1. the strange geometries of computer science James R. Lee University of Washington TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A A A A X Y -X -Y

2. three geometries part I: intrinsic dimensionality and nearest-neighbor search [with Krauthgamer] part II: the heisenberg geometry and sparse cuts in graphs [with A. Naor] part III: negative curvature and traveling salesmen [with Krauthgamer]

3. three geometries part I: intrinsic dimensionality and nearest-neighbor search [with Krauthgamer] part II: the heisenberg geometry and sparse cuts in graphs [with Naor] part III: negative curvature and traveling salesmen [with Krauthgamer]

4. nearest-neighbor search The problem: We are given some set S of n points lying in a huge (possibly infinite) metric space (M, d). Want to preprocess S so that given a query q 2 M, we can efficiently locate the nearest point to q among those in S. (Other concerns: Insertion/deletion of points from S.) (M, d) q S Basic question: For general metric spaces, how efficient can a search algorithm be? For which spaces do efficient algorithms exist?

5. approximate NNS Want to preprocess S so that given a query q 2 M, we can efficiently locate a point a 2 S such that Problem: ( 1 +  )-NNS d(q,a) · ( 1 +  ) d(q,S) Most previous work focused on the special case where M = R d is equipped with some L p norm. Unnatural for many applications. We focus on general spaces and access the query using the distance function as an oracle, e.g. d(q, ¢ ). Application domain: General metric spaces (Ad-hoc networks, manifold data, databases, biological data, …)

6. a hard example Suppose that (S, d) is a uniform metric: d(x,y) = 1 8 x,y 2 S. S 1 1 1  Have to do exhaustive search. Same example works even if S is only a “near-uniform” metric. Need some way of deciding tractability. (lack of information)

7. bounding the geometry r = 5 Definition: A metric space (X,d) is doubling if there exists a constant such that every ball in X can be covered by balls of half the radius. If (X) is the minimum over all such, we define dim(X) = log 2 (X) [GKL03] (“scaled down” notion a la Assouad’83)

8. doubling dimension - dim( R d ) ¼ d (under any norm) - X µ Y ) dim(X) · dim(Y) - dim(X) · log|X| structural properties There exists a ( 1 +  )-NNS data structure using: SPACE: O(n) QUERY TIME: 4 dim(X) O(log n) + O(2/  ) dim(X) UPDATE TIME: O(log n) algorithmic properties [Cole-Gottlieb 06, HarPeled-Mendel 05, KLa 04, KLb 04]

9. nets and dimension A subset N µ X is called an  -net if 1.d(x,y) ¸  8 x  y 2 N 2.X µ [ y 2 N Ball(y,  ) (X,d) Easy lemma: If N is an  -net in X, then |N Å Ball(x,R)| · (2R/  ) dim(X) (bounds the size of “near-uniform” sets in X) x

10. basic algorithm The basic data structure is a sequence of progressively finer  -nets for  = 2 k, 2 k-1, … To make one step, we only need to decide between 4 dim(X) points. Every step reduces distance to the query by a factor of 2.

11. related results efficient NNS data structure $ dim(S) small if we want to answer any possible query from the ambient space M. (M, d) q S - Implementation called “Cover Trees” by BKL used in isomap and beats popular hierarchical search approaches (e.g. ball trees) - Used in many other contexts (e.g. internet routing, peer-to-peer networks, approximation algorithms...)

12. heisenberg geometry part I: intrinsic dimensionality and nearest-neighbor search [with Krauthgamer] part II: the heisenberg geometry and sparse cuts in graphs [with Naor] part III: negative curvature and traveling salesmen [with Krauthgamer]

13. sparsest cut: edge expansion Input: A graph G =( V,E). S E(S, S) For a cut (S,S) let E(S,S) denote the edges crossing the cut. The sparsity of S is the value The SPARSEST CUT problem is to find the cut which minimizes  (S). This problem is NP-hard, so we try to find approximately optimal cuts. (approximation algorithms)

14. semi-definite relaxation Leighton and Rao gave an O(log n) approximation based on Linear Programming/ Multi-commodity flows (1988). To do better, we look at semi-definite programs. integer formulation: SDP formulation: valid 0/1 constraint:

15. semi-definite relaxation add the valid “triangle inequality” constraints: 2 nd eigenvalue of the Laplacian of the graph G (spectral approach) for all u,v,w 2 V combines power of spectral and flow-based approaches final SDP:

16. squared L 2 distance: negative type metrics These constraints imply that the function d(u,v) = (x u -x v ) 2 is a metric (i.e. it satisfies the triangle inequality). These are called negative type or squared-L 2 metrics. Impose strong geometric constraints on the solution: ·   xuxu xvxv xwxw

17. SDP analysis results uniform version: [Arora-Rao-Vazirani 04] non-uniform version: [Arora-L-Naor 05] upper bounds lower bounds non-uniform version: uniform version: [Krauthgamer-Rabani 06] [Devanur-Khot-Saket-Vishnoi 06] Lower bounds (starting with Khot-Vishnoi) were hard because vectors satisfying these constraints are in short supply:

18. the heisenberg group Consider the group of upper triangular matrices with integer entries (under matrix multiplication) Let G( H 3 ) be the Cayley graph of H 3 ( Z ) with generators: X-XY-Y BUT…

19. the heisenberg group X-XY-Y BUT… X Y -X -Y

20. the heisenberg group X-XY-Y X Y -X -Y ALL MOVEMENT IS “INFINITESIMALLY” HORIZONTAL

21. the heisenberg group X-XY-Y The space G( H 3 ) is translation invariant and homogeneous but not isotropic. Using H 3 -analogues of Fourier analysis in the classical setting of R d, we are able to show (with A. Naor) that the shortest-path metric on G( H 3 ) is of negative type. We conjectured that G( H 3 ) does not embed in L 1, and this was proved recently by J. Cheeger and B. Kleiner

22. local rigidity in H 3 Cheeger and Kleiner (wrong attribution and informal version): If S µ H 3 is a set with “small perimeter”, then locally S looks like a vertical hyperplane in R 3. X Y

23. negatively curved spaces part I: intrinsic dimensionality and nearest-neighbor search [with Krauthgamer] part II: the heisenberg geometry and sparse cuts in graphs [with Naor] part III: negative curvature and traveling salesmen [with Krauthgamer]

24. 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]

25. 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.]

26. 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])

27. 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 )

28. 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 of our 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.)

29. what we can say… - 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 (with Krauthgamer)

30. 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 has small doubling dimension. 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

31. 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…

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

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

34. 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.

35. 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

36. 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.

37. conclusion part I: intrinsic dimensionality and nearest-neighbor search [with Krauthgamer] part II: the heisenberg geometry and sparse cuts in graphs [with Naor] part III: negative curvature and traveling salesmen [with Krauthgamer]

38. bounding the geometry Many attempts to study special classes of metric spaces, e.g. Clarkson’99, Karger-Ruhl’02, Hildrum, et al.’04. In the Euclidean case, i.e. M = R d, dimension is a central parameter. Want to impose the restriction that X “behaves like” a low-dimensional normed space: Natural idea: In R d, Vol(ball of radius r) ¼ r d [KR02] impose the following restriction: Suppose that for some constant K, |B(x,2r)| · K |B(x,r)| for all x 2 X, r ¸ 0. Problem: Not very robust. Let’s define dim KR (X) = log 2 (K). - Deletion of points can cause the dimension of a space to increase dramatically. - Insertion of even a single point can cause the dimension to increase dramatically. For a uniform grid in R 2, dim KR (X) = O( 1 ).

39. notions of dimension [KR02] impose the following restriction: Suppose that for some constant K, |B(x,2r)| · K |B(x,r)| for all x 2 X, r ¸ 0. Problem: Not very robust. Let’s define dim KR (X) = log 2 (K). - Deletion of points can cause the dimension of a space to increase dramatically. - Insertion of even a single point can cause the dimension to increase dramatically. For the submetric on red points, dim KR grows significantly…

40. notions of dimension For the submetric on red points, dim KR grows significantly… Want things like - dim(X) · dim(Y) when X µ Y Could define… dim(X) = inf X µ Y dim KR (Y) Hard to utilize an unknown ambient space. “Just want the blue points back.” How to restore uniformity?

41. notions of dimension For the submetric on red points, dim KR grows significantly… Nets (the dark points) How to restore uniformity? A subset N µ X is called an  -net if 1.d(x,y) ¸  8 x  y 2 N 2.X µ [ y 2 N B(y,  ) (a maximal  -separated subset of X) Want things like - dim(X) · dim(Y) when X µ Y

42. notions of dimension For the submetric on red points, dim KR grows significantly… Measures… How to restore uniformity? (associate a mass  (x) to every point x 2 X) Require:  (B(x,2r)) · K ¢  (B(x,r)) Want things like - dim(X) · dim(Y) when X µ Y

43. Notion of efficiency An n-point metric space (S, d) admits an efficient ( 1+  )- NNS scheme if any query against S can be answered in polylog(n) time using only poly(n) space. (After preprocessing) q S The model (unknown terrain) Only requirement: (S [ {q}, d) is a metric space. Efficiency is an intrinsic property of S.

44. results for (1+  )-NNS Notion of efficiency An n-point metric space (S, d) admits an efficient ( 1+  )- NNS scheme if any query against S can be answered in polylog(n) time using only poly(n) space. (After preprocessing) Results [KL04b] For every fixed  < 2/5, (S, d) admits an efficient ( 1+  )-NNS scheme if and only if dim(S) = O(log log n). For example, when dim(X) = O( 1 ), every ( 1 +  )-approximate NNS query can be answered in O(log n) time using O(n 2 ) space.

45. results for (1+  )-NNS Notion of efficiency An n-point metric space (S, d) admits an efficient ( 1+  )- NNS scheme if any query against S can be answered in polylog(n) time using only poly(n) space. (After preprocessing) Results [KL04b] For every fixed  < 2/5, (S, d) admits an efficient ( 1+  )-NNS scheme if and only if dim(S) = O(log log n). For sufficiently fine approximations, there exists a structural characterization. Phase transition When  = 2, there exist spaces (S, d) which admit an efficient ( 1+  )-NNS scheme even though dim(S) =  (log n).

46. results for (1+  )-NNS Are these algorithms actually efficient? Probably not, but… An approach using some of these techniques [KL03a] and implemented by some people who know what they’re doing [BKL04] is remarkably fast. It performs comparably with other popular algorithms [e.g. Clarkson SB data struture, ball trees], and sometimes much better. It speeds up isomap. Approximation algorithms [T’04, GD’05] The internet, routing [KSW’04, S’05, CGMZ’05] Nearest-neighbor search [BKL’04, HP’05, IN’05] Optimal linear extension operators for H d [L’05] Other applications: