Metric Embeddings with Relaxed Guarantees

1. Metric Embeddings with Relaxed Guarantees Ofer Neiman Hebrew University Alex Slivkins Cornell University Joint work with Ittai Abraham, Yair Bartal, Hubert Chan, Kedar Dhamdhere, Anupam Gupta and Jon Kleinberg FOCS 2005

2. Estimating Internet latencies • Latency (round-trip time) is a notion of distance in Internet • distance matrix defined by latencies is almost a metric • Estimate latencies for most node pairs with low load on nodes • linear or near-linear #distance measurements • long line of papers in Systems:Guyton+’95, Francis+’01, Ng+’02, Pias+’03, Ng+’03, Lim+’03, Tang+’03, Lehman+’04, Dabek+’04, Costa+’04 • Setting: large overlay network in Internet • P2P network, file-sharing system, online computer game • Applications: nearest neighbors (servers, file replicas, gamers), diverse node set (fault-tolerant storage), low-stretch P2P routing

3. Estimate latencies via embedding Global Network Positioning (GNP) [Ng+Zhang’02] • select small #nodes as “beacons” • users measure latencies to beacons • embed into low-dim Euclidian space • embed the beacons first • embed non-beacons one by one • magic: 90% node pairs are embedded with relative error <.5 • 900 random nodes, 15 beacons, 7 dimensions • lots of follow-up work: NPS [Ng+ ’03], Vivaldi [Cox+ ’04],Lighthouses[Pias+ ’03], BigBang [Tankel+ ’03], ICS [Lim+ ’03],Virtual Landmarks [Tang+ ’03], PIC [Costa+ ’04], PALM [Lehman+ ’04]

4. How can we explain the magic? • How can we explain GNP theoretically? • assume that latencies form a metric • Well-studied problem: embed a metric into Euclidean space • e.g. any metric can be embedded with distortion O(log n) • all prior work assumes full access to distance matrix • Cannot estimate all distances with small #beacons  allow -slack: guarantees for all but -fraction of node pairs • recall: empirical results ofGNP have -slack, too • new angle for theoretical work on metric embeddings

5. Can we embed any metric with -slack and distortion f()? • any f() and any (non-beacon-based) embedding is non-trivial ? yes • This paper: extend the result of [KSW’04] to all metrics • we achieve f() = O(log 1/) and use only Õ(1/) beacons • (same) target dimension O(log2 1/) • matching lower bound on distortion (even without beacons) Beacon-based embeddings with slack • Thm[Kleinberg, Slivkins, Wexler FOCS’04]for any doubling metric and any >0: embeddinginto lp (p1) • distortion O(log 1/) with -slack • uses only Õ(1/) randomly selected beacons

6. Plan • Introduction • proof sketch • extension Ofer Neiman takes over and talks about: • lower bound theorem • open questions

7. Proof Sketch (1/3) • Beacon-based framework • randomly select small subset S of nodes as “beacons” • coordinates of node = fn(its distances to beacons) • Define a block of coordinates as follows: • for each i,j = 1, 2, ... , (log 1/)(i,j)-th coordinate of node u = distance from u to S(i,j) = { random subset of S of size 2i } • similar to the embedding in [Bourgain’85] which used n (#nodes) instead of 1/

8. short edge u u v v r(v)  n nodes Proof Sketch (2/3) • Def r(v) = min radius of ball around v that contains n nodes • edge (u,v) islong if d(u,v)  4 min[ r(u), r(v) ]short if d(u,v) < min[ r(u), r(v) ]medium-lengthotherwise long edge r(v) d(u,v)  n nodes

9. u v Proof Sketch (2/3) • Def r(v) = min radius of ball around v that contains n nodes • edge (u,v) islong if d(u,v)  4 min[ r(u), r(v) ]short if d(u,v) < min[ r(u), r(v) ]medium-lengthotherwise • Bourgain-style embedding handles long edges: • long edges shrunk by at most O(log 1/), no edge expanded •  n short edges  ignore • medium-length edges ?? • d(u,v)= (r(u)+r(v)) r(v) d(u,v)  n nodes

10. Bourgain new block u v Proof Sketch (3/3) • medium-length edges: d(u,v)= (r(u)+r(v)) • add another block of coordinates such that for any (u,v),the embedded (u,v)-distance in this block is (r(u)+r(v)) u u1 u2 u3 ... ??? r(v) d(u,v) v v1 v2 v3 ... ???  n nodes

11. u v u1 u2 u3 ... +r(u) +r(u) –r(u) v1 v2 v3 ... +r(v) +r(v) –r(v) Bourgain new block u v r(v) d(u,v)  n nodes Proof Sketch (3/3) • medium-length edges: d(u,v)= (r(u)+r(v)) • add another block of coordinates such that for any (u,v),the embedded (u,v)-distance in this block is (r(u)+r(v)) • how? each coordinate of u is r(u), sign chosen at random • Beacon-based solution: estimate r(u) from distances to beacons

12. Plan • Introduction • proof sketch • extension Ofer Neiman takes over and talks about: • lower bound theorem • open questions

13. decomposable metrics include doubling metrics and shortest-paths metrics of graphs excluding a fixed minor Extension: one embedding for all  • Thm for decomposable metrics, embedding into lp , p1: for any >0,distortion O(log 1/)1/p on all but -fraction of edges • graceful degradation: one embedding works for any ! • target dimension as small as O(log2 n) • extends a result from [KSW’04] on growth-constrained metrics

14. Extension: one embedding for all  • Thm for decomposable metrics, embedding into lp , p1: for any >0,distortion O(log 1/)1/p on all but -fraction of edges • graceful degradation: one embedding works for any ! • target dimension as small as O(log2 n) • extends a result from [KSW’04] on growth-constrained metrics • Can we extend this to all metrics?Partial result: embedding into l1 with distortion O(log 1/)and target dimension as large as O(n2)

15. Plan • Introduction • proof sketch • extension Ofer Neiman takes over and talks about: • lower bound theorem • open questions

16. Lower Bound Results • Given a family of metric spaces H, having lower bound D(n)on the distortion of a regular embedding into some family X, for some |H|=n. • We show lower bound of for ε-slack embedding of H into X.

17. Lower Bounds General idea: • Take a metric HєHwhich is “hard” for a regular embedding into some family X. • Replace every point in Hby a set, creating H’. • H’ contain many isomorphic copies of H. • In any ε-slack embedding into X, at least one copy will incur high distortion.

18. H’ H Cu Cv d u1 v1 d u v u2 v2 d uk vk Metric Composition • Choose and create a metric , for anyxєH. • Distances in Cx are bounded by δ.

19. Lower Bound Letf be any ε-slack embedding ignoring the edgesE, by definition|E|<εn2/2. Assume that distortion(f)=R

20. vx vy • For each xєH, pickvxєCx∩T. t T: set of vertices intersecting less than edges in E. Finding a copy Cy Cx in T • For each pair (vx , vy),find tєCy such that(vx , t), (vy , t)E in T

21. Main corollaries • distortion for ε-slack embedding into lp. • distortion for ε-slack embedding into trees. • distortion for randomized ε-slack embedding • into a distribution of trees. • distortion for ε-slack embedding of • doubling or l1metrics into l2.

22. distortion for slack embedding into a tree • metric. Follow-up work • gracefully degrading distortion into • lp , with dimension O(log(n)). implies constant average distortion!

23. Open problems • Gracefully degrading embedding into a tree with distortion • Algorithmic applications… • Volume respecting slack embeddings. • Finding more gracefully degrading embeddings (l1→l2 ??).

24. Extra: general embedding theorem Thm embedding into lp space (p1) with distortion D(n) any >0: embedding with distortion D(Õ(1/)) and -slack • for any family of finite metrics closed under taking subsets • beacon-based algorithm using Õ(1/) beacons • similar theorem about lower bounds Some applications: • decomposable metrics into lp: distortion O(log 1/)1/p • negative type metrics (e.g. l1) into l2: distortion Õ(log 1/)1/2

25. Extra: distortion with -slack Cannot estimate all distances with small #beacons • partition nodes into k equal-size clusters, k>#beacons;distance 0 within each cluster, 1 between clusters • in clusters without beacons, distance to every beacon is 1  no way to tell which node lies in which cluster • We allow guarantees for all but -fraction of node pairs • recall: empirical results ofGNP have -slack, too • new angle for theoretical work on metric embeddings