101 research outputs found
Multi-Embedding of Metric Spaces
Metric embedding has become a common technique in the design of algorithms.
Its applicability is often dependent on how high the embedding's distortion is.
For example, embedding finite metric space into trees may require linear
distortion as a function of its size. Using probabilistic metric embeddings,
the bound on the distortion reduces to logarithmic in the size.
We make a step in the direction of bypassing the lower bound on the
distortion in terms of the size of the metric. We define "multi-embeddings" of
metric spaces in which a point is mapped onto a set of points, while keeping
the target metric of polynomial size and preserving the distortion of paths.
The distortion obtained with such multi-embeddings into ultrametrics is at most
O(log Delta loglog Delta) where Delta is the aspect ratio of the metric. In
particular, for expander graphs, we are able to obtain constant distortion
embeddings into trees in contrast with the Omega(log n) lower bound for all
previous notions of embeddings.
We demonstrate the algorithmic application of the new embeddings for two
optimization problems: group Steiner tree and metrical task systems
Fat Polygonal Partitions with Applications to Visualization and Embeddings
Let be a rooted and weighted tree, where the weight of any node
is equal to the sum of the weights of its children. The popular Treemap
algorithm visualizes such a tree as a hierarchical partition of a square into
rectangles, where the area of the rectangle corresponding to any node in
is equal to the weight of that node. The aspect ratio of the
rectangles in such a rectangular partition necessarily depends on the weights
and can become arbitrarily high.
We introduce a new hierarchical partition scheme, called a polygonal
partition, which uses convex polygons rather than just rectangles. We present
two methods for constructing polygonal partitions, both having guarantees on
the worst-case aspect ratio of the constructed polygons; in particular, both
methods guarantee a bound on the aspect ratio that is independent of the
weights of the nodes.
We also consider rectangular partitions with slack, where the areas of the
rectangles may differ slightly from the weights of the corresponding nodes. We
show that this makes it possible to obtain partitions with constant aspect
ratio. This result generalizes to hyper-rectangular partitions in
. We use these partitions with slack for embedding ultrametrics
into -dimensional Euclidean space: we give a -approximation algorithm for embedding -point ultrametrics
into with minimum distortion, where denotes the spread
of the metric, i.e., the ratio between the largest and the smallest distance
between two points. The previously best-known approximation ratio for this
problem was polynomial in . This is the first algorithm for embedding a
non-trivial family of weighted-graph metrics into a space of constant dimension
that achieves polylogarithmic approximation ratio.Comment: 26 page
On metric Ramsey-type phenomena
The main question studied in this article may be viewed as a nonlinear
analogue of Dvoretzky's theorem in Banach space theory or as part of Ramsey
theory in combinatorics. Given a finite metric space on n points, we seek its
subspace of largest cardinality which can be embedded with a given distortion
in Hilbert space. We provide nearly tight upper and lower bounds on the
cardinality of this subspace in terms of n and the desired distortion. Our main
theorem states that for any epsilon>0, every n point metric space contains a
subset of size at least n^{1-\epsilon} which is embeddable in Hilbert space
with O(\frac{\log(1/\epsilon)}{\epsilon}) distortion. The bound on the
distortion is tight up to the log(1/\epsilon) factor. We further include a
comprehensive study of various other aspects of this problem.Comment: 67 pages, published versio
Online Embeddings
13th International Workshop, APPROX 2010, and 14th International Workshop, RANDOM 2010, Barcelona, Spain, September 1-3, 2010. ProceedingsWe initiate the study of on-line metric embeddings. In such an embedding we are given a sequence of n points X = x [subscript 1],...,x [subscript n] one by one, from a metric space M = (X,D). Our goal is to compute a low-distortion embedding of M into some host space, which has to be constructed in an on-line fashion, so that the image of each x i depends only on x [subscript 1],...,x [subscript i] . We prove several results translating existing embeddings to the on-line setting, for the case of embedding into ℓ [subscript p] spaces, and into distributions over ultrametrics
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
Tropical Principal Component Analysis and its Application to Phylogenetics
Principal component analysis is a widely-used method for the dimensionality
reduction of a given data set in a high-dimensional Euclidean space. Here we
define and analyze two analogues of principal component analysis in the setting
of tropical geometry. In one approach, we study the Stiefel tropical linear
space of fixed dimension closest to the data points in the tropical projective
torus; in the other approach, we consider the tropical polytope with a fixed
number of vertices closest to the data points. We then give approximative
algorithms for both approaches and apply them to phylogenetics, testing the
methods on simulated phylogenetic data and on an empirical dataset of
Apicomplexa genomes.Comment: 28 page
Euclidean quotients of finite metric spaces
This paper is devoted to the study of quotients of finite metric spaces. The
basic type of question we ask is: Given a finite metric space M, what is the
largest quotient of (a subset of) M which well embeds into Hilbert space. We
obtain asymptotically tight bounds for these questions, and prove that they
exhibit phase transitions. We also study the analogous problem for embedings
into l_p, and the particular case of the hypercube.Comment: 36 pages, 0 figures. To appear in Advances in Mathematic
Embedding Metrics into Ultrametrics and Graphs into Spanning Trees with Constant Average Distortion
This paper addresses the basic question of how well can a tree approximate
distances of a metric space or a graph. Given a graph, the problem of
constructing a spanning tree in a graph which strongly preserves distances in
the graph is a fundamental problem in network design. We present scaling
distortion embeddings where the distortion scales as a function of ,
with the guarantee that for each the distortion of a fraction
of all pairs is bounded accordingly. Such a bound implies, in
particular, that the \emph{average distortion} and -distortions are
small. Specifically, our embeddings have \emph{constant} average distortion and
-distortion. This follows from the following
results: we prove that any metric space embeds into an ultrametric with scaling
distortion . For the graph setting we prove that any
weighted graph contains a spanning tree with scaling distortion
. These bounds are tight even for embedding in arbitrary
trees.
For probabilistic embedding into spanning trees we prove a scaling distortion
of , which implies \emph{constant}
-distortion for every fixed .Comment: Extended abstrat apears in SODA 200
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