volume distortion for subsets of Rn

James R. Lee

Institute for Advanced Study & University of Washington

Symposium on Computational Geometry, 2006; Sedona, AZ

graph bandwidth

Given a graph G=(V,E), we seek a permutation : V ! {1,2,...,n}.

The bandwidth of is

The bandwidth of G is bw(G) = min bw().

1 j=u nk=v

Goal: Efficiently compute an ordering for which bw() ¼ bw(G).

embeddings & projections

1) Embed G into a Euclidean space Rk (preserving distances)

2) Project onto a random line and take the induced linear ordering.

embeddings & projections

Analysis: Count the number of points that fall into an interval, and use this to upper bound the stretch.

Problem: Only have control on the expectation, but there could be costly correlations...

embeddings & projections

Analysis: Count the number of points that fall into an interval, and use this to upper bound the stretch.

Problem: Only have control on the expectation, but there could be costly correlations...

Feige gave an example where this approach will take a graph ofbandwidth 2, and with high probability yield a solution of bandwidth

embeddings & projections

Analysis: Count the number of points that fall into an interval, and use this to upper bound the stretch.

Feige gave an example where this approach will take a graph ofbandwidth 2, and with high probability yield a solution of bandwidth

embeddings & projections

Volume of a set of points controls the probabilitythey project close together.

Conditioned on the projection of the three redpoints, the projection of the blue point stillvaries proportional to the distance to the

affine hull of the red points(false, but essentially true in high dimensions)

volume distortion

This leads to a new notion of embedding [Feige 00], one which tries to maximize the volume of e.g. all triangles in the image.

non-expansive

We would like to get as close as possible to the best possiblevolume for all the triples in our metric space

(e.g. shortest path on our graph)

higher-dimensional distortion

Given a metric space (X,d), a number k, and a non-expansive mappingf : X ! Rm, we define the k-dimensional distortion of f as thesmallest number D such that...

(note: 1-dimensional distortion recovers the “standard” notion)

previous results

General n-point metric spaces

[Feige 97]

[Rao 99]

[Krauthgamer-L-Mendel-Naor 04]

Subsets of Euclidean spaces (important to analyze bandwidth SDP)

[Rao 99]

[Krauthgamer-Linial-Magen 03 Dunagan-Vempala 01]

New results:(k=3)

main result

Given an n-point subset X µ Rn, there exists a non-expansivemapping f : X ! Rn such that...

This embedding maximizes the volume of every k-point subset withinfactor ¼ (log n)k/2.

construction of the embedding

Three phases

1)Randomized reduction to a collection of polylog(n) easier problemsRandom partitions, random sampling, gluing via smooth bump functions,

measured descent, ... [KLMN, L, ALN]

construction of the embedding

2) Reduction to a continuous problem in the “right” dimensionDimension reduction (Johnson-Lindenstrauss) and

Kirszbraun’s extension theorem

Rk

Lipschitz extension problem: Given S µ X and a non-expansive map f : S ! Rk, does there exist an non-expansive extension f : X ! Rk ?

Answer: Yes if X is a subset of Euclidean space

kirszbraun’s theorem

construction of the embedding

3) Solution of the continuous problem

If you think of Fd(x) as being a real-valued random variable forevery x, then we are saying that

(standard deviation)

construction of the embedding

If you think of Fd(x) as being a real-valued random variable forevery x, then we are saying that

xQ

y

open problems

– remove the O(log log n) terms (here and SparsestCut) / simplify analysis

– improve the approximation ratio for bandwidth best known is ¼ O(log n)3 [Feige, Dunagan-Vempala] best known for trees is O(log n)2.5 [Gupta] conjectured optimal bound: O(log n)