Sparsity: Graphs, Structures, and Algorithms
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It seems likely that these problems will be amenable to algorithms based on randomized projections that dramatically reduce the effective dimensionality of the underlying problems. Such techniques has recently proven highly effective for the related problems of how to find approximate lists of nearest neighbors for clouds of points in high dimensional spaces, and for constructing approximate low-rank factorizations of large matrices. In both cases, a key observation is that the problem of distortions of distances that is inherent to Abstract The study of computational problems on graphs has long been a central area of research in computer science.
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This motivates a theoretical study of the abilities and limitations of sparsity-based methods. However, a priori it is not clear how to even define sparsity formally.
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Multiple sparsity-oriented paradigms have been studied in the literature, e. However, many of those paradigms suffer from being either too restrictive to model real-life applications, or too general to yield strong tractability results.
The central notions of their framework are bounded expansion and nowhere dense classes. It quickly turned out that the proposed notions can be used to build a mathematical theory of sparse graphs that offers a wealth of tools, leading to new techniques and powerful results. This theory has been extensively developed in the recent years.
This defines what data type is provided by the srcEdgeData and dstEdgeData parameters. However, the majority of these methods require the sampling patterns in training, which limits their application to a specific problem class. Function nvgraphDestroyGraphDescr. This function sets index of edge data where mask for edges will be stored. The eigenvectors are stored in row-major order and normalized by row and by column. In the case of highly sparse graphs we can leverage sparse libraries, and an even faster solution for both dense and sparse graph is to perform the computations on a GPU.
It is particularly remarkable that the concepts of classes of bounded expansion and nowhere dense classes can be connected to fundamental ideas from multiple other fields of computer science, often in a surprising way, providing several complementary viewpoints on the subject. On one hand, foundations of the area are grounded in structural graph theory, which aims at describing structure in graphs through various decompositions and auxiliary parameters.
On the other hand, nowhere denseness seems to delimit the border of algorithmic tractability of first-order logic, providing a link to finite model theory and its computational aspects. Finally, there is a fruitful transfer of ideas to and from the field of algorithm design: sparsity-based methods can be used to design new, efficient algorithms, especially in the paradigms of parameterized complexity, approximation algorithms, and distributed computing.
