Linus Källberg

Fast and robust approximation of smallest enclosing balls in arbitrary dimensions

In this paper, an algorithm is introduced that computes an arbitrarily fine approximation of the smallest enclosing ball of a point set in any dimension. This operation is important in, for example, classification, clustering, and data mining. The algorithm is very simple to implement, gives reliable results, and gracefully handles large problem instances in low and high dimensions, as confirmed by both theoretical arguments and empirical evaluation. For example, using a CPU with eight cores, it takes less than two seconds to compute a 1.001‐approximation of the smallest enclosing ball of one million points uniformly distributed in a hypercube in dimension 200. Furthermore, the presented approach extends to a more general class of input objects, such as ball sets.

Parallel computation of optimal enclosing balls by iterative orthant scan

We propose an algorithm for computing the exact minimum enclosing ball of large point sets in general dimensions. It aims to reduce the number of passes by retrieving a well-balanced set of outliers in each linear search through the input by decomposing the space into orthants. The experimental evidence indicates that the convergence rate in terms of the required number of linear passes is superior compared to previous exact methods, and substantially faster execution times are observed in dimensions d ? 16 . In the important three-dimensional case, the execution times indicate real-time performance. Furthermore, we show how the algorithm can be adapted for parallel execution on both CPU and GPU architectures using OpenMP, AVX, and CUDA. For large datasets, our CUDA solution is superior. For example, the benchmark results show that optimal bounding spheres for inputs with tens of millions of points can be computed in just a few milliseconds. Graphical abstractDisplay Omitted HighlightsA fast minimum enclosing ball algorithm for general dimensions.An effective heuristic drastically reducing the number of algorithmic steps.Exhaustive study of parallelization opportunities for different platforms.Real-time exact bounding sphere computation in 3D.

Improved pruning of large data sets for the minimum enclosing ball problem

Display Omitted We develop pruning strategies to accelerate minimum enclosing ball computations.We give improved pruning bounds that are valid in a number of existing algorithms.Using these bounds, we achieve twice the effect compared to earlier approaches.We demonstrate substantial speedups of several state-of-the-art algorithms. Minimum enclosing ball algorithms are studied extensively as a tool in approximation and classification of multidimensional data. We present pruning techniques that can accelerate several existing algorithms by continuously removing interior points from the input. By recognizing a key property shared by these algorithms, we derive tighter bounds than have previously been presented, resulting in twice the effect on performance. Furthermore, only minor modifications are required to incorporate the pruning procedure. The presented bounds are independent of the dimension, and empirical evidence shows that the pruning procedure remains effective in dimensions up to at least 200. In some cases, performance improvements of two orders of magnitude are observed for large data sets.

Faster Approximation of Minimum Enclosing Balls by Distance Filtering and GPU Parallelization

Minimum enclosing balls are used extensively to speed up multidimensional data processing in, e.g., machine learning, spatial databases, and computer graphics. We present a case study of several acceleration techniques that are applicable in enclosing ball algorithms based on repeated farthest-point queries. Two different distance filtering heuristics are proposed aiming at reducing the cost of the farthest-point queries as much as possible by exploiting lower and upper distance bounds. Furthermore, auto-tunable GPU solutions using CUDA are developed for both low- and high-dimensional cases. Empirical tests apply these techniques to two recent algorithms and demonstrate substantial speedups of the ball computations. Our results also indicate that a combination of the approaches has the potential to give further performance improvements.

Ray Tracing using Hierarchies of Slab Cut Balls

In this paper, bounding volume trees of slab cut balls are evaluated and compared with other types of trees for raytracing. A novel tree construction algorithm is proposed, which utilizes a relative orientation heuristic betweenparent and child nodes. Also, a fast intersection test between a ray and a slab cut ball is presented. Experimentalcomparisons to other commonly used enclosing shapes reveal that the slab cut ball is attractive. In particular, theslab cut ball outperforms the sphere in all tested scenes with speed-up factors between 1 and 4.

An External Memory Algorithm for the Minimum Enclosing Ball Problem

In this article we present an external memory algorithm for computing the exact minimum enclosing ball of a massive set of points in any dimension. We test the performance of the algorithm on real-life three-dimensional data sets and demonstrate for the first time the practical efficiency of exact out-of-core algorithms. By use of simple heuristics, we achieve near-optimal I/O in all our test cases.

A Filtering Heuristic for the Computation of Minimum-Volume Enclosing Ellipsoids

We study heuristics to accelerate existing state-of-the-art algorithms for the minimum-volume enclosing ellipsoid problem. We propose a new filtering heuristic that can significantly reduce the number of distance computations performed in algorithms derived from Khachiyan’s first-order algorithm. Our experiments indicate that in high dimensions, the filtering heuristic is more effective than the elimination heuristic proposed by Harman and Pronzato. In lower dimensions, the elimination heuristic is superior.

Analysing switch-case code with abstract execution

Constructing the control-flow graph (CFG) of machine code is made difficult by dynamic transfers of control (DTC), where the address of the next instruction is computed at run-time. Switchcase statements make compilers generate a large variety of machine-code forms with DTC. Two analysis approaches are commonly used: pattern-matching methods identify predefined instruction patterns to extract the target addresses, while analytical methods try to compute the set of target addresses using a general value-analysis. We tested the abstract execution method of the SWEET tool as a value analysis for switch-case code. SWEET is here used as a plugin to the Bound-T tool: thus our work can also be seen as an experiment in modular tool design, where a general value-analysis tool is used to aid the CFG construction in a WCET analysis tool. We find that the abstract-execution analysis works at least as well as the switch-case analyses in Bound-T itself, which are mostly based on pattern-matching. However, there are still some weaknesses: the abstract domains available in SWEET are not well suited to representing sets of DTC target addresses, which are small but sparse and irregular. Also, in some cases the abstract-execution analysis fails because the used domain is not relational, that is, does not model arithmetic relationships between the values of different variables. Future work will be directed towards the design of abstract domains eliminating these weaknesses.