20 research outputs found
Ramsey-type theorems for metric spaces with applications to online problems
A nearly logarithmic lower bound on the randomized competitive ratio for the
metrical task systems problem is presented. This implies a similar lower bound
for the extensively studied k-server problem. The proof is based on Ramsey-type
theorems for metric spaces, that state that every metric space contains a large
subspace which is approximately a hierarchically well-separated tree (and in
particular an ultrametric). These Ramsey-type theorems may be of independent
interest.Comment: Fix an error in the metadata. 31 pages, 0 figures. Preliminary
version in FOCS '01. To be published in J. Comput. System Sc
On Metric Ramsey-type Dichotomies
The classical Ramsey theorem, states that every graph contains either a large
clique or a large independent set. Here we investigate similar dichotomic
phenomena in the context of finite metric spaces. Namely, we prove statements
of the form "Every finite metric space contains a large subspace that is nearly
quilateral or far from being equilateral". We consider two distinct
interpretations for being "far from equilateral". Proximity among metric spaces
is quantified through the metric distortion D. We provide tight asymptotic
answers for these problems. In particular, we show that a phase transition
occurs at D=2.Comment: 14 pages, 0 figure
Unbounded lower bound for k-server against weak adversaries
We study the resource augmented version of the -server problem, also known
as the -server problem against weak adversaries or the -server
problem. In this setting, an online algorithm using servers is compared to
an offline algorithm using servers, where . For uniform metrics, it
has been known since the seminal work of Sleator and Tarjan (1985) that for any
, the competitive ratio drops to a constant if . This result was later generalized to weighted stars (Young 1994) and
trees of bounded depth (Bansal et al. 2017). The main open problem for this
setting is whether a similar phenomenon occurs on general metrics.
We resolve this question negatively. With a simple recursive construction, we
show that the competitive ratio is at least , even as
. Our lower bound holds for both deterministic and randomized
algorithms. It also disproves the existence of a competitive algorithm for the
infinite server problem on general metrics.Comment: To appear in STOC 202
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
Unbounded lower bound for k-server against weak adversaries
We study the resource augmented version of the k-server problem, also known as the k-server problem against weak adversaries or the (h,k)-server problem. In this setting, an online algorithm using k servers is compared to an offline algorithm using h servers, where h ≤ k. For uniform metrics, it has been known since the seminal work of Sleator and Tarjan (1985) that for any ">0, the competitive ratio drops to a constant if k=(1+") · h. This result was later generalized to weighted stars (Young 1994) and trees of bounded depth (Bansal et al. 2017). The main open problem for this setting is whether a similar phenomenon occurs on general metrics. We resolve this question negatively. With a simple recursive construction, we show that the competitive ratio is at least ω(loglogh), even as k→∞. Our lower bound holds for both deterministic and randomized algorithms. It also disproves the existence of a competitive algorithm for the infinite server problem on general metrics
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