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
$
)
'(
&
%
&
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$
: 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
)
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, and
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1)
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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
>
?=: ;
>
<=: ;
Host Spaces
Tree metrics.
>
@=: ;
Ultrametrics.
#
8
>
>A
>
0
<=: ;
0
DC# > ! =?: ;
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
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probabilistic, ,
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Prop.: Suppose that
and Then,
on
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Apply
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Sample an embedding
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:
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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
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a
c d
a
a`
“Path":
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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)
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x
y
p
t
s '(
Minimize:
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u
,
m
DsM
3.
.
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mY
(J K
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mY
poly-time
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Using probab’-approx’: GSP on
-point metric space has
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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
&
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2
Let
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in
2
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p
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path approx’ of
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be an
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&
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n
p
.
an Euler tour of .
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.
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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
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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
} }~
B ^
B ; ;
DC# > DC# >
Multi-embedding into Ultrametrics
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\
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K (J K
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|
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:
9
The dependence on is much better than in probab’-embedding, for which it is .
on . /
, and
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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.
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8
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Thm. Any -point metric space has probabilistic multi , for which the embedding to spaces of size at most . path-approx’ is at most
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(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
b
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Thm. Let . Any point metric path space has points, approximation by a UM with and Proof:
b
B
\
a
into
A
∆
Partition the diameter equal width shells.
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(J K
X
S
S= A (intersection) B ∆
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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
