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# Tighter local versus global properties of metric spaces - PowerPoint PPT Presentation

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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Presentation Transcript

Moses Charikar

Joint work with

Konstantin Makarychev

Yury Makarychev

Princeton University

• 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

• 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

• 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 ?

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Results

• Constant degree expander

• High global distortion

• Subgraphs of expander are sparse

• Sparse graphs embed well

• Different metric on expander

• 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

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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

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P3

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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

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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)

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S3

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• 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

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

• (log(n/k)) path decomposable

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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)

• 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/)/

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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

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Global distortion

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

• Distortion

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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)

• 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

• 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

• 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.

Rx,m

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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

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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)

• 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)

• 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)

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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