In this paper we propose a special computational device which uses light rays for solving the Hamiltonian path problem on a directed graph. The device has a graph-like representation and the light is traversing it by following the routes given by the connections between nodes. In each node the rays are uniquely marked so that they can be easily identified. At the destination node we will search only for particular rays that have passed only once through each node. We show that the proposed device can solve small and medium instances of the problem in reasonable time.
A recent paper @cite_25 introduces the idea of sorting by using some properties of light. The method called Rainbow Sort is based on the physical concepts of refraction and dispersion. It is inspired by the observation that light that traverses a prism is sorted by wavelength (see Figure ). For implementing the Rainbow Sort one need to perform the following steps:
@cite_25
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Rainbow Sort is an unconventional method for sorting, which is based on the physical concepts of refraction and dispersion. It is inspired by the observation that light that traverses a prism is sorted by wavelength. At first sight this "rainbow effect" that appears in nature has nothing to do with a computation in the classical sense, still it can be used to design a sorting method that has the potential of running in ? (n) with a space complexity of ? (n), where n denotes the number of elements that are sorted. In Section 1, some upper and lower bounds for sorting are presented in order to provide a basis for comparisons. In Section 2, the physical background is outlined, the setup and the algorithm are presented and a lower bound for Rainbow Sort of ? (n) is derived. In Section 3, we describe essential difficulties that arise when Rainbow Sort is implemented. Particularly, restrictions that apply due to the Heisenberg uncertainty principle have to be considered. Furthermore, we sketch a possible implementation that leads to a running time of O(n+m), where m is the maximum key value, i.e., we assume that there are integer keys between 0 and m. Section 4 concludes with a summary of the complexity and some remarks on open questions, particularly on the treatment of duplicates and the preservation of references from the keys to records that contain the actual data. In Appendix A, a simulator is introduced that can be used to visualise Rainbow Sort. Sorting is a kernel algorithm for a wide range of applications. We present a new algorithm, GPU-Warpsort, to perform comparison-based parallel sort on Graphics Processing Units (GPUs). It mainly consists of a bitonic sort followed by a merge sort. Our algorithm achieves high performance by efficiently mapping the sorting tasks to GPU architectures. Firstly, we take advantage of the synchronous execution of threads in a warp to eliminate the barriers in bitonic sorting network. We also provide sufficient homogeneous parallel operations for all the threads within a warp to avoid branch divergence. Furthermore, we implement the merge sort efficiently by assigning each warp independent pairs of sequences to be merged and by exploiting totally coalesced global memory accesses to eliminate the bandwidth bottleneck. Our experimental results indicate that GPU-Warpsort works well on different kinds of input distributions, and it achieves up to 30 higher performance than previous optimized comparison-based GPU sorting algorithm on input sequences with millions of elements. A subgraph of an edge-coloured complete graph is called rainbow if all its edges have different colours. In 1980 Hahn conjectured that every properly edge-coloured complete graph @math has a rainbow Hamiltonian path. Although this conjecture turned out to be false, it was widely believed that such a colouring always contains a rainbow cycle of length almost @math . In this paper, improving on several earlier results, we confirm this by proving that every properly edge-coloured @math has a rainbow cycle of length @math . One of the main ingredients of our proof, which is of independent interest, shows that a random subgraph of a properly edge-coloured @math formed by the edges of a random set of colours has a similar edge distribution as a truly random graph with the same edge density. In particular it has very good expansion properties. This paper addresses light transport through a discrete random medium, which we define as a volume filled with macroscopic scattering geometry generated by a random process. This formulation is more general than standard radiative transport, because it can be applied to media that are made up of closely packed scatterers. A new approach to rendering these media is introduced, based on precomputed solutions to a local multiple scattering problem, including a new algorithm for generating paths through random media that moves through the interior of the medium in large strides without considering individual scattering events. A method for rendering homogeneous isotropic random media is described that generates paths using precomputed scattering solutions compressed and randomly sampled using Nonnegative Matrix Factorization. It can efficiently render discrete media, such as a large pile of glass objects, in which the individual scatterers are visible. The method is demonstrated on scenes containing tens of thousands of transparent, specular objects that are nearly impossible to render with standard global illumination techniques.