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Based on the NilHecke algebra $\mathsf{NH}_n$, the odd NilHecke algebra developed by Ellis, Khovanov and Lauda and Kang, Kashiwara and Tsuchioka's quiver Hecke superalgebra, we develop the Clifford Hecke superalgebra $\mathsf{NH}\mathfrak{C}_n$ as another super-algebraic analogue of $\mathsf{NH}_n$. We show that there is a notion of symmetric polynomials fitting in this picture, and we prove that these are generated by an appropriate analogue of elementary symmetric polynomials, whose properties we shall discuss in this text.

This work presents a producer-consumer link between two independent clock domains. The link allows for metastability-free, low-latency, high-throughput communication by slight adjustments to the clock frequencies of the producer and consumer domains steered by a controller circuit. Any such controller cannot deterministically avoid, detect, nor resolve metastability. Typically, this is addressed by synchronizers, incurring a larger dead time in the control loop. We follow the approach of Friedrichs et al. (TC 2018) who proposed metastability-containing circuits. The result is a simple control circuit that may become metastable, yet deterministically avoids buffer underrun or overflow. More specifically, the controller output may become metastable, but this may only affect oscillator speeds within specific bounds. In contrast, communication is guaranteed to remain metastability-free. We formally prove correctness of the producer-consumer link and a possible implementation that has only small overhead. With SPICE simulations of the proposed implementation we further substantiate our claims. The simulation uses 65nm process running at roughly 2GHz., Comment: 12 page journal article

The vast majority of hardware architectures use a carefully timed reference signal to clock their computational logic. However, standard distribution solutions are not fault-tolerant. In this work, we present a simple grid structure as a more reliable clock propagation method and study it by means of simulation experiments. Fault-tolerance is achieved by forwarding clock pulses on arrival of the second of three incoming signals from the previous layer. A key question is how well neighboring grid nodes are synchronized, even without faults. Analyzing the clock skew under typical-case conditions is highly challenging. Because the forwarding mechanism involves taking the median, standard probabilistic tools fail, even when modeling link delays just by unbiased coin flips. Our statistical approach provides substantial evidence that this system performs surprisingly well. Specifically, in an "infinitely wide" grid of height~$H$, the delay at a pre-selected node exhibits a standard deviation of $O(H^{1/4})$ ($\approx 2.7$ link delay uncertainties for $H=2000$) and skew between adjacent nodes of $o(\log \log H)$ ($\approx 0.77$ link delay uncertainties for $H=2000$). We conclude that the proposed system is a very promising clock distribution method. This leads to the open problem of a stochastic explanation of the tight concentration of delays and skews. More generally, we believe that understanding our very simple abstraction of the system is of mathematical interest in its own right., Comment: 16 pages, 11 figures

Consider an arbitrary network of communicating modules on a chip, each requiring a local signal telling it when to execute a computational step. There are three common solutions to generating such a local clock signal: (i) by deriving it from a single, central clock source, (ii) by local, free-running oscillators, or (iii) by handshaking between neighboring modules. Conceptually, each of these solutions is the result of a perceived dichotomy in which (sub)systems are either clocked or fully asynchronous, suggesting that the designer's choice is limited to deciding where to draw the line between synchronous and asynchronous design. In contrast, we take the view that the better question to ask is how synchronous the system can and should be. Based on a distributed clock synchronization algorithm, we present a novel design providing modules with local clocks whose frequency bounds are almost as good as those of corresponding free-running oscillators, yet neighboring modules are guaranteed to have a phase offset substantially smaller than one clock cycle. Concretely, parameters obtained from a 15nm ASIC implementation running at 2GHz yield mathematical worst-case bounds of 30ps on phase offset for a 32x32 node grid network.

For a semisimple complex Lie algebra $\mathfrak g$, the BGG category $\mathcal{O}$ is of particular interest in representation theory. It is known that Irving's shuffling functors $\mathrm{Sh}_{w}$, indexed by elements $w\in W$ of the Weyl group, induce an action of the braid group $B_W$ associated to $W$ on the derived categories $D^\mathrm{b}(\mathcal{O}_\lambda)$ of blocks of $\mathcal{O}$. We show that for maximal parabolic subalgebras $\mathfrak{p}$ of $\mathfrak{sl}_n$ corresponding to the parabolic subgroup $W_\mathfrak{p}=S_{n-1}\times S_1$ of $S_n$, the derived shuffling functors $\mathbf{L}\mathrm{Sh}{s_i}$ are instances of Seidel and Thomas' spherical twist functors. Namely, we show that certain parabolic indecomposable projectives $P^\mathfrak{p}(w)$ are spherical objects, and the associated twist functors are naturally isomorphic to $\mathbf{L}\mathrm{Sh}{w}[1]$ as auto-equivalences of $D^\mathrm{b}(\mathcal{O}^\mathfrak{p})$. We give an overview of the main properties of the BGG category $\mathcal{O}$, the construction of shuffling and spherical twist functors, and give some examples how to determine images of both. To this end, we employ the equivalence of blocks of $\mathcal{O}$ and the module categories of certain path algebras., Comment: 31 pages

In their seminal paper from 2004, Kuhn, Moscibroda, and Wattenhofer (KMW) proved a hardness result for several fundamental graph problems in the LOCAL model: For any (randomized) algorithm, there are input graphs with $n$ nodes and maximum degree $\Delta$ on which $\Omega(\min\{\sqrt{\log n/\log \log n},\log \Delta/\log \log \Delta\})$ (expected) communication rounds are required to obtain polylogarithmic approximations to a minimum vertex cover, minimum dominating set, or maximum matching. Via reduction, this hardness extends to symmetry breaking tasks like finding maximal independent sets or maximal matchings. Today, more than $15$ years later, there is still no proof of this result that is easy on the reader. Setting out to change this, in this work, we provide a fully self-contained and $\mathit{simple}$ proof of the KMW lower bound. The key argument is algorithmic, and it relies on an invariant that can be readily verified from the generation rules of the lower bound graphs., Comment: 20 pages, 6 figures

Friedrichs et al. (TC 2018) showed that metastability can be contained when sorting inputs arising from time-to-digital converters, i.e., measurement values can be correctly sorted without resolving metastability using synchronizers first. However, this work left open whether this can be done by small circuits. We show that this is indeed possible, by providing a circuit that sorts Gray code inputs (possibly containing a metastable bit) and has asymptotically optimal depth and size. Our solution utilizes the parallel prefix computation (PPC) framework (JACM 1980). We improve this construction by bounding its fan-out by an arbitrary $f \geq 3$, without affecting depth and increasing circuit size by a small constant factor only. Thus, we obtain the first PPC circuits with asymptotically optimal size, constant fan-out, and optimal depth. To show that applying the PPC framework to the sorting task is feasible, we prove that the latter can, despite potential metastability, be decomposed such that the core operation is associative. We obtain asymptotically optimal metastability-containing sorting networks. We complement these results with simulations, independently verifying the correctness as well as small size and delay of our circuits., Comment: This article generalizes and extends work presented at DATE 2018

In many models for large-scale computation, decomposition of the problem is key to efficient algorithms. For distance-related graph problems, it is often crucial that such a decomposition results in clusters of small diameter, while the probability that an edge is cut by the decomposition scales linearly with the length of the edge. There is a large body of literature on low diameter graph decomposition with small edge cutting probabilities, with all existing techniques heavily building on single source shortest paths (SSSP) computations. Unfortunately, in many theoretical models for large-scale computations, the SSSP task constitutes a complexity bottleneck. Therefore, it is desirable to replace exact SSSP computations with approximate ones. However this imposes a fundamental challenge since the existing constructions of such decompositions inherently rely on the subtractive form of the triangle inequality. The current paper overcomes this obstacle by developing a technique termed blurry ball growing. By combining this technique with a clever algorithmic idea of Miller et al. (SPAA 13), we obtain a construction of low diameter decompositions with small edge cutting probabilities which replaces exact SSSP computations by (a small number of) approximate ones. The utility of our approach is showcased by deriving efficient algorithms that work in the Congest, PRAM, and semi-streaming models of computation. As an application, we obtain metric tree embedding algorithms in the vein of Bartal (FOCS 96) whose computational complexities in these models are optimal up to polylogarithmic factors. Our embeddings have the additional useful property that the tree can be mapped back to the original graph such that each edge is "used" only O(log n) times, which is of interest for capacitated problems and simulating Congest algorithms on the tree into which the graph is embedded.

We study parallel algorithms for the classical balls-into-bins problem, in which $m$ balls acting in parallel as separate agents are placed into $n$ bins. Algorithms operate in synchronous rounds, in each of which balls and bins exchange messages once. The goal is to minimize the maximal load over all bins using a small number of rounds and few messages. While the case of $m=n$ balls has been extensively studied, little is known about the heavily loaded case. In this work, we consider parallel algorithms for this somewhat neglected regime of $m\gg n$. The naive solution of allocating each ball to a bin chosen uniformly and independently at random results in maximal load $m/n+\Theta(\sqrt{m/n\cdot \log n})$ (for $m\geq n \log n$) w.h.p. In contrast, for the sequential setting Berenbrink et al (SIAM J. Comput 2006) showed that letting each ball join the least loaded bin of two randomly selected bins reduces the maximal load to $m/n+O(\log\log m)$ w.h.p. To date, no parallel variant of such a result is known. We present a simple parallel threshold algorithm that obtains a maximal load of $m/n+O(1)$ w.h.p. within $O(\log\log (m/n)+\log^* n)$ rounds. The algorithm is symmetric (balls and bins all "look the same"), and balls send $O(1)$ messages in expectation per round. The additive term of $O(\log^* n)$ in the complexity is known to be tight for such algorithms (Lenzen and Wattenhofer Distributed Computing 2016). We also prove that our analysis is tight, i.e., algorithms of the type we provide must run for $\Omega(\min\{\log\log (m/n),n\})$ rounds w.h.p. Finally, we give a simple asymmetric algorithm (i.e., balls are aware of a common labeling of the bins) that achieves a maximal load of $m/n + O(1)$ in a constant number of rounds w.h.p. Again, balls send only a single message per round, and bins receive $(1+o(1))m/n+O(\log n)$ messages w.h.p.

Many graph problems are locally checkable: a solution is globally feasible if it looks valid in all constant-radius neighborhoods. This idea is formalized in the concept of locally checkable labelings (LCLs), introduced by Naor and Stockmeyer (1995). Recently, Chang et al. (2016) showed that in bounded-degree graphs, every LCL problem belongs to one of the following classes: - "Easy": solvable in $O(\log^* n)$ rounds with both deterministic and randomized distributed algorithms. - "Hard": requires at least $\Omega(\log n)$ rounds with deterministic and $\Omega(\log \log n)$ rounds with randomized distributed algorithms. Hence for any parameterized LCL problem, when we move from local problems towards global problems, there is some point at which complexity suddenly jumps from easy to hard. For example, for vertex coloring in $d$-regular graphs it is now known that this jump is at precisely $d$ colors: coloring with $d+1$ colors is easy, while coloring with $d$ colors is hard. However, it is currently poorly understood where this jump takes place when one looks at defective colorings. To study this question, we define $k$-partial $c$-coloring as follows: nodes are labeled with numbers between $1$ and $c$, and every node is incident to at least $k$ properly colored edges. It is known that $1$-partial $2$-coloring (a.k.a. weak $2$-coloring) is easy for any $d \ge 1$. As our main result, we show that $k$-partial $2$-coloring becomes hard as soon as $k \ge 2$, no matter how large a $d$ we have. We also show that this is fundamentally different from $k$-partial $3$-coloring: no matter which $k \ge 3$ we choose, the problem is always hard for $d = k$ but it becomes easy when $d \gg k$. The same was known previously for partial $c$-coloring with $c \ge 4$, but the case of $c < 4$ was open.

To date, infrared interferometry at best achieved contrast ratios of a few times $10^{-4}$ on bright targets. GRAVITY, with its dual-field mode, is now capable of high contrast observations, enabling the direct observation of exoplanets. We demonstrate the technique on HR8799, a young planetary system composed of four known giant exoplanets. We used the GRAVITY fringe tracker to lock the fringes on the central star, and integrated off-axis on the HR8799e planet situated at 390 mas from the star. Data reduction included post-processing to remove the flux leaking from the central star and to extract the coherent flux of the planet. The inferred K band spectrum of the planet has a spectral resolution of 500. We also derive the astrometric position of the planet relative to the star with a precision on the order of 100$\,\mu$as. The GRAVITY astrometric measurement disfavors perfectly coplanar stable orbital solutions. A small adjustment of a few degrees to the orbital inclination of HR 8799 e can resolve the tension, implying that the orbits are close to, but not strictly coplanar. The spectrum, with a signal-to-noise ratio of $\approx 5$ per spectral channel, is compatible with a late-type L brown dwarf. Using Exo-REM synthetic spectra, we derive a temperature of $1150\pm50$\,K and a surface gravity of $10^{4.3\pm0.3}\,$cm/s$^{2}$. This corresponds to a radius of $1.17^{+0.13}_{-0.11}\,R_{\rm Jup}$ and a mass of $10^{+7}_{-4}\,M_{\rm Jup}$, which is an independent confirmation of mass estimates from evolutionary models. Our results demonstrate the power of interferometry for the direct detection and spectroscopic study of exoplanets at close angular separations from their stars., Comment: published in A&A

Synchronizing clocks in distributed systems is well-understood, both in terms of fault-tolerance in fully connected systems and the dependence of local and global worst-case skews (i.e., maximum clock difference between neighbors and arbitrary pairs of nodes, respectively) on the diameter of fault-free systems. However, so far nothing non-trivial is known about the local skew that can be achieved in topologies that are not fully connected even under a single Byzantine fault. Put simply, in this work we show that the most powerful known techniques for fault-tolerant and gradient clock synchronization are compatible, in the sense that the best of both worlds can be achieved simultaneously. Concretely, we combine the Lynch-Welch algorithm [Welch1988] for synchronizing a clique of $n$ nodes despite up to $f