algorithms on negatively curved spaces james r. lee university of washington robert krauthgamer ibm...

algorithms on negatively curved spaces

James R. LeeUniversity of Washington

Robert KrauthgamerIBM Research (Almaden)

why negative curvature?

- Extensive theory of computational geometry in Rd. What about other classical geometries? (e.g. hyperbolic) Eppstein: Is there an analogue of Arora’s TSP alg for H2?

- 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]

what’s negative curvature?

Gromov -hyperbolicity

For a metric space (X,d) with fixed basedpoint r 2 X, we definethe 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.]

what’s negative curvature? (geodesic spaces)

Thin triangles

A geodesic space is -hyperbolic (for some ) if and onlyif 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])

what’s negative curvature? (geodesic spaces)

Exponential divergence of geodesics

A geodesic space is -hyperbolic (for some ) if and onlyevery pair of geodesics “diverges” at an exponential rate.

z

x

y

threshold

t=t0t=t1

P

length(P) ¸ exp(t1-t0)

results

Make various assumptions on the space locally - locally doubling (every small ball has poly volume growth) - locally Euclidean (every small ball embeds in Rk for some k)and globally - geodesic (every pair of points connected by a path) - -hyperbolic for some ¸ 0e.g. bounded degree hyperbolic graphs, simply connected manifoldswith neg. sectional curvature (e.g. Hk), word hyperbolic groups

Most algorithms are intrinsic in the sense that they only needaccess to a distance function d (not a particular representationof the points or geodesics, etc.)

results

- Nearest neighbor search data structure with O(log n) query time, O(n2) space

- Linear-sized (1+)-spanners, compact routing schemes, etc.

- PTAS (approx. scheme) for TSP, and other Arora-type problems

random tesellations: how’s the view from infinity?

Bonk and Schramm: If the space is locally nice (e.g. locally Euclidean orbounded degree graph), then 1H2 is doubling (poly volume growth)

boundary at infinity 1H2

equivalence classes of geodesic raysemenating from the origin

- Two rays are equivalent if they stay within bounded distance forever- Natural metric structure on 1H2

random tessellations: how’s the view from infinity?

Use hierarchical random partitions of 1X to construct random tessellations of X.

Now let’s see how to use this for finding near-optimal TSP tours…

the approximate TSP algorithm

Tree doubling ain’t gonna cut it…

MST OPT

log

n

1

n/2

log n

differ by 2-o(1) factor

the approximate TSP algorithm

tree of metric spaces:family of metric spaces gluedtogether in a tree-like fashion

metric spaces

the approximate TSP algorithm

For every >0, and d¸1, there exists a number D(,d) such that everyfinite subset X µ Hd admits a (1+)-embedding into a distributionover dominating trees of metric spaces where the constituent spaces admiteach admit an embedding into Rd with distortion D(,d).

THEOREM.

the approximate TSP algorithm

For every >0, and d¸1, there exists a number D(,d) such that everyfinite subset X µ Hd admits a (1+)-embedding into a distributionover dominating trees of metric spaces where the constituent spaces admiteach admit an embedding into Rd with distortion D(,d).

THEOREM.

- In other words, we have a random map f : X ! T({Xi}) where T({Xi}) is a random tree of metric spaces with induced metric dT whose constituent spaces are the {Xi}.

- For every x,y 2 X we have dT(f(x),f(y)) ¸ d(x,y).

- For every x,y 2 X we have

x2x2x2

E [dT (f (x); f (y))] · (1 + ")d(x;y)

the approximate TSP algorithm

ALGORITHM.

- Sample a random map f : X ! T({X1, X2, …, Xm})- For each k=1,2,…,m, use Arora’ to compute a near- optimal salesman tour for every distorted Euclidean piece Xk.

X

- Output the induced tour on X.

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?