Approximation Algorithms for Low-Distortion Embeddings into the Line/Plane

1. Approximation Algorithms for Low-Distortion Embeddings into the Line/Plane Piotr Indyk (MIT) With: Mihai Badoiu, Julia Chuzhoy, Kedar Dhamdhere, Anupam Gupta, Harald Raecke, Yuri Rabinovich, R. Ravi, Tasos Sidiropoulos

2. Low-Distortion Embeddings • Consider metrics (X,DX) and (Y,DY) • (X,DX)c-embeds into (Y,DY) if there is a mapping f:XY such that, for all p,q  X : DX(p,q) ≤ DY( f(p),f(q) ) ≤ cDX(p,q) • Example: • (X,DX): subset of (R2,l2) • (Y,DY) = (R, l2)

3. Embeddings • Absolute bounds: for a metric M’ and a class of metrics C, show that for every MC, Mc-embeds into M’ • E.g., any n-point metric log n -embeds into l2 [Bourgain’84] “Embeddings for Approximation Algorithms” • Relative bounds: give an algorithm that, given MC as an input: • if Mc-embeds into M’, • then it finds an (a*c)-embedding of M into M’ “Approximation Algorithms for Embeddings”

4. Why? • Theory: • Relative bounds meaningful even if worst-case distortion large. • E.g., embedding metrics into R1 • Every absolute embedding problem has its relative counterpart • Practice: Multi-Dimensional Scaling, ISOMAP, LLE • Iterative algorithms • Existence of “non-global minimizers” is an issue

5. Related work (pre 2004) • Absolute bound implies relative bound (since distortion ≥1). In particular, every metric is n-embeddable into the line [Matousek’90] • Smallest distortion embedding of any metric into l2[Linial,London,Rabinovich’94] • O(log n)-approximate algorithm for embedding un-weighted graph metrics into sub-trees [Emek,Peleg’04] • Variations: • Additive distortion: Victor Chepoi’s talk • Average distortion: • [Dhamdhere,Gupta, Ravi’04] [Badoiu,Indyk,Rabinovich]: min distance sum for non-contracting mappings into the line (constant factor) • One-to-one mappings • [Kenyon,Rabani,Sinclair’04]: exact algorithms for distortion =O(1)

6. Results or RO(1)

7. Sphere  Plane O(c)

8. Sphere  Plane • Given X  S2, |X|=n, approximate the min distortion of f: X  R2 • The distortion could be Ω(n1/2) • Take 1/n1/2 -net of S2

9. Algorithm • Find largest empty cap B(p,r) • “Cut” B(p,r) • “Unwrap” the sphere • Distortion: O(1/r)

10. Analysis – Lower bound • The set X is an r-net of S2 • Consider optimal f: X  R2, assume non-expansion • Extendf to (non-expanding) g: S2  R2 • Borsuk-Ulam: there exist antipodal p,q for which g(p)=g(q) • There exists p’,q’ X with |p’-p|≤r,|q-q’| ≤r

11. Lower bound ctd p p’ g(q’) g(q) g(p) g(p’) q’ q • Distortion is at least ||p’-q’|| / ||g(p’)-g(q’)||  (2-2r)/2r = Ω(1/r)

12. Unweighted graphs  line O(c2)

13. Unweighted graphs  line a • Given G=(V,E) inducing (V,D) • Find a mapping f:VR • Non-contracting • Minimizes expansion • Similar problem - bandwidth: • Minimize expansion of the mapping f • Ignore contraction (but |f(x)-f(y)|  1) • Comb: • Bandwidth: a • Distortion: b • Distortion from by bandwidth order: ab b

14. Embeddings into a line • Intuition: • Assume we want to embed an “almost line metric” induced by (V,E) • Metric should be “long and thin” • Distances from one endpoint should be a good approximation of the embedding 1 0 2 3 3 4 4 5 6 7 8 9

15. Algorithm • Assume optimal embedding f:V  R • Guess: • v0= leftmost node in f(V) • vL = rightmost node in f(V) • Compute the shortest path P=v0, v1, … vL from v0 to vL • Vi = {vV: D(v,vi)=D(v,P) } Vi vL vi v0

16. Algorithm ctd. Vi • Compute our embedding g: where each g(Vi) is a walk of a spanning tree of Vi • Implications: • Distortion of g is O(|Vi|) • Suffices to show that |Vi| is small g(V1) g(V2) …

17. Diameter of Vi Claim: For any xVi , D(x,vi) ≤ c/2. Assume D(x,vi) > c/2. Then: • |f(x)-f(vj)|  D(x,vj)  D(x,vi) > c/2 • |f(x)-f(vj+1)|  …  c/2 • |f(vj)-f(vj+1)| = | f(vj)-f(x)| + |f(x)-f(vj+1)| >c • But D(vj,vj+1)=1 - contradiction vi f(vj+1) f(vj) f(x)

18. Size of Vi Vi Claim: |S| ≤ diam(S)*c+1 • Otherwise maxvVi f(v) – minvVi f(v)  |Vi|-1 > diam(Vi)*c min f(v) max f(v)

19. Final result • Given an un-weighted graph metric that is c-embeddable into the line, the algorithm construct embedding with distortion O(c2) • Used together with O(n)-embedding gives a O(n1/2)-approximation

20. General metrics  line • The algorithm for un-weighted graphs fails

21. Ultra-metrics  Plane O(c3)

22. Ultra-metrics • Definitions: • Metric induced on leaves of a tree, where all leaves have the same depth • D(p,q) ≤ max [ D(p,r), D(r,q) ] • Property: • Ultra-metric 2-embeds into 2-HST (after scaling): • Edges at level i have weight 2i

23. Examples of embeddings 4 2 • Full 2-HST, each node has 4 children • Θ(log n) distortion • Full 2-HST, each node has 2 children • O(1) distortion … … 2 8 4

24. Distortion lower bound • Consider: • Node v at level i • Non-contracting embedding f • L(v)=L(v1 )U … U L(vl) • B(v)= UuL(v) B(f(u),2i) • Claim: The distortion is at least (Omega of) Diameter(B(v))/2i -2 ≥ |B(v)|1/2 /2i -2 v 2i …… vl v1

25. Volume lower bound v 2i We have: |B(v)| = ∑j |UuL(v) B(f(u),2i)| =∑j |B(vj)B(2i/2) | ≥ ∑j [|B(vj)|1/2 +( π 22(i-1) )1/2 )] …… vl v1 2 Minkowski sum: AB = {a+b: aA, bB} [Brunn-Minkowski]: |A  B|1/2≥ |A|1/2 +|B|1/2

26. Lower bound - example • Full 2-HST, each node has 4 children • For a vertex v at the i-th level we have Bi1/2 =|B(v)|1/2≥ 41/2 [ Bi-11/2 + π1/2 2(i-1) ] ≥ 2 [Bi-11/2 + 2i-1] • By induction: Bi1/2 ≥ i 2i • Distortion: i 2i / 2i • Distortion at least (log n)

27. Algorithm (sketch) • Each sub-tree is embedded into a square • Squares of the children are recursively packed into the parents’ square • Add DMZ to prevent contraction • Elongate while preserving volume • Ensure that elongations at different levels cancel out

28. Open problems • Essentially any embedding problem can be rephrased in the relative/approximation setting • Specific questions: • cO(1) distortion for general metrics into line • Better (constant?) factor for un-weighted graphs into line • NP-hardness of sphere to plane