

The construction and its analysis are much simpler than for probab’-embedding.

Multi-Embedding and Path Approximation of Metric Spaces – p.19/4

Probabilistic Multi Embedding It is possible to combine multi-embedding with probab’-embedding. 

8J ( K (J K (J K

(J K

I

8 

8

 V Ž

8

Thm. Any -point metric space has probabilistic multi , for which the embedding to spaces of size at most . path-approx’ is at most

8 

(J K

There is a lower bound of of this type of embedding.

9

We thus obtain a slight improvement w.r.t. approx’ factor for these problems.

8

The reductions for MTS and GSP also hold for this type of embedding. in the

on the path-approx’

Multi-Embedding and Path Approximation of Metric Spaces – p.20/4

Multi-embedding into Ultrametrics 

Πb 


8J ( K 8 Œ ” \

( JX K

V

 

’

I

8



’

“

Thm. Let . Any point metric path space has points, approximation by a UM with and Proof:

b

B

š›

\

–—

a

into

A

∆

’

˜

Partition the diameter equal width shells.

™




’ (J K ˜

8 

(J K

‘ 

 •

X

–—

S

S= A (intersection) B ∆

&

Pick one shell , and duplicate it.

W

,

Construct recursively UMs for the inner shells , and for the outer shells .

B

™

Join them with a new root labelled with .

A

a

S’

S"

b

Multi-Embedding and Path Approximation of Metric Spaces – p.21/4

Summary Definition of a metric multi-embedding. Has very low “distortion" for expanders. Applicable to MTS and GSP. Improves on probab’-embedding into UM. May have very low “distortion" embedding into trees.

Multi-Embedding and Path Approximation of Metric Spaces – p.22/4

Open Problems What is the trade off between the blow-up and the path-approx’ in multi-embeddings into trees. More applications. Tight bounds on [probabilistic] path-approx’ into UM. Is probab’ multi embedding really necessary? Other types of “embeddings" or “distortions".

Multi-Embedding and Path Approximation of Metric Spaces – p.23/4