Tighter local versus global properties of metric spaces

1. Tighter local versus global properties of metric spaces Moses Charikar Joint work with Konstantin Makarychev Yury Makarychev Princeton University

2. 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

3. 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

4. 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 ?

5. µ µ ¶ ¶ 2 ( = ) l l k µ ¶ ( = ) ( = ) k l ­ k 1 2 o g o g n n ( ( ( ( ( ) ( = ( = = ) ) ) ) ) ) l l l k k k ­ O O D D D o g n ~ o g n o o n g g n n ­ ­ ­ D ( = ) l l k ± 1 l o g o g o g n Results

6. Lower bound: Roadmap • Constant degree expander • High global distortion • Subgraphs of expander are sparse • Sparse graphs embed well • Different metric on expander

7. 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

8. 1 L ¡ ( ) ¹ O 1 + e = L ( ) 1 1 ¡ ¡ ¹ 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

9. P1 P2 P3 L L 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 • 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

10. u v 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 S2

11. 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 P1 P3 H

12. µ ¶ 1 O 1 + ( = ) l k o g n Expanders have sparse subgraphs • [ABLT]3-regular expander, girth (log n), every subset of size k is sparse • (log(n/k)) path decomposable

13. µ ¶ ( = ) l k o g n ­ ( = ) l ± 1 o g 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)

14. 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/)/

15. 1 L ¡ ( ) ¹ ± O 1 1 · + + e = L ( ) 1 1 ¡ ¡ ¹ 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

16. µ ¶ µ ¶ ( = ) l k 1 o g n ­ ­ = ( = ) l ± 1 ¹ o g Global distortion • min distortion for embedding expander into l1 is(avg distance/length of edge) • Distortion

17. µ ¶ l o g n ­ l l l k + o g o g n o g 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)

18. 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

19. 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

20. 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.

21. 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

22. ( ) d x y ; ( ( ) ( ) ) ( ) f 6 f l P O · £ r x y o g m = R R + x m y m ; ; 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)

23. ( ) d x y ; ( ( ) ( ) ) ( ) h 6 h l P O · £ r x y o g m = R R + x m y m ; ; • 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)

24. 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)

25. 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)

26. µ ¶ l o g n ( ( = ) ) l k O o g n ­ l l l k + o g o g n o g Conclusion and Questions • Almost tight connections between local and global distortion of finite metrics • Every subset of size k isometrically embeddable into l1 versus