A Repair Framework for Scalar MDS Codes

Several works have developed vector-linear maximum-distance separable (MDS) storage codes that min- imize the total communication cost required to repair a single coded symbol after an erasure, referred to as repair bandwidth (BW). Vector codes allow communicating fewer sub-symbols per node, instead of the entire content. This allows non trivial savings in repair BW. In sharp contrast, classic codes, like Reed- Solomon (RS), used in current storage systems, are deemed to suffer from naive repair, i.e. downloading the entire stored message to repair one failed node. This mainly happens because they are scalar-linear. In this work, we present a simple framework that treats scalar codes as vector-linear. In some cases, this allows significant savings in repair BW. We show that vectorized scalar codes exhibit properties that simplify the design of repair schemes. Our framework can be seen as a finite field analogue of real interference alignment. Using our simplified framework, we design a scheme that we call clique-repair which provably identifies the best linear repair strategy for any scalar 2-parity MDS code, under some conditions on the sub-field chosen for vectorization. We specify optimal repair schemes for specific (5,3)- and (6,4)-Reed- Solomon (RS) codes. Further, we present a repair strategy for the RS code currently deployed in the Facebook Analytics Hadoop cluster that leads to 20% of repair BW savings over naive repair which is the repair scheme currently used for this code.Comment: 10 Pages; accepted to IEEE JSAC -Distributed Storage 201

An updated survey on rainbow connections of graphs - a dynamic survey

The concept of rainbow connection was introduced by Chartrand, Johns, McKeon and Zhang in 2008. Nowadays it has become a new and active subject in graph theory. There is a book on this topic by Li and Sun in 2012, and a survey paper by Li, Shi and Sun in 2013. More and more researchers are working in this field, and many new papers have been published in journals. In this survey we attempt to bring together most of the new results and papers that deal with this topic. We begin with an introduction, and then try to organize the work into the following categories, rainbow connection coloring of edge-version, rainbow connection coloring of vertex-version, rainbow $k$-connectivity, rainbow index, rainbow connection coloring of total-version, rainbow connection on digraphs, rainbow connection on hypergraphs. This survey also contains some conjectures, open problems and questions for further study

08431 Abstracts Collection -- Moderately Exponential Time Algorithms

From $19/10/2008$ to $24/10/2008$, the Dagstuhl Seminar 08431 ``Moderately Exponential Time Algorithms \u27\u27 was held in Schloss Dagstuhl~--~Leibniz Center for Informatics. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available

Fundamental Limits of Nanophotonic Design

Nanoscale fabrication techniques, computational inverse design, and fields from silicon photonics to metasurface optics are enabling transformative use of an unprecedented number of structural degrees of freedom in nanophotonics. A critical need is to understand the extreme limits to what is possible by engineering nanophotonic structures. This thesis establishes the first general theoretical framework identifying fundamental limits to light--matter interactions. It derives bounds for applications across nanophotonics, including far-field scattering, optimal wavefront shaping, optical beam switching, and wave communication, as well as the miniaturization of optical components, including perfect absorbers, linear optical analog computing units, resonant optical sensors, multilayered thin films, and high-NA metalenses. The bounds emerge from an infinite set of physical constraints that have to be satisfied by polarization fields in response to an excitation. The constraints encode power conservation in single-scenario scattering and requisite field correlations in multi-scenario scattering. The framework developed in this thesis, encompassing general linear wave scattering dynamics, offers a new way to understand optimal designs and their fundamental limits, in nanophotonics and beyond.Comment: PhD thesi

Distance-regular graphs

This is a survey of distance-regular graphs. We present an introduction to distance-regular graphs for the reader who is unfamiliar with the subject, and then give an overview of some developments in the area of distance-regular graphs since the monograph 'BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A., Distance-Regular Graphs, Springer-Verlag, Berlin, 1989] was written.Comment: 156 page

Extracting Data-Level Parallelism in High-Level Synthesis for Reconfigurable Architectures

High-Level Synthesis (HLS) tools are a set of algorithms that allow programmers to obtain implementable Hardware Description Language (HDL) code from specifications written high-level, sequential languages such as C, C++, or Java. HLS has allowed programmers to code in their preferred language while still obtaining all the benefits hardware acceleration has to offer without them needing to be intimately familiar with the hardware platform of the accelerator. In this work we summarize and expand upon several of our approaches to improve the automatic memory banking capabilities of HLS tools targeting reconfigurable architectures, namely Field-Programmable Gate Arrays or FPGA\u27s. We explored several approaches to automatically find the optimal partition factor and a usable banking scheme for stencil kernels including a tessellation based approach using multiple families of hyperplanes to do the partitioning which was able to find a better banking factor than current state-of-the-art methods and a graph theory methodology that allowed us to mathematically prove the optimality of our banking solutions. For non-stencil kernels we relaxed some of the conditions in our graph-based model to propose a best-effort solution to arbitrarily reduce memory access conflicts (simultaneous accesses to the same memory bank). We also proposed a non-linear transformation using prime factorization to convert a small subset of non-stencil kernels into stencil memory accesses, allowing us to use all previous work in memory partition to them. Our approaches were able to obtain better results than commercial tools and state-of-the-art algorithms in terms of reduced resource utilization and increased frequency of operation. We were also able to obtain better partition factors for some stencil kernels and usable baking schemes for non-stencil kernels with better performance than any applicable existing algorithm