6,941 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
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
Using genetic algorithms to create meaningful poetic text
Work carried out when all authors were at the University of Edinburgh.Peer reviewedPostprin
Metrical Service Systems with Multiple Servers
We study the problem of metrical service systems with multiple servers
(MSSMS), which generalizes two well-known problems -- the -server problem,
and metrical service systems. The MSSMS problem is to service requests, each of
which is an -point subset of a metric space, using servers, with the
objective of minimizing the total distance traveled by the servers.
Feuerstein initiated a study of this problem by proving upper and lower
bounds on the deterministic competitive ratio for uniform metric spaces. We
improve Feuerstein's analysis of the upper bound and prove that his algorithm
achieves a competitive ratio of . In the randomized
online setting, for uniform metric spaces, we give an algorithm which achieves
a competitive ratio , beating the deterministic lower
bound of . We prove that any randomized algorithm for
MSSMS on uniform metric spaces must be -competitive. We then
prove an improved lower bound of on
the competitive ratio of any deterministic algorithm for -MSSMS, on
general metric spaces. In the offline setting, we give a pseudo-approximation
algorithm for -MSSMS on general metric spaces, which achieves an
approximation ratio of using servers. We also prove a matching
hardness result, that a pseudo-approximation with less than servers is
unlikely, even for uniform metric spaces. For general metric spaces, we
highlight the limitations of a few popular techniques, that have been used in
algorithm design for the -server problem and metrical service systems.Comment: 18 pages; accepted for publication at COCOON 201
Randomized online computation with high probability guarantees
We study the relationship between the competitive ratio and the tail
distribution of randomized online minimization problems. To this end, we define
a broad class of online problems that includes some of the well-studied
problems like paging, k-server and metrical task systems on finite metrics, and
show that for these problems it is possible to obtain, given an algorithm with
constant expected competitive ratio, another algorithm that achieves the same
solution quality up to an arbitrarily small constant error a with high
probability; the "high probability" statement is in terms of the optimal cost.
Furthermore, we show that our assumptions are tight in the sense that removing
any of them allows for a counterexample to the theorem. In addition, there are
examples of other problems not covered by our definition, where similar high
probability results can be obtained.Comment: 20 pages, 2 figure
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