1,230 research outputs found
Optimal Random Matchings, Tours, and Spanning Trees in Hierarchically Separated Trees
We derive tight bounds on the expected weights of several combinatorial
optimization problems for random point sets of size distributed among the
leaves of a balanced hierarchically separated tree. We consider {\it
monochromatic} and {\it bichromatic} versions of the minimum matching, minimum
spanning tree, and traveling salesman problems. We also present tight
concentration results for the monochromatic problems.Comment: 24 pages, to appear in TC
Optimal Geo-Indistinguishable Mechanisms for Location Privacy
We consider the geo-indistinguishability approach to location privacy, and
the trade-off with respect to utility. We show that, given a desired degree of
geo-indistinguishability, it is possible to construct a mechanism that
minimizes the service quality loss, using linear programming techniques. In
addition we show that, under certain conditions, such mechanism also provides
optimal privacy in the sense of Shokri et al. Furthermore, we propose a method
to reduce the number of constraints of the linear program from cubic to
quadratic, maintaining the privacy guarantees and without affecting
significantly the utility of the generated mechanism. This reduces considerably
the time required to solve the linear program, thus enlarging significantly the
location sets for which the optimal mechanisms can be computed.Comment: 13 page
Random Weighting, Asymptotic Counting, and Inverse Isoperimetry
For a family X of k-subsets of the set 1,...,n, let |X| be the cardinality of
X and let Gamma(X,mu) be the expected maximum weight of a subset from X when
the weights of 1,...,n are chosen independently at random from a symmetric
probability distribution mu on R. We consider the inverse isoperimetric problem
of finding mu for which Gamma(X,mu) gives the best estimate of ln|X|. We prove
that the optimal choice of mu is the logistic distribution, in which case
Gamma(X,mu) provides an asymptotically tight estimate of ln|X| as k^{-1}ln|X|
grows. Since in many important cases Gamma(X,mu) can be easily computed, we
obtain computationally efficient approximation algorithms for a variety of
counting problems. Given mu, we describe families X of a given cardinality with
the minimum value of Gamma(X,mu), thus extending and sharpening various
isoperimetric inequalities in the Boolean cube.Comment: The revision contains a new isoperimetric theorem, some other
improvements and extensions; 29 pages, 1 figur
06481 Abstracts Collection -- Geometric Networks and Metric Space Embeddings
The Dagstuhl Seminar 06481 ``Geometric Networks and Metric Space
Embeddings\u27\u27 was held from November~26 to December~1, 2006 in the
International Conference and Research Center (IBFI), Schloss
Dagstuhl. During the seminar, several participants presented their
current research, and ongoing work and open problems were discussed.
In this paper we describe the seminar topics, we have compiled a
list of open questions that were posed during the seminar, there is
a list of all talks and there are abstracts of the presentations
given during the seminar. Links to extended abstracts or full
papers are provided where available
Simpler, faster and shorter labels for distances in graphs
We consider how to assign labels to any undirected graph with n nodes such
that, given the labels of two nodes and no other information regarding the
graph, it is possible to determine the distance between the two nodes. The
challenge in such a distance labeling scheme is primarily to minimize the
maximum label lenght and secondarily to minimize the time needed to answer
distance queries (decoding). Previous schemes have offered different trade-offs
between label lengths and query time. This paper presents a simple algorithm
with shorter labels and shorter query time than any previous solution, thereby
improving the state-of-the-art with respect to both label length and query time
in one single algorithm. Our solution addresses several open problems
concerning label length and decoding time and is the first improvement of label
length for more than three decades.
More specifically, we present a distance labeling scheme with label size (log
3)/2 + o(n) (logarithms are in base 2) and O(1) decoding time. This outperforms
all existing results with respect to both size and decoding time, including
Winkler's (Combinatorica 1983) decade-old result, which uses labels of size
(log 3)n and O(n/log n) decoding time, and Gavoille et al. (SODA'01), which
uses labels of size 11n + o(n) and O(loglog n) decoding time. In addition, our
algorithm is simpler than the previous ones. In the case of integral edge
weights of size at most W, we present almost matching upper and lower bounds
for label sizes. For r-additive approximation schemes, where distances can be
off by an additive constant r, we give both upper and lower bounds. In
particular, we present an upper bound for 1-additive approximation schemes
which, in the unweighted case, has the same size (ignoring second order terms)
as an adjacency scheme: n/2. We also give results for bipartite graphs and for
exact and 1-additive distance oracles
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