Tighter local versus global properties of metric spaces

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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
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 ?
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Results
• 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.
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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)
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