Tight Bounds for Randomized Load Balancing on Arbitrary Network Topologies

We consider the problem of balancing load items (tokens) in networks. Starting with an arbitrary load distribution, we allow nodes to exchange tokens with their neighbors in each round. The goal is to achieve a distribution where all nodes have nearly the same number of tokens. For the continuous case where tokens are arbitrarily divisible, most load balancing schemes correspond to Markov chains, whose convergence is fairly well-understood in terms of their spectral gap. However, in many applications, load items cannot be divided arbitrarily, and we need to deal with the discrete case where the load is composed of indivisible tokens. This discretization entails a non-linear behavior due to its rounding errors, which makes this analysis much harder than in the continuous case. We investigate several randomized protocols for different communication models in the discrete case. As our main result, we prove that for any regular network in the matching model, all nodes have the same load up to an additive constant in (asymptotically) the same number of rounds as required in the continuous case. This generalizes and tightens the previous best result, which only holds for expander graphs, and demonstrates that there is almost no difference between the discrete and continuous cases. Our results also provide a positive answer to the question of how well discrete load balancing can be approximated by (continuous) Markov chains, which has been posed by many researchers.Comment: 74 pages, 4 figure

Projected Richardson varieties and affine Schubert varieties

Let $G$ be a complex quasi-simple algebraic group and $G/P$ be a partial flag variety. The projections of Richardson varieties from the full flag variety form a stratification of $G/P$. We show that the closure partial order of projected Richardson varieties agrees with that of a subset of Schubert varieties in the affine flag variety of $G$. Furthermore, we compare the torus-equivariant cohomology and $K$-theory classes of these two stratifications by pushing or pulling these classes to the affine Grassmannian. Our work generalizes results of Knutson, Lam, and Speyer for the Grassmannian of type $A$.Comment: 20 pages. To appear in Annales de l'institut fourie

Is it appropriate to model turbidity currents with the three-equation model?

The three-equation model (TEM) was developed in the 1980s to model turbidity currents (TCs) and has been widely used ever since. However, its physical justification was questioned because self-accelerating TCs simulated with the steady TEM seemed to violate the turbulent kinetic energy balance. This violation was considered as a result of very strong sediment erosion that consumes more turbulent kinetic energy than is produced. To confine bed erosion and thus remedy this issue, the four-equation model (FEM) was introduced by assuming a proportionality between the bed shear stress and the turbulent kinetic energy. Here we analytically proof that self-accelerating TCs simulated with the original steady TEM actually never violate the turbulent kinetic energy balance, provided that the bed drag coefficient is not unrealistically low. We find that stronger bed erosion, surprisingly, leads to more production of turbulent kinetic energy due to conversion of potential energy of eroded material into kinetic energy of the current. Furthermore, we analytically show that, for asymptotically supercritical flow conditions, the original steady TEM always produces self-accelerating TCs if the upstream boundary conditions ("ignition" values) are chosen appropriately, while it never does so for asymptotically subcritical flow conditions. We numerically show that our novel method to obtain the ignition values even works for Richardson numbers very near to unity. Our study also includes a comparison of the TEM and FEM closures for the bed shear stress to simulation data of a coupled Large Eddy and Discrete Element Model of sediment transport in water, which suggests that the TEM closure might be more realistic than the FEM closure

A Comprehensive Study of Automatic Program Repair on the QuixBugs Benchmark

Automatic program repair papers tend to repeatedly use the same benchmarks. This poses a threat to the external validity of the findings of the program repair research community. In this paper, we perform an empirical study of automatic repair on a benchmark of bugs called QuixBugs, which has been little studied. In this paper, 1) We report on the characteristics of QuixBugs; 2) We study the effectiveness of 10 program repair tools on it; 3) We apply three patch correctness assessment techniques to comprehensively study the presence of overfitting patches in QuixBugs. Our key results are: 1) 16/40 buggy programs in QuixBugs can be repaired with at least a test suite adequate patch; 2) A total of 338 plausible patches are generated on the QuixBugs by the considered tools, and 53.3% of them are overfitting patches according to our manual assessment; 3) The three automated patch correctness assessment techniques, RGT_Evosuite, RGT_InputSampling and GT_Invariants, achieve an accuracy of 98.2%, 80.8% and 58.3% in overfitting detection, respectively. To our knowledge, this is the largest empirical study of automatic repair on QuixBugs, combining both quantitative and qualitative insights. All our empirical results are publicly available on GitHub in order to facilitate future research on automatic program repair

Asymptotic properties of maximum likelihood estimators in models with multiple change points

Models with multiple change points are used in many fields; however, the theoretical properties of maximum likelihood estimators of such models have received relatively little attention. The goal of this paper is to establish the asymptotic properties of maximum likelihood estimators of the parameters of a multiple change-point model for a general class of models in which the form of the distribution can change from segment to segment and in which, possibly, there are parameters that are common to all segments. Consistency of the maximum likelihood estimators of the change points is established and the rate of convergence is determined; the asymptotic distribution of the maximum likelihood estimators of the parameters of the within-segment distributions is also derived. Since the approach used in single change-point models is not easily extended to multiple change-point models, these results require the introduction of those tools for analyzing the likelihood function in a multiple change-point model.Comment: Published in at http://dx.doi.org/10.3150/09-BEJ232 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Poly(oxime-ester) Vitrimers with Catalyst-Free Bond Exchange.

Vitrimers are network polymers that undergo associative bond exchange reactions in the condensed phase above a threshold temperature, dictated by the exchangeable bonds comprising the vitrimer. For vitrimers, chemistries reliant on poorly nucleophilic bond exchange partners (e.g., hydroxy-functionalized alkanes) or poorly electrophilic exchangeable bonds, catalysts are required to lower the threshold temperature, which is undesirable in that catalyst leaching or deactivation diminishes its influence over time and may compromise reuse. Here we show how to access catalyst-free bond exchange reactions in catalyst-dependent polyester vitrimers by obviating conventional ester bonds in favor of oxime-esters. Poly(oxime-ester) (POE) vitrimers are synthesized using thiol-ene click chemistry, affording high stretchability and malleability. POE vitrimers are readily recycled with little degradation of their initial mechanical properties, suggesting exciting opportunities for sustainable plastics