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?