Ittai Abraham, Yair Bartal, Ofer Neiman The Hebrew University

1. Embedding Metrics into Ultrametrics and Graphs into Spanning Trees with Constant Average Distortion Ittai Abraham, Yair Bartal, Ofer Neiman The Hebrew University

2. Embedding Metric Spaces • Metric spaces MX=(X,dX), MY=(Y,dy) • Embedding is a function f : X→Y • For u,v in X, non-contracting embedding f :distf(u,v)= dy(f(u),f(v)) / dx(u,v) • Distortion : dist(f)=max{u,v  X} distf(u,v)

3. Two Schemes • Embedding a graph into a spanning tree of the graph. • Embedding a metric into an ultrametric Δ(A) • Metric on leaves of rooted labeled tree. • 0 ≤ Δ(D) ≤ Δ(B) ≤ Δ(A). • d(x,y) = Δ(lca(x,y)). d(x,y) = Δ(D). d(x,w) = Δ(B). d(w,z) = Δ(A). Δ(C) Δ(B) Given a weighted graph, the distance between 2 points is the length of the shortest path between them Δ(D) z w y x

4. Motivation • Simple and compact representation of a metric space. • Ultrametric embedding provides approximation algorithms to numerous NP-hard problems. • Constructing a spanning tree is a well studied network design objective.

5. Previous Results • For embedding n point metric into ultrametrics: • A single ultrametric/tree requires Θ(n) distortion. [Bartal 96/BLMN 03/HM 05/RR 95]. • Probabilistic embedding with Θ(log n) expected distortion. [Bartal 96,98,04, FRT 03] • Embedding into spanning trees: • Minimum Spanning Tree: n-1 distortion. • Probabilistic embedding with Õ(log2n) expected distortion. [EEST 05]

6. Average Distortion • Average distortion : • lq-distortion : • Any metric embeds intoHilbert spacewith constant average distortion [ABN 06]. • Any metric probabilistically embeds into ultrametrics with constant average distortion[ABN 05/06, CDGKS 05]. • Also:Simultaneously tight lq-distortion for all q. l∞-dist = distortion l1-dist = average distortion.

7. Our Results • An embedding of any n point metric into a single ultrametric. • An embedding of any graph on n vertices into a spanning tree of the graph. • Average distortion = O(1). • l2-distortion = • lq-distortion = Θ(n1-2/q), for 2<q≤∞

8. Embeddings with scaling distortion • Definition:f has scaling distortionα, if for everyε there exist at least pairs (u,v) such that distf(u,v) ≤α(ε). Thm: Every metric space embeds into an ultrametric and every graph has a spanning tree with scaling distortion • For ε=¼, ¾ of pairs • have distortion < c·2 • For ε=1/16, 15/16 of pairs • have distortion < c·4 • … • For ε=1/n2, all pairs • have distortion < c·n

9. Additional Result • Thm:Any graph probabilistically embeds into a distribution of spanning trees with expected scaling distortionÕ(log2(1/ε)). • Implies that the lq-distortion is bounded by O(1)for any fixed 1≤q<∞. • For q=∞ matches the [EEST 05] result.

10. Embedding into an ultrametric • Partition X into 2 sets X1, X2 • Create a root labeled Δ = diam(X). • The children of the root are created recursively on X1, X2 • Plan : show for all ε,at most ε fraction of distances are distorted “too much”. • Using induction, for all 0<ε≤1 simultaneously: Bε – distorted distances for current level and ε. X X1 X2 A separated pair (x,y) is distorted “too much” if Δ X1 X2 | Bε|≤ ε|X1||X2|

11. Partition Algorithm • Fix some point u, such that |B(u,Δ/2)|<n/2 fix a constant c = 1/150. • Goal: find r>0, define X1=B(u,r), X2=X\X1 . • Such that for all ε>0 : (the set of possible “bad” pairs) X2 X1 r u S1 A separated pair (x,y) is distorted if S2

12. Partition Algorithm • Let • Choose r from the interval • Claim 1: The interval is “sparse”, contains at most points. • Claim 2: Any r in the interval is good for all • Proof: • By the maximality of , • Clearly |S1|≤|X1|.

13. Small values of ε • Claim 3: There exists some r in the interval which is good for all simultaneously. • While there exists uncolored r in the interval which is “bad” for some : • Take uncolored ri with largest bad . • Color the segment of length around ri. r is bad for ε if letting X1=B(u,r) will imply |Bε|>ε|X1|·|X2| r3 r1 r2 u

14. Every point can be at most at 2 bad segments Small values of ε Bound on the length of all the bad segments • T = number of points in all bad segments. By claim 1 the interval contains at most points S2 S1 A bad segment contains at least points Otherwise |Bε|is bounded by r1 r2 u

15. Embedding into a Spanning Tree • The spanning tree is created by a hierarchical star decomposition that uses ideas from [EEST 05]. • The decomposition for ultrametrics is in the heart of the star decomposition. • Furthermore, the spanning tree construction requires some additional ideas.

16. Star Decomposition Apoint z is in the cone with radius r if d(z,x1)+d(x1,x0)-d(z,x0)≤r • Let R be the radius for x0. • Cut a central ball X0 with radius≈R/2. • While un-assigned points exist: • Let xi with a neighbor yi. • Apply decompose algorithm with cone-radius αkR. • (k=level of recursion). • Add edges (xi,yi) to the tree. • Continue recursively inside each cluster. x1 y1 x0 y2 x2

17. Cone-radius If u,v are separated then dT(u,v)<2rad(T[X]) • Cone-radius αkR = loss of 1/αk in distortion. • Tree radius blow-up = • EESTchose α=1/log n • To ensure small blow-up and scaling distortion take as long as • rad(X) decreases geometrically. • Work for all ε<εlim n = size of original metric Δ = radius of original metric x1 y1 x0 y2 x2 Reset the parameters and k when this fails

18. Conclusion • An scaling approximation of • Metrics by ultrametrics. • Graphs by spanning trees. • Implies constant approximation on average. • Implies l2-distortion. • A Õ(log2(1/ε)) scaling probabilistic approximation of graphs by a random spanning tree. • Implies constant lq-distortion for all fixed q<∞.