178,907 research outputs found
Algorithmic patterns for -matrices on many-core processors
In this work, we consider the reformulation of hierarchical ()
matrix algorithms for many-core processors with a model implementation on
graphics processing units (GPUs). matrices approximate specific
dense matrices, e.g., from discretized integral equations or kernel ridge
regression, leading to log-linear time complexity in dense matrix-vector
products. The parallelization of matrix operations on many-core
processors is difficult due to the complex nature of the underlying algorithms.
While previous algorithmic advances for many-core hardware focused on
accelerating existing matrix CPU implementations by many-core
processors, we here aim at totally relying on that processor type. As main
contribution, we introduce the necessary parallel algorithmic patterns allowing
to map the full matrix construction and the fast matrix-vector
product to many-core hardware. Here, crucial ingredients are space filling
curves, parallel tree traversal and batching of linear algebra operations. The
resulting model GPU implementation hmglib is the, to the best of the authors
knowledge, first entirely GPU-based Open Source matrix library of
this kind. We conclude this work by an in-depth performance analysis and a
comparative performance study against a standard matrix library,
highlighting profound speedups of our many-core parallel approach
Algorithmic and Combinatorial Results in Selection and Computational Geometry
This dissertation investigates two sets of algorithmic and combinatorial problems. Thefirst part focuses on the selection problem under the pairwise comparison model. For the classic “median of medians” scheme, contrary to the popular belief that smaller group sizes cause superlinear behavior, several new linear time algorithms that utilize small groups are introduced. Then the exact number of comparisons needed for an optimal selection algorithm is studied. In particular, the implications of a long standing conjecture known as Yao’s hypothesis are explored. For the multiparty model, we designed low communication complexity protocols for selecting an exact or an approximate median of data that is distributed among multiple players.
In the second part, three computational geometry problems are studied. For the longestspanning tree with neighborhoods, approximation algorithms are provided. For the stretch factor of polygonal chains, upper bounds are proved and almost matching lower bound constructions in \mathbb{R}^2 and higher dimensions are developed. For the piercing number τ and independence number ν of a family of axis-parallel rectangles in the plane, a lower bound construction for ν = 4 that matches Wegner’s conjecture is analyzed. The previous matching construction for ν = 3, due to Wegner himself, dates back to 1968
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