68 research outputs found
Measured descent: A new embedding method for finite metrics
We devise a new embedding technique, which we call measured descent, based on
decomposing a metric space locally, at varying speeds, according to the density
of some probability measure. This provides a refined and unified framework for
the two primary methods of constructing Frechet embeddings for finite metrics,
due to [Bourgain, 1985] and [Rao, 1999]. We prove that any n-point metric space
(X,d) embeds in Hilbert space with distortion O(sqrt{alpha_X log n}), where
alpha_X is a geometric estimate on the decomposability of X. As an immediate
corollary, we obtain an O(sqrt{(log lambda_X) \log n}) distortion embedding,
where \lambda_X is the doubling constant of X. Since \lambda_X\le n, this
result recovers Bourgain's theorem, but when the metric X is, in a sense,
``low-dimensional,'' improved bounds are achieved.
Our embeddings are volume-respecting for subsets of arbitrary size. One
consequence is the existence of (k, O(log n)) volume-respecting embeddings for
all 1 \leq k \leq n, which is the best possible, and answers positively a
question posed by U. Feige. Our techniques are also used to answer positively a
question of Y. Rabinovich, showing that any weighted n-point planar graph
embeds in l_\infty^{O(log n)} with O(1) distortion. The O(log n) bound on the
dimension is optimal, and improves upon the previously known bound of O((log
n)^2).Comment: 17 pages. No figures. Appeared in FOCS '04. To appeaer in Geometric &
Functional Analysis. This version fixes a subtle error in Section 2.
On the optimality of gluing over scales
We show that for every , there exist -point metric spaces
(X,d) where every "scale" admits a Euclidean embedding with distortion at most
, but the whole space requires distortion at least . This shows that the scale-gluing lemma [Lee, SODA 2005] is tight,
and disproves a conjecture stated there. This matching upper bound was known to
be tight at both endpoints, i.e. when and , but nowhere in between.
More specifically, we exhibit -point spaces with doubling constant
requiring Euclidean distortion ,
which also shows that the technique of "measured descent" [Krauthgamer, et.
al., Geometric and Functional Analysis] is optimal. We extend this to obtain a
similar tight result for spaces with .Comment: minor revision
Metric structures in L_1: Dimension, snowflakes, and average distortion
We study the metric properties of finite subsets of L_1. The analysis of such
metrics is central to a number of important algorithmic problems involving the
cut structure of weighted graphs, including the Sparsest Cut Problem, one of
the most compelling open problems in the field of approximation algorithms.
Additionally, many open questions in geometric non-linear functional analysis
involve the properties of finite subsets of L_1.Comment: 9 pages, 1 figure. To appear in European Journal of Combinatorics.
Preliminary version appeared in LATIN '0
Vertex Sparsifiers: New Results from Old Techniques
Given a capacitated graph and a set of terminals ,
how should we produce a graph only on the terminals so that every
(multicommodity) flow between the terminals in could be supported in
with low congestion, and vice versa? (Such a graph is called a
flow-sparsifier for .) What if we want to be a "simple" graph? What if
we allow to be a convex combination of simple graphs?
Improving on results of Moitra [FOCS 2009] and Leighton and Moitra [STOC
2010], we give efficient algorithms for constructing: (a) a flow-sparsifier
that maintains congestion up to a factor of , where , (b) a convex combination of trees over the terminals that maintains
congestion up to a factor of , and (c) for a planar graph , a
convex combination of planar graphs that maintains congestion up to a constant
factor. This requires us to give a new algorithm for the 0-extension problem,
the first one in which the preimages of each terminal are connected in .
Moreover, this result extends to minor-closed families of graphs.
Our improved bounds immediately imply improved approximation guarantees for
several terminal-based cut and ordering problems.Comment: An extended abstract appears in the 13th International Workshop on
Approximation Algorithms for Combinatorial Optimization Problems (APPROX),
2010. Final version to appear in SIAM J. Computin
Maximum gradient embeddings and monotone clustering
Let (X,d_X) be an n-point metric space. We show that there exists a
distribution D over non-contractive embeddings into trees f:X-->T such that for
every x in X, the expectation with respect to D of the maximum over y in X of
the ratio d_T(f(x),f(y)) / d_X(x,y) is at most C (log n)^2, where C is a
universal constant. Conversely we show that the above quadratic dependence on
log n cannot be improved in general. Such embeddings, which we call maximum
gradient embeddings, yield a framework for the design of approximation
algorithms for a wide range of clustering problems with monotone costs,
including fault-tolerant versions of k-median and facility location.Comment: 25 pages, 2 figures. Final version, minor revision of the previous
one. To appear in "Combinatorica
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