122,405 research outputs found
Fractal Dimension and Lower Bounds for Geometric Problems
No full textWe study the complexity of geometric problems on spaces of low fractal dimension. It was recently shown by [Sidiropoulos & Sridhar, SoCG 2017] that several problems admit improved solutions when the input is a pointset in Euclidean space with fractal dimension smaller than the ambient dimension. In this paper we prove nearly-matching lower bounds, thus establishing nearly-optimal bounds for various problems as a function of the fractal dimension.
More specifically, we show that for any set of n points in d-dimensional Euclidean space, of fractal dimension delta in (1,d), for any epsilon>0 and c >= 1, any c-spanner must have treewidth at least Omega(n^{1-1/(delta - epsilon)} / c^{d-1}), matching the previous upper bound. The construction used to prove this lower bound on the treewidth of spanners, can also be used to derive lower bounds on the running time of algorithms for various problems, assuming the Exponential Time Hypothesis. We provide two prototypical results of this type:
- For any delta in (1,d) and any epsilon >0, d-dimensional Euclidean TSP on n points with fractal dimension at most delta cannot be solved in time 2^{O(n^{1-1/(delta - epsilon)})}. The best-known upper bound is 2^{O(n^{1-1/delta} log n)}.
- For any delta in (1,d) and any epsilon >0, the problem of finding k-pairwise non-intersecting d-dimensional unit balls/axis parallel unit cubes with centers having fractal dimension at most delta cannot be solved in time f(k)n^{O (k^{1-1/(delta - epsilon)})} for any computable function f. The best-known upper bound is n^{O(k^{1-1/delta} log n)}. The above results nearly match previously known upper bounds from [Sidiropoulos & Sridhar, SoCG 2017], and generalize analogous lower bounds for the case of ambient dimension due to [Marx & Sidiropoulos, SoCG 2014]
Constant-Factor Approximations for Asymmetric TSP on Nearly-Embeddable Graphs
No full textIn the Asymmetric Traveling Salesperson Problem (ATSP) the goal is to find a closed walk of minimum cost in a directed graph visiting every vertex. We consider the approximability of ATSP on topologically restricted graphs. It has been shown by Oveis Gharan and Saberi [SODA, 2011] that there exists polynomial-time constant-factor approximations on planar graphs and more generally graphs of constant orientable genus. This result was extended to non-orientable genus by Erickson and Sidiropoulos [SoCG, 2014].
We show that for any class of nearly-embeddable graphs, ATSP admits a polynomial-time constant-factor approximation. More precisely, we show that for any fixed non-negative k, there exist positive alpha and beta, such that ATSP on n-vertex k-nearly-embeddable graphs admits an alpha-approximation in time O(n^beta). The class of k-nearly-embeddable graphs contains graphs with at most k apices, k vortices of width at most k, and an underlying surface of either orientable or non-orientable genus at most k. Prior to our work, even the case of graphs with a single apex was open. Our algorithm combines tools from rounding the Held-Karp LP via thin trees with dynamic programming.
We complement our upper bounds by showing that solving ATSP exactly on graphs of pathwidth k (and hence on k-nearly embeddable graphs) requires time n^{Omega(k)}, assuming the Exponential-Time Hypothesis (ETH). This is surprising in light of the fact that both TSP on undirected graphs and Minimum Cost Hamiltonian Cycle on directed graphs are FPT parameterized by treewidth
Mathematical Programming Algorithms for Regression-Based Nonlinear Filtering in R^N
No full textConstrained regression problems appear in the context of optimal nonlinear filtering, as well as in a variety of other contexts, e.g., chromatographic analysis in chemometrics and manufacturing, and spectral estimation. This paper presents novel mathematical programming algorithms for some important constrained regression problems in IR N . For brevity, we focus on four key problems, namely, locally monotonic regression (the optimal counterpart of iterated median filtering) , and the related problem of piecewise monotonic regression, runlength-constrained regression (a useful segmentation and edge detection technique), and uni- and oligo-modal regression (of interest in chromatography and spectral estimation). The proposed algorithms are exact and efficient, and they also naturally suggest slightly suboptimal but very fast approximate algorithms, which may be preferable in practice. N.D. Sidiropoulos is with the Institute for Systems Research, University of Maryland, College Park, MD..
Sustainability Aspects of Electrical Machines for E-Mobility Applications Part I: A Design with Reduced Rare-earth Elements
No full textThis paper presents a study on different permanent magnet motor topologies that can be used in e-mobility applications. Considering the environmental effects of rare earth elements used in permanent magnet materials, focus in this paper is placed on the solutions with less rare earth element magnets. Additionally, challenges in using non rare earth magnets in the rotor structure of permanent magnet motors are reviewed and discussed. The risk of demagnetization under short-circuit condition is considered as one of the main challenges in using non rare earth magnets that needs attention when designing the electric propulsion system. As shown in this paper, proper rotor design can significantly improve the risk of demagnetization allowing short-circuit in larger operation range. In addition, obtaining an acceptable level of performance and efficiency as using rare earth elements is possible to achieve
Algorithmic Interpretations of Fractal Dimension
No full textWe study algorithmic problems on subsets of Euclidean space of low fractal dimension. These spaces are the subject of intensive study in various branches of mathematics, including geometry, topology, and measure theory. There are several well-studied notions of fractal dimension for sets and measures in Euclidean space. We consider a definition of fractal dimension for finite metric spaces which agrees with standard notions used to empirically estimate the fractal dimension of various sets. We define the fractal dimension of some metric space to be the infimum delta>0, such that for any eps>0, for any ball B of radius r >= 2eps, and for any eps-net N, we have |B cap N|=O((r/eps)^delta).
Using this definition we obtain faster algorithms for a plethora of classical problems on sets of low fractal dimension in Euclidean space. Our results apply to exact and fixed-parameter algorithms, approximation schemes, and spanner constructions. Interestingly, the dependence of the performance of these algorithms on the fractal dimension nearly matches the currently best-known dependence on the standard Euclidean dimension. Thus, when the fractal dimension is strictly smaller than the ambient dimension, our results yield improved solutions in all of these settings.
We remark that our definition of fractal definition is equivalent up to constant factors to the well-studied notion of doubling dimension.
However, in the problems that we consider, the dimension appears in the exponent of the running time, and doubling dimension is not precise enough for capturing the best possible such exponent for subsets of Euclidean space. Thus our work is orthogonal to previous results on spaces of low doubling dimension; while algorithms on spaces of low doubling dimension seek to extend results from the case of low dimensional Euclidean spaces to more general metric spaces, our goal is to obtain faster algorithms for special pointsets in Euclidean space
Preanalytic Variables and Tissue Stewardship for Reliable Next-Generation Sequencing (NGS) Clinical Analysis
No full textPattern recognition systems design on parallel GPU architectures for breast lesions characterisation employing multimodality images
No full textThis thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University London.The aim of this research was to address the computational complexity in designing multimodality Computer-Aided Diagnosis (CAD) systems for characterising breast lesions, by harnessing the general purpose computational potential of consumer-level Graphics Processing Units (GPUs) through parallel programming methods. The complexity in designing such systems lies on the increased dimensionality of the problem, due to the multiple imaging modalities involved, on the inherent complexity of optimal design methods for securing high precision, and on assessing the performance of the design prior to deployment in a clinical environment, employing unbiased system evaluation methods. For the purposes of this research, a Pattern Recognition (PR)-system was designed to provide highest possible precision by programming in parallel the multiprocessors of the NVIDIA’s GPU-cards, GeForce 8800GT or 580GTX, and using the CUDA programming framework and C++. The PR-system was built around the Probabilistic Neural Network classifier and its performance was evaluated by a re-substitution method, for estimating the system’s highest accuracy, and by the external cross validation method, for assessing the PR-system’s unbiased accuracy to new, “unseen” by the system, data. Data comprised images of patients with histologically verified (benign or malignant) breast lesions, who underwent both ultrasound (US) and digital mammography (DM). Lesions were outlined on the images by an experienced radiologist, and textural features were calculated. Regarding breast lesion classification, the accuracies for discriminating malignant from benign lesions were, 85.5% using US-features alone, 82.3% employing DM-features alone, and 93.5% combining US and DM features. Mean accuracy to new “unseen” data for the combined US and DM features was 81%. Those classification accuracies were about 10% higher than accuracies achieved on a single CPU, using sequential programming methods, and 150-fold faster. In addition, benign lesions were found smoother, more homogeneous, and containing larger structures. Additionally, the PR-system design was adapted for tackling other medical problems, as a proof of its generalisation. These included classification of rare brain tumours, (achieving 78.6% for overall accuracy (OA) and 73.8% for estimated generalisation accuracy (GA), and accelerating system design 267 times), discrimination of patients with micro-ischemic and multiple sclerosis lesions (90.2% OA and 80% GA with 32-fold design acceleration), classification of normal and pathological knee cartilages (93.2% OA and 89% GA with 257-fold design acceleration), and separation of low from high grade laryngeal cancer cases (93.2% OA and 89% GA, with 130-fold design acceleration). The proposed PR-system improves breast-lesion discrimination accuracy, it may be redesigned on site when new verified data are incorporated in its depository, and it may serve as a second opinion tool in a clinical environment
Computing Bi-Lipschitz Outlier Embeddings into the Line
No full textThe problem of computing a bi-Lipschitz embedding of a graphical metric into the line with minimum distortion has received a lot of attention. The best-known approximation algorithm computes an embedding with distortion O(c²), where c denotes the optimal distortion [Bădoiu et al. 2005]. We present a bi-criteria approximation algorithm that extends the above results to the setting of outliers.
Specifically, we say that a metric space (X,ρ) admits a (k,c)-embedding if there exists K ⊂ X, with |K| = k, such that (X⧵ K, ρ) admits an embedding into the line with distortion at most c. Given k ≥ 0, and a metric space that admits a (k,c)-embedding, for some c ≥ 1, our algorithm computes a (poly(k, c, log n), poly(c))-embedding in polynomial time. This is the first algorithmic result for outlier bi-Lipschitz embeddings. Prior to our work, comparable outlier embeddings where known only for the case of additive distortion
The Hirsch spectrum: a novel tool for analysing scientific journals
No full textThis paper introduces the Hirsch spectrum (h-spectrum) for analyzing the academic reputation of a scientific journal. h-Spectrum is a novel tool based on the Hirsch (h) index. It is easy to construct: considering a specific journal in a specific interval of time, h-spectrum is defined as the distribution representing the h-indexes associated to the authors of the journal articles. This tool allows defining a reference profile of the typical author of a journal, compare different journals within the same scientific field, and provide a rough indication of prestige/reputation of a journal in the scientific community. h-Spectrum can be associated to every journal. Ten specific journals in the Quality Engineering/Quality Management field are analyzed so as to preliminarily investigate the h-spectrum characteristic
Quasimetric Embeddings and Their Applications
No full textWe study generalizations of classical metric embedding results to the case of quasimetric spaces; that is, spaces that do not necessarily satisfy symmetry. Quasimetric spaces arise naturally from the shortest-path distances on directed graphs. Perhaps surprisingly, very little is known about low-distortion embeddings for quasimetric spaces.
Random embeddings into ultrametric spaces are arguably one of the most successful geometric tools in the context of algorithm design. We extend this to the quasimetric case as follows. We show that any n-point quasimetric space supported on a graph of treewidth t admits a random embedding into quasiultrametric spaces with distortion O(t*log^2(n)), where quasiultrametrics are a natural generalization of ultrametrics. This result allows us to obtain t*log^{O(1)}(n)-approximation algorithms for the Directed Non-Bipartite Sparsest-Cut and the Directed Multicut problems on n-vertex graphs of treewidth t, with running time polynomial in both n and t.
The above results are obtained by considering a generalization of random partitions to the quasimetric case, which we refer to as random quasipartitions. Using this definition and a construction of [Chuzhoy and Khanna 2009] we derive a polynomial lower bound on the distortion of random embeddings of general quasimetric spaces into quasiultrametric spaces. Finally, we establish a lower bound for embedding the shortest-path quasimetric of a graph G into graphs that exclude G as a minor. This lower bound is used to show that several embedding results from the metric case do not have natural analogues in the quasimetric setting
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