Embedding Metrics into Embedding Metrics into Ultrametrics and Graphs Ultrametrics and Graphs into Spanning Trees with into Spanning Trees with

Constant Average DistortionConstant Average DistortionIttai Abraham, Yair Bartal, Ofer NeimanIttai Abraham, Yair Bartal, Ofer Neiman

The Hebrew UniversityThe Hebrew University

Embedding Metric SpacesEmbedding Metric Spaces

Metric spaces Metric spaces MMXX=(X,d=(X,dXX), M), MYY=(Y,d=(Y,dyy)) EmbeddingEmbedding is a function is a function f : Xf : X→→YY For For u,v u,v in in X, X, non-contracting embedding non-contracting embedding f :f :

distdistff(u,v)(u,v)= d= dyy(f(u),f(v)) / d(f(u),f(v)) / dxx(u,v) (u,v) Distortion Distortion : : dist(f)=dist(f)= maxmax{u,v {u,v X} X} dist distff(u,v)(u,v)

Two SchemesTwo Schemes

1.1. Embedding a graph into a Embedding a graph into a spanning treespanning tree of the graph.of the graph.

2.2. Embedding a metric into an Embedding a metric into an ultrametricultrametric

xyz

Δ(A)

Δ(B)Δ(C)

Δ(D)

w

Metric on leaves of rooted Metric on leaves of rooted labeled tree.labeled tree.

0 ≤ 0 ≤ ΔΔ((DD) ≤ ) ≤ ΔΔ((BB) ≤ ) ≤ ΔΔ(A).(A). dd((x,yx,y) = ) = ΔΔ((lcalca((x,yx,y)).)).

dd((x,yx,y) = ) = ΔΔ((DD).).

dd((x,wx,w) = ) = ΔΔ((BB).).

dd((w,zw,z) = ) = ΔΔ((AA).).

Given a weighted graph, the distance between 2 points is the length of the shortest path between them

MotivationMotivation

SimpleSimple and and compactcompact representation of a metric representation of a metric space.space.

Ultrametric embedding provides Ultrametric embedding provides approximation approximation algorithmsalgorithms to numerous NP-hard problems. to numerous NP-hard problems.

Constructing a spanning tree is a well studied Constructing a spanning tree is a well studied network design objectivenetwork design objective..

Previous ResultsPrevious Results

For embeddingFor embedding n n point metric into point metric into ultrametricsultrametrics: : A single ultrametric/tree requires A single ultrametric/tree requires ΘΘ(n)(n) distortion. distortion. [Bartal [Bartal

96/BLMN 03/HM 05/RR 95].96/BLMN 03/HM 05/RR 95]. Probabilistic embedding with Probabilistic embedding with ΘΘ(log n)(log n) expected expected

distortion. distortion. [Bartal 96,98,04, FRT 03][Bartal 96,98,04, FRT 03]

Embedding into Embedding into spanning treesspanning trees:: Minimum Spanning Tree: n-1 distortion.Minimum Spanning Tree: n-1 distortion. Probabilistic embedding with Probabilistic embedding with Õ(logÕ(log22n)n) expected expected

distortion. distortion. [EEST 05][EEST 05]

Average DistortionAverage Distortion Average distortion :Average distortion :

llqq-distortion :-distortion :

Any metric embeds intoAny metric embeds into Hilbert spaceHilbert space with with constant average distortion constant average distortion [ABN 06].[ABN 06].

Any metric Any metric probabilisticallyprobabilistically embeds into embeds into ultrametricsultrametrics with constant average distortion with constant average distortion [ABN 05/06, CDGKS 05].[ABN 05/06, CDGKS 05].

Also:Also: Simultaneously tightSimultaneously tight l lqq-distortion for all -distortion for all q.q.

Xvuf vudist

nfavgdist ),(

2

1

qqfq vudistfdist

1),(Ε

ll∞∞-dist = distortion-dist = distortion

ll11-dist = average distortion.-dist = average distortion.

Our ResultsOur Results

An embedding of any n point metric into a An embedding of any n point metric into a single single ultrametricultrametric..

An embedding of any graph on n vertices An embedding of any graph on n vertices into a into a spanning treespanning tree of the graph. of the graph. Average distortion = Average distortion = O(1).O(1). ll22-distortion = -distortion =

llqq-distortion = -distortion = ΘΘ(n(n1-2/q1-2/q), ), forfor 2<q 2<q≤∞≤∞

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Embeddings with scaling Embeddings with scaling distortiondistortion

DefinitionDefinition:: ff has has scaling distortionscaling distortion αα, , if for if for everyevery εε there exist at least pairs there exist at least pairs (u,v) (u,v) such that such that distdistff(u,v) (u,v) ≤≤ αα((εε).).

Thm: Every metric space embeds into an ultrametric and every graph has a spanning tree with scaling distortion 1O

• For ε=¼, ¾ of pairs have distortion < c·2• For ε=1/16, 15/16 of pairs have distortion < c·4…• For ε=1/n2, all pairs have distortion < c·n

12

132

4

11

4

11

001

Occavgdisti

ii

iii

21

n

Additional ResultAdditional Result Thm:Thm: Any graph probabilistically embeds into a Any graph probabilistically embeds into a

distribution of spanning trees with expected scaling distribution of spanning trees with expected scaling distortiondistortion Õ(logÕ(log22(1/(1/εε)).)).

Implies that theImplies that the l lqq-distortion is bounded-distortion is bounded by by O(1)O(1) for any for any

fixedfixed 1≤q<∞. 1≤q<∞. ForFor q=∞ q=∞ matches the matches the [EEST 05][EEST 05] result result..

Embedding into an ultrametricEmbedding into an ultrametric

Partition Partition X X into 2 sets into 2 sets XX11, X, X22

Create a root labeled Create a root labeled ΔΔ = = diamdiam((XX).). The children of the root are created The children of the root are created

recursively on recursively on XX11, X, X22

PlanPlan : show for all : show for all εε,, at most at most εε fraction fraction of distances are distorted “too much”.of distances are distorted “too much”.

Using induction, for all 0<Using induction, for all 0<ε≤ε≤1 1 simultaneously:simultaneously:

BBεε – distorted distances for current level – distorted distances for current level

and and εε..

XXXX11 XX22

Δ

XX11 XX22

222

21 XB

XX

| | BBεε |≤ |≤ εε|X|X11||X||X22||

A separated pair (x,y) is distorted “too much” if

cyxd ,

Partition AlgorithmPartition Algorithm

Fix some pointFix some point u, u, such that such that |B(u,|B(u,ΔΔ/2)|<n/2/2)|<n/2 fix a constant fix a constant c = 1/150. c = 1/150.

Goal:Goal: find find r>0r>0, define , define XX11=B(u,r), X=B(u,r), X22=X\X=X\X11 . . Such that for all Such that for all εε>0>0 : :

(the set of possible “bad” pairs)(the set of possible “bad” pairs)

2121 XXSS

uurrc2 crwudXwS ,:22

crwudXwS ,:11

XX11

SS11

SS22

XX22

A separated pair (x,y) is distorted if

cyxd ,

Partition AlgorithmPartition Algorithm

LetLet Choose Choose rr from the interval from the interval

Claim 1: Claim 1: The interval is “sparse”, contains at The interval is “sparse”, contains at most points. most points.

Claim 2: Claim 2: Any Any rr in the interval is good for all in the interval is good for all Proof:Proof:

By the maximality of , By the maximality of , Clearly Clearly |S|S11||≤≤|X|X11|.|.

2,4

nuB 4,:max

32n4

42,,2 uBcruBS

22 2 XnS

2

1

2 Xn

Xn

nuBuB 444,2,

Small values of Small values of εε

Claim 3:Claim 3: There exists some r in the interval which is good There exists some r in the interval which is good for all for all simultaneouslysimultaneously. .

While there exists uncolored While there exists uncolored rr in the interval which is “bad” in the interval which is “bad” for some :for some : Take uncolored Take uncolored rrii with largest bad . with largest bad .

Color the segment of length around Color the segment of length around rrii..

32

uu24 22 c

rr11

12 c

rr22

ic 2

32 32i

r is bad for ε if letting X1=B(u,r) will imply |Bε|>ε|X1|·|X2|

rr33

Small values of Small values of εε

T = number of points in all bad segments. T = number of points in all bad segments.

cct

ii 242

1

nTnt

ii 42

2

1

1

uu222 c

rr11

12 c

rr22

4

A bad segment contains at least points Otherwise |Bε |is bounded by

212 2 XXn

n2

By claim 1 the interval contains at most points

4

n4

Bound on the length of all the bad segments

121

t

ii

SS11 SS22

Every point can be at most at 2 bad segments

Embedding into a Spanning TreeEmbedding into a Spanning Tree

The spanning tree is created by a The spanning tree is created by a hierarchical hierarchical star decompositionstar decomposition that uses that uses ideas from ideas from [EEST 05].[EEST 05].

The decomposition for ultrametrics is in the The decomposition for ultrametrics is in the heart of the star decomposition.heart of the star decomposition.

Furthermore, the spanning tree Furthermore, the spanning tree construction requires some additional construction requires some additional ideas. ideas.

y2

x1 y1

Star DecompositionStar Decomposition

Let Let RR be the radius for be the radius for xx00..

Cut a central ball Cut a central ball XX00 with with

radiusradius≈≈R/R/22.. While un-assigned points While un-assigned points

exist:exist: LetLet x xii with a neighbor with a neighbor yyii.. Apply decompose algorithm Apply decompose algorithm

with cone-radius with cone-radius ααkkRR.. ((kk=level of recursion).=level of recursion).

Add edges Add edges (x(xii,y,yii)) to the tree. to the tree. Continue recursively inside Continue recursively inside

each cluster.each cluster.

x0

x2

A point z is in the cone with radius r ifd(z,x1)+d(x1,x0)-d(z,x0)≤r

y2

x1 y1

Cone-radiusCone-radius

Cone-radius Cone-radius ααkkRR = loss of = loss of

1/1/ααkk in distortion. in distortion. Tree radius blow-up =Tree radius blow-up = EESTEEST chose chose αα=1/log n=1/log n To ensure small blow-up To ensure small blow-up

and scaling distortion takeand scaling distortion take

as long asas long as rad(X)rad(X) decreases decreases

geometrically.geometrically. Work for all Work for all εε<<εεlimlim

x0

x2

k

k1

nXk lim

XnX rad

n = size of original metricΔ = radius of original metric

Reset the parameters and k when this fails

If u,v are separated thendT(u,v)<2rad(T[X])

ConclusionConclusion

An scaling approximation ofAn scaling approximation of Metrics by Metrics by ultrametricsultrametrics.. Graphs by Graphs by spanning treesspanning trees..

Implies constant approximation on average.Implies constant approximation on average. Implies Implies ll22-distortion. -distortion.

A A Õ(logÕ(log22(1/(1/εε)))) scaling probabilistic approximation scaling probabilistic approximation of graphs by a of graphs by a random spanning treerandom spanning tree..

Implies constant Implies constant llqq-distortion for all fixed -distortion for all fixed q<q<∞∞..

1

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