94,979 research outputs found
Solving Geometric Problems in Space-Conscious Models
When dealing with massive data sets, standard algorithms may
easily ``run out of memory''. In this thesis, we design efficient
algorithms in space-conscious models. In particular, in-place
algorithms, multi-pass algorithms, read-only algorithms, and
stream-sort algorithms are studied, and the focus is on
fundamental geometric problems, such as 2D convex hulls, 3D convex
hulls, Voronoi diagrams and nearest neighbor queries, Klee's
measure problem, and low-dimensional linear programming.
In-place algorithms only use O(1) extra space besides the input
array. We present a data structure for 2D nearest neighbor queries
and algorithms for Klee's measure problem in this model.
Algorithms in the multi-pass model only make read-only sequential
access to the input, and use sublinear working space and small
(usually a constant) number of passes on the input. We present
algorithms and lower bounds for many problems, including
low-dimensional linear programming and convex hulls, in this
model.
Algorithms in the read-only model only make read-only random
access to the input array, and use sublinear working space. We
present algorithms for Klee's measure problem and 2D convex hulls
in this model.
Algorithms in the stream-sort model use sorting as a primitive
operation. Each pass can either sort the data or make sequential
access to the data. As in the multi-pass model, these algorithms
can only use sublinear working space and a small (usually a
constant) number of passes on the data. We present algorithms for
constructing convex hulls and polygon triangulation in this model
Towards Tight Bounds for the Streaming Set Cover Problem
We consider the classic Set Cover problem in the data stream model. For
elements and sets () we give a -pass algorithm with a
strongly sub-linear space and logarithmic
approximation factor. This yields a significant improvement over the earlier
algorithm of Demaine et al. [DIMV14] that uses exponentially larger number of
passes. We complement this result by showing that the tradeoff between the
number of passes and space exhibited by our algorithm is tight, at least when
the approximation factor is equal to . Specifically, we show that any
algorithm that computes set cover exactly using passes
must use space in the regime of .
Furthermore, we consider the problem in the geometric setting where the
elements are points in and sets are either discs, axis-parallel
rectangles, or fat triangles in the plane, and show that our algorithm (with a
slight modification) uses the optimal space to find a
logarithmic approximation in passes.
Finally, we show that any randomized one-pass algorithm that distinguishes
between covers of size 2 and 3 must use a linear (i.e., ) amount of
space. This is the first result showing that a randomized, approximate
algorithm cannot achieve a space bound that is sublinear in the input size.
This indicates that using multiple passes might be necessary in order to
achieve sub-linear space bounds for this problem while guaranteeing small
approximation factors.Comment: A preliminary version of this paper is to appear in PODS 201
A Study in function optimization with the breeder genetic algorithm
Optimization is concerned with the finding of global optima
(hence the name) of problems that can be cast in the form of a
function of several variables and constraints thereof. Among the
searching methods, {em Evolutionary Algorithms} have been shown to be
adaptable and general tools that have often outperformed traditional
{em ad hoc} methods. The {em Breeder Genetic Algorithm} (BGA)
combines a direct representation with a nice conceptual
simplicity. This work contains a general description of the algorithm
and a detailed study on a collection of function optimization
tasks. The results show that the BGA is a powerful and reliable
searching algorithm. The main discussion concerns the choice of
genetic operators and their parameters, among which the family of
Extended Intermediate Recombination (EIR) is shown to stand out. In
addition, a simple method to dynamically adjust the operator is
outlined and found to greatly improve on the already excellent overall
performance of the algorithm.Postprint (published version
Topological Optimization of the Evaluation of Finite Element Matrices
We present a topological framework for finding low-flop algorithms for
evaluating element stiffness matrices associated with multilinear forms for
finite element methods posed over straight-sided affine domains. This framework
relies on phrasing the computation on each element as the contraction of each
collection of reference element tensors with an element-specific geometric
tensor. We then present a new concept of complexity-reducing relations that
serve as distance relations between these reference element tensors. This
notion sets up a graph-theoretic context in which we may find an optimized
algorithm by computing a minimum spanning tree. We present experimental results
for some common multilinear forms showing significant reductions in operation
count and also discuss some efficient algorithms for building the graph we use
for the optimization
Algorithmic Algebraic Geometry and Flux Vacua
We develop a new and efficient method to systematically analyse four
dimensional effective supergravities which descend from flux compactifications.
The issue of finding vacua of such systems, both supersymmetric and
non-supersymmetric, is mapped into a problem in computational algebraic
geometry. Using recent developments in computer algebra, the problem can then
be rapidly dealt with in a completely algorithmic fashion. Two main results are
(1) a procedure for calculating constraints which the flux parameters must
satisfy in these models if any given type of vacuum is to exist; (2) a stepwise
process for finding all of the isolated vacua of such systems and their
physical properties. We illustrate our discussion with several concrete
examples, some of which have eluded conventional methods so far.Comment: 41 pages, 4 figure
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