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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
- What do local properties tell us about global properties ?
- Property of interest: embeddability in normed spaces

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

- 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 ?

Lower bound: Roadmap

- Constant degree expander
- High global distortion
- Subgraphs of expander are sparse
- Sparse graphs embed well
- Different metric on expander

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

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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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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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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- [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)

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/)/

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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)

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

- 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

- 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

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

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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

- 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

- 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

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