17,029 research outputs found
Helly-Type Theorems in Property Testing
Helly's theorem is a fundamental result in discrete geometry, describing the
ways in which convex sets intersect with each other. If is a set of
points in , we say that is -clusterable if it can be
partitioned into clusters (subsets) such that each cluster can be contained
in a translated copy of a geometric object . In this paper, as an
application of Helly's theorem, by taking a constant size sample from , we
present a testing algorithm for -clustering, i.e., to distinguish
between two cases: when is -clusterable, and when it is
-far from being -clusterable. A set is -far
from being -clusterable if at least
points need to be removed from to make it -clusterable. We solve
this problem for and when is a symmetric convex object. For , we
solve a weaker version of this problem. Finally, as an application of our
testing result, in clustering with outliers, we show that one can find the
approximate clusters by querying a constant size sample, with high probability
Multi-Step Processing of Spatial Joins
Spatial joins are one of the most important operations for combining spatial objects of several relations. In this paper, spatial join processing is studied in detail for extended spatial objects in twodimensional data space. We present an approach for spatial join processing that is based on three steps. First, a spatial join is performed on the minimum bounding rectangles of the objects returning a set of candidates. Various approaches for accelerating this step of join processing have been examined at the last year’s conference [BKS 93a]. In this paper, we focus on the problem how to compute the answers from the set of candidates which is handled by
the following two steps. First of all, sophisticated approximations
are used to identify answers as well as to filter out false hits from
the set of candidates. For this purpose, we investigate various types
of conservative and progressive approximations. In the last step, the
exact geometry of the remaining candidates has to be tested against
the join predicate. The time required for computing spatial join
predicates can essentially be reduced when objects are adequately
organized in main memory. In our approach, objects are first decomposed
into simple components which are exclusively organized
by a main-memory resident spatial data structure. Overall, we
present a complete approach of spatial join processing on complex
spatial objects. The performance of the individual steps of our approach
is evaluated with data sets from real cartographic applications.
The results show that our approach reduces the total execution
time of the spatial join by factors
Testing surface area with arbitrary accuracy
Recently, Kothari et al.\ gave an algorithm for testing the surface area of
an arbitrary set . Specifically, they gave a randomized
algorithm such that if 's surface area is less than then the algorithm
will accept with high probability, and if the algorithm accepts with high
probability then there is some perturbation of with surface area at most
. Here, is a dimension-dependent constant which is
strictly larger than 1 if , and grows to as .
We give an improved analysis of Kothari et al.'s algorithm. In doing so, we
replace the constant with for arbitrary. We
also extend the algorithm to more general measures on Riemannian manifolds.Comment: 5 page
The Optimal Mechanism in Differential Privacy
We derive the optimal -differentially private mechanism for single
real-valued query function under a very general utility-maximization (or
cost-minimization) framework. The class of noise probability distributions in
the optimal mechanism has {\em staircase-shaped} probability density functions
which are symmetric (around the origin), monotonically decreasing and
geometrically decaying. The staircase mechanism can be viewed as a {\em
geometric mixture of uniform probability distributions}, providing a simple
algorithmic description for the mechanism. Furthermore, the staircase mechanism
naturally generalizes to discrete query output settings as well as more
abstract settings. We explicitly derive the optimal noise probability
distributions with minimum expectation of noise amplitude and power. Comparing
the optimal performances with those of the Laplacian mechanism, we show that in
the high privacy regime ( is small), Laplacian mechanism is
asymptotically optimal as ; in the low privacy regime
( is large), the minimum expectation of noise amplitude and minimum
noise power are and as , while the expectation of
noise amplitude and power using the Laplacian mechanism are
and , where is
the sensitivity of the query function. We conclude that the gains are more
pronounced in the low privacy regime.Comment: 40 pages, 5 figures. Part of this work was presented in DIMACS
Workshop on Recent Work on Differential Privacy across Computer Science,
October 24 - 26, 201
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