1,748 research outputs found
On Range Searching with Semialgebraic Sets II
Let be a set of points in . We present a linear-size data
structure for answering range queries on with constant-complexity
semialgebraic sets as ranges, in time close to . It essentially
matches the performance of similar structures for simplex range searching, and,
for , significantly improves earlier solutions by the first two authors
obtained in~1994. This almost settles a long-standing open problem in range
searching.
The data structure is based on the polynomial-partitioning technique of Guth
and Katz [arXiv:1011.4105], which shows that for a parameter , , there exists a -variate polynomial of degree such that
each connected component of contains at most points
of , where is the zero set of . We present an efficient randomized
algorithm for computing such a polynomial partition, which is of independent
interest and is likely to have additional applications
Quantum query complexity of minor-closed graph properties
We study the quantum query complexity of minor-closed graph properties, which
include such problems as determining whether an -vertex graph is planar, is
a forest, or does not contain a path of a given length. We show that most
minor-closed properties---those that cannot be characterized by a finite set of
forbidden subgraphs---have quantum query complexity \Theta(n^{3/2}). To
establish this, we prove an adversary lower bound using a detailed analysis of
the structure of minor-closed properties with respect to forbidden topological
minors and forbidden subgraphs. On the other hand, we show that minor-closed
properties (and more generally, sparse graph properties) that can be
characterized by finitely many forbidden subgraphs can be solved strictly
faster, in o(n^{3/2}) queries. Our algorithms are a novel application of the
quantum walk search framework and give improved upper bounds for several
subgraph-finding problems.Comment: v1: 25 pages, 2 figures. v2: 26 page
Competitive Boolean Function Evaluation: Beyond Monotonicity, and the Symmetric Case
We study the extremal competitive ratio of Boolean function evaluation. We
provide the first non-trivial lower and upper bounds for classes of Boolean
functions which are not included in the class of monotone Boolean functions.
For the particular case of symmetric functions our bounds are matching and we
exactly characterize the best possible competitiveness achievable by a
deterministic algorithm. Our upper bound is obtained by a simple polynomial
time algorithm.Comment: 15 pages, 1 figure, to appear in Discrete Applied Mathematic
Get the Most out of Your Sample: Optimal Unbiased Estimators using Partial Information
Random sampling is an essential tool in the processing and transmission of
data. It is used to summarize data too large to store or manipulate and meet
resource constraints on bandwidth or battery power. Estimators that are applied
to the sample facilitate fast approximate processing of queries posed over the
original data and the value of the sample hinges on the quality of these
estimators.
Our work targets data sets such as request and traffic logs and sensor
measurements, where data is repeatedly collected over multiple {\em instances}:
time periods, locations, or snapshots.
We are interested in queries that span multiple instances, such as distinct
counts and distance measures over selected records. These queries are used for
applications ranging from planning to anomaly and change detection.
Unbiased low-variance estimators are particularly effective as the relative
error decreases with the number of selected record keys.
The Horvitz-Thompson estimator, known to minimize variance for sampling with
"all or nothing" outcomes (which reveals exacts value or no information on
estimated quantity), is not optimal for multi-instance operations for which an
outcome may provide partial information.
We present a general principled methodology for the derivation of (Pareto)
optimal unbiased estimators over sampled instances and aim to understand its
potential. We demonstrate significant improvement in estimate accuracy of
fundamental queries for common sampling schemes.Comment: This is a full version of a PODS 2011 pape
- …