Computational geometry

Exact algorithms for dealing with geometric objects are complicated, hard to implement in practice, and slow. Over the last 20 years a theory of geometric approximation algorithms has emerged. These algorithms tend to be simple, fast, and more robust than their exact counterparts. This book is the first to cover geometric approximation algorithms in detail. In addition, more traditional computational geometry techniques that are widely used in developing such algorithms, like sampling, linear programming, etc., are also surveyed. Other topics covered include approximate nearest-neighbor search, shape approximation, coresets, dimension reduction, and embeddings. The topics covered are relatively independent and are supplemented by exercises. Close to 200 color figures are included in the text to illustrate proofs and ideas.

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Geometric Approximation Algorithms

Data center networks (DCNs) form the backbone infrastructure of many large-scale enterprise applications as well as emerging cloud computing providers. This paper describes the design, implementation and evaluation of OSA, a novel Optical Switching Architecture for DCNs. Leveraging runtime reconfigurable optical devices, OSA dynamically changes its topology and link capacities, thereby achieving unprecedented flexibility to adapt to dynamic traffic patterns. Extensive analytical simulations using both real and synthetic traffic patterns demonstrate that OSA can deliver high bisection bandwidth (60%-100% of the non-blocking architecture). Implementation and evaluation of a small-scale functional prototype further demonstrate the feasibility of OSA.

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OSA: an optical switching architecture for data center networks with unprecedented flexibility

In this paper, a novel Delaunay-based variational approach to isotropic tetrahedral meshing is presented. To achieve both robustness and efficiency, we minimize a simple mesh-dependent energy through global updates of both vertex positions and connectivity. As this energy is known to be the ∠1 distance between an isotropic quadratic function and its linear interpolation on the mesh, our minimization procedure generates well-shaped tetrahedra. Mesh design is controlled through a gradation smoothness parameter and selection of the desired number of vertices. We provide the foundations of our approach by explaining both the underlying variational principle and its geometric interpretation. We demonstrate the quality of the resulting meshes through a series of examples.

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Variational tetrahedral meshing

A detailed examination of evolving traffic characteristics, operator requirements, and network technology trends suggests a move away from nonblocking interconnects in data center networks (DCNs). As a result, recent efforts have advocated oversubscribed networks with the capability to adapt to traffic requirements on-demand. In this paper, we present the design, implementation, and evaluation of OSA, a novel Optical Switching Architecture for DCNs. Leveraging runtime reconfigurable optical devices, OSA dynamically changes its topology and link capacities, thereby achieving unprecedented flexibility to adapt to dynamic traffic patterns. Extensive analytical simulations using both real and synthetic traffic patterns demonstrate that OSA can deliver high bisection bandwidth (60%-100% of the nonblocking architecture). Implementation and evaluation of a small-scale functional prototype further demonstrate the feasibility of OSA.

/pdf/osa-an-optical-switching-architecture-for-data-center-2xutrj78ve.pdf

A sliver is a tetrahedron whose four vertices lie close to a plane and whose perpendicular projection to that plane is a convex quadrilateral with no short edge. Slivers axe both undesirable and ubiquitous in 3-dimensional Delaunay triangulations. Even when the point-set is well-spaced, slivers may result. This paper shows that such a point set permits a small perturbation whose Delaunay triangulation contains no slivers. It also gives deterministic algorithms that compute the perturbation of n points in time O(n logn) with one processor and in time O(log n) with O(n) processors.

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Smoothing and cleaning up slivers

We present an algorithm to construct meshes suitable for spacetime discontinuous Galerkin finite-element methods. Our method generalizes and improves the ‘Tent Pitcher’ algorithm of Ungor and Sheffer. Given an arbitrary simplicially meshed domain X of any dimension and a time interval [0, T], our algorithm builds a simplicial mesh of the spacetime domain X × [0, T], in constant time per element. Our algorithm avoids the limitations of previous methods by carefully adapting the durations of spacetime elements to the local quality and feature size of the underlying space mesh.

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Building spacetime meshes over arbitrary spatial domains

A key step in the nite element method is to generate a high quality mesh that is as small as possible for an input domain. Several meshing methods and heuristics have been developed and implemented. Methods based on advancing front, Delaunay triangulations, and quadtrees/octrees are among the most popular ones. Advancing front uses simple data structures and is eecient. Unfortunately, in general, it does not provide any guarantee on the size and quality of the mesh it produces. On the other hand, the circle-packing based Delaunay methods generate a well-shaped mesh whose size is within a constant factor of the optimal. In this paper, we present a new meshing algorithm, the biting method, which combines the strengths of advancing front and circle packing. We prove that it generates a high quality 2D mesh, and the size of the mesh is within a constant factor of the optimal.

Biting: advancing front meets sphere packing

We show it is possible to tile three-dimensional space using only tetrahedra with acute dihedral angles. We present several constructions to achieve this, including one in which all dihedral angles are less than $77.08^\circ$, and another which tiles a slab in space.

Tiling space and slabs with acute tetrahedra

We show it is possible to tile three-dimensional space using only tetrahedra with acute dihedral angles. We present several constructions to achieve this, including one in which all dihedral angles are less than 74.21°, and another which tiles a slab in space.

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Alper Üngör

Papers

Smoothing and cleaning up slivers

Building spacetime meshes over arbitrary spatial domains

Biting: advancing front meets sphere packing

Tiling space and slabs with acute tetrahedra

Tiling space and slabs with acute tetrahedra