1. algorithms on negatively curved spaces James R. Lee University of Washington Robert Krauthgamer IBM Research (Almaden) TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A A A A

2. why negative curvature? - Extensive theory of computational geometry in R d. What about other classical geometries? (e.g. hyperbolic) Eppstein: Is there an analogue of Arora’s TSP alg for H 2 ? - Class of “low-dimensional” spaces with exponential volume growth, in contrast with other notions of “intrinsic” dimension (e.g. doubling spaces) - Natural family of spaces that seem to arise in applied settings (e.g. networking, vision, databases) Modeling internet topology [ST’04], genomic data [BW’05] Similarity between 2-D objects (non-positive curvature) [SM’04]

3. what’s negative curvature? Gromov  -hyperbolicity Aside: How do we represent our manifold…? For a metric space (X,d) with fixed basedpoint r 2 X, we define the Gromov product (x|y) = [d(x,r) + d(y,r) – d(x,y)]/2. [For a tree with root r, (x|y) = d(r, lca(x,y)).] r x y (x|y) (X,d) is said to be  -hyperbolic if, for every x,y,z 2 X, we have (x|y) ¸ min{(x|z), (y|z)} -  [A tree is 0-hyperbolic.]

4. what’s negative curvature? (geodesic spaces) Thin triangles A geodesic space is  -hyperbolic (for some  ) if and only if every geodesic triangle is  -thin (for some  ). z x y geodesics [x,y], [y,z], [x,z]  -thin: every point of [x,y] is within  of [y,z] [ [x,z] (and similarly for [y,z] and [x,z])

5. what’s negative curvature? (geodesic spaces) Exponential divergence of geodesics A geodesic space is  -hyperbolic (for some  ) if and only every pair of geodesics “diverges” at an exponential rate. z x y threshold t=t0t=t0 t=t1t=t1 P length(P) ¸ exp(t 1 -t 0 )

6. results Make various assumptions on the space locally - locally doubling (every small ball has poly volume growth) - locally Euclidean (every small ball embeds in R k for some k) and globally - geodesic (every pair of points connected by a path) -  -hyperbolic for some  ¸ 0 e.g. bounded degree hyperbolic graphs, simply connected manifolds with neg. sectional curvature (e.g. H k ), word hyperbolic groups Most algorithms are intrinsic in the sense that they only need access to a distance function d (not a particular representation of the points or geodesics, etc.)

7. results - Nearest neighbor search data structure with O(log n) query time, O(n 2 ) space - Linear-sized (1+  )-spanners, compact routing schemes, etc. - PTAS (approx. scheme) for TSP, and other Arora-type problems

8. random tesellations: how’s the view from infinity? Bonk and Schramm: If the space is locally nice (e.g. locally Euclidean or bounded degree graph), then  1 H 2 is doubling (poly volume growth) boundary at infinity  1 H 2 equivalence classes of geodesic rays emenating from the origin - Two rays are equivalent if they stay within bounded distance forever - Natural metric structure on  1 H 2

9. random tessellations: how’s the view from infinity? Use hierarchical random partitions of  1 X to construct random tessellations of X. Now let’s see how to use this for finding near-optimal TSP tours…

10. the approximate TSP algorithm Tree doubling ain’t gonna cut it… MSTOPT log n 1 n/2 differ by 2-o(1) factor

11. the approximate TSP algorithm tree of metric spaces: family of metric spaces glued together in a tree-like fashion metric spaces

12. the approximate TSP algorithm For every  > 0, and d ¸ 1, there exists a number D( ,d) such that every finite subset X µ H d admits a ( 1 +  )-embedding into a distribution over dominating trees of metric spaces where the constituent spaces admit each admit an embedding into R d with distortion D( ,d). T HEOREM.

13. the approximate TSP algorithm For every  > 0, and d ¸ 1, there exists a number D( ,d) such that every finite subset X µ H d admits a ( 1 +  )-embedding into a distribution over dominating trees of metric spaces where the constituent spaces admit each admit an embedding into R d with distortion D( ,d). T HEOREM. - In other words, we have a random map f : X ! T({X i }) where T({X i }) is a random tree of metric spaces with induced metric d T whose constituent spaces are the {X i }. - For every x,y 2 X we have d T (f(x),f(y)) ¸ d(x,y). - For every x,y 2 X we have

14. the approximate TSP algorithm A LGORITHM. - Sample a random map f : X ! T({X 1, X 2, …, X m }) - For each k = 1,2,…,m, use Arora’ to compute a near- optimal salesman tour for every distorted Euclidean piece X k. X - Output the induced tour on X.

15. open questions - Can these results be extended to non-positively curved manifolds? What about planar graphs (simply connected, 2-d manifolds)? - Can the NNS data structure be made dynamic? linear space? - Is there a PTAS for TSP in doubling spaces?