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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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slide1
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
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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 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

P2

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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)
slide23
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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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