Tighter local versus global properties of metric spaces
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Tighter local versus global properties of metric spaces. Moses Charikar. Joint work with. Konstantin Makarychev. Yury Makarychev. Princeton University. Local versus Global. Local properties: properties of subsets Global properties: properties of entire set

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Tighter local versus global properties of metric spaces

Moses Charikar

Joint work with

Konstantin Makarychev

Yury Makarychev

Princeton University


Local versus global
Local versus Global

  • Local properties: properties of subsets

  • Global properties: properties of entire set

  • What do local properties tell us about global properties ?

  • Property of interest: embeddability in normed spaces


Motivations
Motivations

  • Natural mathematical question

  • Questions of similar flavor

    • Embedding into l2n

    • Characterization of tree metrics

    • Helly’s theorem

    • Ramsey theory

    • Graph minors work

      • minor exclusion is local property, what does it mean for entire graph

    • Property testing

      • infer properties of entire set from sample

  • Lift-and-project methods in optimization

    • Can guarantee local properties

    • Need guarantee on global property


Local versus global distortion
Local versus global distortion

  • Metric on n points

  • Property : Embeddability into l1

  • Dloc : distortion for embedding any subset of size k

  • Dglob : distortion for embedding entire metric

  • What is the relationship between Dloc and Dglob ?


Results

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Results


Lower bound roadmap
Lower bound: Roadmap

  • Constant degree expander

  • High global distortion

  • Subgraphs of expander are sparse

  • Sparse graphs embed well

  • Different metric on expander


Sparse graphs
Sparse graphs

  • G is  sparse if every subgraph on k vertices has at most k edges

  • G is -path decomposable if

    • every 2-connected subgraph H contains a path of length 

    • vertices of path have degree 2 in H

  • [ABLT]1+O(1/ ) sparse graph and girth ()  -path decomposable


Embedding sparse graphs

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Embedding sparse graphs

  • G: -path decomposable, L = /9,   1/L,(u,v) = 1-(1-)d(u,v)embeds into l1 with distortion 1+O(e-L)

  • Distribution on multicuts:

    • d(u,v)  L, Pr(u,v separated) = 1-(1-)d(u,v)

    • d(u,v) > L, Pr(u,v separated)  1-(1-)L

  • Distortion


Distribution on multicuts

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Distribution on multicuts

  • d(u,v)  L, Pr(u,v separated) = 1-(1-)d(u,v)

  • d(u,v) > L, Pr(u,v separated)  1-(1-)L

  • Can be done for path of length 3L(endpoints separated with probability 1)

  • Cut edges independently with probability 

  • Decisions for P1 and P3 not independent

  • By induction

    • G has a cut vertex

    • G has a path of length = 9L


Distribution on multicuts1

u

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Distribution on multicuts

  • G has cut vertex c

  • Sample multicuts independently in Si

  • Pr[u,v not separated] = Pr[u,c not separated] Pr[v,c not separated]= (1-)d(u,c) (1- )d(v,c) = (1- )d(u,v)

c

S1

S3

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Distribution on multicuts2
Distribution on multicuts

  • G has a path of length = 9L

  • Divide path into 3 parts P1, P2, P3

  • Sample multicuts independently in H,P1, P2, P3

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Expanders have sparse subgraphs

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Expanders have sparse subgraphs

  • [ABLT]3-regular expander, girth (log n), every subset of size k is sparse

    • (log(n/k)) path decomposable


Local versus global distortion1

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Local versus global distortion

  • Every embedding of (X,) into l1 requires distortion

  • Every subset of X of size k embeds into l1with distortion 1+

  • Expander from[ABLT] with new metric

  • (u,v) = 1-(1-)d(u,v)


Picking parameters
Picking parameters

  • 3-regular expander

  • Subset X of size k

  • H: vertices within distance  of X.|H|  k.3

  • Pick   (log(n/k)), so that log(n/k.3)  

  • H is  path decomposable

  • Metric (u,v) = 1-(1-)d(u,v)=c.log(1/)/


Bounding local distortion

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Bounding local distortion

  • Subset X of size k

  • H: vertices within distance  of X.

    • u,v  X

    • dH(u,v)  dG(u,v)

    • dH(u,v) = dG(u,v) if dG(u,v) 

  • H is  path decomposable

  • Embedding  of H into l1 :

    • dH(u,v)  L, ||(u)- (v)||1 = 1-(1-)d(u,v)

    • dH(u,v) > L, ||(u)- (v)||1  1-(1-)L

  • Embedding of (u,v) = 1-(1-)dG(u,v)

  • Distortion


Global distortion

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Global distortion

  • min distortion for embedding expander into l1 is(avg distance/length of edge)

  • Distortion


Isometric local embeddings

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Isometric local embeddings

  • Every subset of size k embeds isometrically into l1

  • Entire metric requires distortion

  • Modification of distortion 1+ distortion construction for =1/(k.log n)


Near isometric to isometric
Near-isometric to isometric

  • Metric space (X,)

  • M: ratio of largest to smallest distance

  • Every subset of (X,) size k embeds into l1 with distortion 1+1/(2kM)

  •  : smallest distance

  • Metric ’(u,v) = (u,v) + 

  • Every subset of (X,’) size k embeds isometrically into l1

    • Original embedding + almost uniform metric


Upper bound
Upper bound

  • Every size k subset of (X,d) embeddable into l1 with distortion D (X,d) embeddable into l1 with distortion O(D.log(n/k))

  • Sum of two embeddings

    • handle large and small distances separately


Upper bound overview
Upper bound: Overview

  • x  X, m = n/k

  • Rx,m = distance of m closest point to x

  • Pick subset S of size k

  • Every x  X within distance 2Rx,m of some point in S

  • First embedding:Distortion D embedding of S + random mapping of X to S

  • Second embedding: First log(n/k) scales of Bourgain’s embedding.


Hitting set construction

Rx,m

Ry,m

Hitting Set Construction

  • Subset S of size k, every x  X within distance 2Rx,m of some point in S

  • U = {B(x, Rx,m) : x  X }

  • Repeat

    • Pick ball of min radius in U

    • Delete balls that intersect chosen ball from U

  • S : centers of chosen balls

  • At least n/k balls deleted in each step  |S|  k

  • g : X  Sd(x,g(x))  2 Rx,m


Randomized clustering

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Randomized clustering

  • Random mapping f : X  X

  • d(x,f(x))  Rx,m(always)

  • [CKR, FRT]

  • Pick  R (0,1)

  • Pick random order of X

  • f(x) = min point in B(x, .Rx,m)


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  • Random mapping f : X  X

  • g : X  S, |S|=k, d(x,g(x))  2 Rx,m

  • h(x) = g(f(x))

  • d(x,h(x))  5 Rx,m(always)

  • E[d(h(x),h(y)]  O(log m) d(x,y)

  • E[d(h(x),h(y)]  d(x,y) – 5(Rx,m + Ry,m)


Embedding large scales
Embedding large scales

  • Every size k subset of (X,d) embeddable into lp with distortion D

  • Embedding  : X  lp

  • ||(x)- (y)||p  D·O(log m)·d(x,y)

  • ||(x)- (y)||p  d(x,y) – (7D+2)(Rx,m + Ry,m)


Bourgain s embedding for small scales
Bourgain’s embedding for small scales

  • metric space (X,d), any m

  • embedding  : X  lp

  • ||(x) - (y)||p  O(log m)·d(x,y)

  • ||(x) - (y)||p  min(d(x,y), Rx,m + Ry,m)

  • ||(x)- (y)||p  D·O(log m)·d(x,y)

  • ||(x)- (y)||p  d(x,y) – (7D+2)(Rx,m + Ry,m)


Conclusion and questions

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Conclusion and Questions

  • Almost tight connections between local and global distortion of finite metrics

  • Every subset of size k isometrically embeddable into l1 versus


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