Fault Tolerance in Networks of Bounded Degree

Achieving processor cooperation in the presence of faults is a major problem in distributed systems. Popular paradigms such as Byzantine agreement have been studied principally in the context of a complete network. Indeed, Dolev [J. Algorithms, 3 (1982), pp. 14–30] and Hadzilacos [Issues of Fault Tolerance in Concurrent Computations, Ph.D. thesis, Harvard University, Cambridge, MA, 1984] have shown that Ω(t) connectivity is necessary if the requirement is that all nonfaulty processors decide unanimously, where t is the number of faults to be tolerated. We believe that in forseeable technologies the number of faults will grow with the size of the network while the degree will remain practically fixed. We therefore raise the question whether it is possible to avoid the connectivity requirements by slightly lowering our expectations. In many practical situations we may be willing to “lose” some correct processors and settle for cooperation between the vast majority of the processors. Thus motivated, we present a general simulation technique by which vertices (processors) in almost any network of bounded degree can simulate an algorithm designed for the complete network. The simulation has the property that although some correct processors may be cut off from the majority of the network by faulty processors, the vast majority of the correct processors will be able to communicate among themselves undisturbed by the (arbitrary) behavior of the faulty nodes. We define a new paradigm for distributed computing, almost-everywhere agreement, in which we require only that almost all correct processors reach consensus. Unlike the traditional Byzantine agreement problem, almost-everywhere agreement can be solved on networks of bounded degree. Specifically, we can simulate any sufficiently resilient Byzantine agreement algorithm on a network of bounded degree using our communication scheme described above. Although we “lose” some correct processors, effectively treating them as faulty, the vast majority of correct processors decide on a common value

Calibrationless Multi-coil Magnetic Resonance Imaging with Compressed Sensing

We present a method for combining the data retrieved by multiple coils of a Magnetic Resonance Imaging (MRI) system with the a priori assumption of compressed sensing to reconstruct a single image. The final image is the result of an optimization problem that only includes constraints based on fundamental physics (Maxwell's equations and the Biot-Savart law) and accepted phenomena (e.g. sparsity in the Wavelet domain). The problem is solved using an alternating minimization approach: two convex optimization problems are alternately solved, one with the Fast Iterative Shrinkage Threshold Algorithm (FISTA) and the other with the Primal-Dual Hybrid Gradient (PDHG) method. We show results on simulated data as well as data of the knee, brain, and ankle. In all cases studied, results from the new algorithm show higher quality and increased detail when compared to conventional reconstruction algorithms

Global epigenomic reconfiguration during mammalian brain development

DNA methylation is implicated in mammalian brain development and plasticity underlying learning and memory. We report the genome-wide composition, patterning, cell specificity, and dynamics of DNA methylation at single-base resolution in human and mouse frontal cortex throughout their lifespan. Widespread methylome reconfiguration occurs during fetal to young adult development, coincident with synaptogenesis. During this period, highly conserved non-CG methylation (mCH) accumulates in neurons, but not glia, to become the dominant form of methylation in the human neuronal genome. Moreover, we found an mCH signature that identifies genes escaping X-chromosome inactivation. Last, whole-genome single-base resolution 5-hydroxymethylcytosine (hmC) maps revealed that hmC marks fetal brain cell genomes at putative regulatory regions that are CG-demethylated and activated in the adult brain and that CG demethylation at these hmC-poised loci depends on Tet2 activity