Non-empty Bins Research Articles

Near-Optimal Stochastic Bin-Packing in Large Service Systems with Time-Varying Item Sizes

In modern computing systems, jobs' resource requirements often vary over time. Accounting for this temporal variability during job scheduling is essential for meeting performance goals. However, theoretical understanding on how to schedule jobs with time-varying resource requirements is limited. Motivated by this gap, we propose a new setting of the stochastic bin-packing problem in service systems that allows for time-varying job resource requirements, also referred to as 'item sizes' in traditional bin-packing terms. In this setting, a job or 'item' must be dispatched to a server or 'bin' upon arrival. Its resource requirement varies over time as a Markov chain while in service. Our goal is to minimize the expected number of active servers, or 'non-empty bins', in the steady state. Under our problem formulation, we develop a job dispatch policy, named Join-Requesting-Server (JRS). We show that in the asymptotic regime where the job arrival rate scales large linearly with respect to a scaling factor r, JRS achieves an additive optimality gap of O(√r) in the objective value, where the optimal objective value is Θ(r). Our approach highlights a novel policy conversion framework that utilizes the solution of a single-server problem.

Near-Optimal Stochastic Bin-Packing in Large Service Systems with Time-Varying Item Sizes

In modern computing systems, jobs' resource requirements often vary over time. Accounting for this temporal variability during job scheduling is essential for meeting performance goals. However, theoretical understanding on how to schedule jobs with time-varying resource requirements is limited. Motivated by this gap, we propose a new setting of the stochastic bin-packing problem in service systems that allows for time-varying job resource requirements, also referred to as 'item sizes' in traditional bin-packing terms. In this setting, a job or 'item' must be dispatched to a server or 'bin' upon arrival. Its resource requirement may vary over time while in service, following a Markovian assumption. Once the job's service is complete, it departs from the system. Our goal is to minimize the expected number of active servers, or 'non-empty bins', in steady state. Under our problem formulation, we develop a job dispatch policy, named Join-Reqesting-Server (JRS). Broadly, JRS lets each server independently evaluate its current job configuration and decide whether to accept additional jobs, balancing the competing objectives of maximizing throughput and minimizing the risk of resource capacity overruns. The JRS dispatcher then utilizes these individual evaluations to decide which server to dispatch each arriving job to. The theoretical performance guarantee of JRS is in the asymptotic regime where the job arrival rate scales large linearly with respect to a scaling factor r. We show that JRS achieves an additive optimality gap of O(√r) in the objective value, where the optimal objective value is Θ(r). When specialized to constant job resource requirements, our result improves upon the state-of-the-art o(r) optimality gap. Our technical approach highlights a novel policy conversion framework that reduces the policy design problem into a single-server problem.

Histogram modification in adaptive bi-histogram equalization for contrast enhancement on digital images

In this paper, a novel histogram modification-based bi histogram equalization (HE) approach for contrast enhancement on digital images is presented. At first, a power-logarithm transformation function is used to change the histogram of the input image. The logarithm operation reduces the input histogram’s excessive peaks, while the power function restores the histogram structure. The adjusted histogram is then separated into two sub-histograms at the threshold limit, using the input image’s minimum intensity and standard deviation. Sub-histograms are clipped based on their individual plateau limit parameters, which are based on the median value of the individual sub-histogram, to control the over-enhanced outcomes. The clipped pixels are then redistributed evenly across the histogram’s non-empty bins. Finally, the intended outcome is achieved by applying the HE procedure on the updated individual sub-histogram. Simulation results show that the proposed HE approach successfully improves the visual quality of the images. Quantitative measurements such as entropy, feature similarity index measure (FSIM), spectral residual similarity index measure (SR-SIM), gradient magnitude similarity deviation (GMSD), visual saliency-induced index (VSI) and multi scale structural similarity index measure (MS-SSIM) efficiently confirm the effectiveness of the suggested method when compared with the existing enhancement techniques. Furthermore, a comparative study of the different approaches is performed using a dataset of 300 images, demonstrating the superiority of the suggested method over the other state-of-the-art techniques.

Propagation of chaos for a general balls into bins dynamics

Consider $N$ balls initially placed in $L$ bins. At each time step take a ball from each non-empty bin and \emph{randomly} reassign the balls into the bins.We call this finite Markov chain \emph{General Repeated Balls into Bins} process. It is a discrete time interacting particles system with parallel updates. Assuming a \emph{quantitative} chaotic condition on the reassignment rule we prove a \emph{quantitative} propagation of chaos for this model. We furthermore study some equilibrium properties of the limiting nonlinear process.

Mixing time for the Repeated Balls into Bins dynamics

We consider a nonreversible finite Markov chain called Repeated Balls-into-Bins (RBB) process. This process is a discrete time conservative interacting particle system with parallel updates. Place initially in $L$ bins $rL$ balls, where $r$ is a fixed positive constant. At each time step a ball is removed from each non-empty bin. Then all these removed balls are uniformly reassigned into bins. We prove that the mixing time of the RBB process is of order $L$. Furthermore we show that if the initial configuration has $o(L)$ balls per site the equilibrium is attained in $o(L)$ steps.

Propagation of chaos for a balls into bins model

Consider a finite number of balls initially placed in $L$ bins. At each time step a ball is taken from each non-empty bin. Then all the balls are uniformly reassigned into bins. This finite Markov chain is called Repeated Balls-into-Bins process and is a discrete time interacting particle system with parallel updating. We prove that, starting from a suitable (chaotic) set of initial states, as $L\to+\infty$, the numbers of balls in each bin becomes independent from the rest of the system i.e. we have propagation of chaos. We furthermore study some equilibrium properties of the limiting nonlinear process.

Self-Stabilizing Balls and Bins in Batches

A fundamental problem in distributed computing is the distribution of requests to a set of uniform servers without a centralized controller. Classically, such problems are modeled as static balls into bins processes, where m balls (tasks) are to be distributed among n bins (servers). In a seminal work, Azar et al. (SIAM J Comput 29(1):180–200, 1999. https://doi.org/10.1137/S0097539795288490 ) proposed the sequential strategy $$\textsc {Greedy}[{d}]$$ for $$n=m$$ . Each ball queries the load of d random bins and is allocated to a least loaded of them. Azar et al. (1999) showed that $$d=2$$ yields an exponential improvement compared to $$d=1$$ . Berenbrink et al. (SIAM J Comput 35(6):1350–1385, 2006. https://doi.org/10.1137/S009753970444435X ) extended this to $$m\gg n$$ , showing that for $$d=2$$ the maximal load difference is independent of m (in contrast to the $$d=1$$ case). We propose a new variant of an infinite balls-into-bins process. In each round an expected number of $$\lambda n$$ new balls arrive and are distributed (in parallel) to the bins. Subsequently, each non-empty bin deletes one of its balls. This setting models a set of servers processing incoming requests, where clients can query a server’s current load but receive no information about parallel requests. We study the $$\textsc {Greedy}[{d}]$$ distribution scheme in this setting and show a strong self-stabilizing property: for any arrival rate $$\lambda =\lambda (n)<1$$ , the system load is time-invariant. Moreover, for any (even super-exponential) round t, the maximum system load is (w.h.p.) for $$d=1$$ and for $$d=2$$ . In particular, $$\textsc {Greedy}[{2}]$$ has an exponentially smaller system load for high arrival rates.

Self-stabilizing repeated balls-into-bins

We study the following synchronous process that we call repeated balls-into-bins. The process is started by assigning n balls to n bins in an arbitrary fashion. In every subsequent round, one ball is extracted from each non-empty bin according to some fixed strategy (random, FIFO, etc), and re-assigned to one of the n bins uniformly at random. We define a configuration legitimate if its maximum load is mathcal {O}(log n). We prove that, starting from any configuration, the process converges to a legitimate configuration in linear time and then only takes on legitimate configurations over a period of length bounded by any polynomial in n, with high probability (w.h.p.). This implies that the process is self-stabilizing and that every ball traverses all bins within mathcal {O}(nlog ^2 n) rounds, w.h.p.

Signatures versus histograms: Definitions, distances and algorithms

The aim of this paper is to present a new method to compare histograms. The main advantage is that there is an important time-complexity reduction with respect to the methods presented before. This reduction is statistically and analytically demonstrated in the paper. The distances between histograms that we presented are defined on a structure called signature, which is a lossless representation of histograms. Moreover, the type of the elements of the sets that the histograms represent are ordinal, nominal and modulo. We show that the computational cost of these distances is O ( z ′ ) for the ordinal and nominal types and O ( z ′ 2 ) for the modulo one, z ′ being the number of non-empty bins of the histograms. The computational cost of the algorithms presented in the literature depends on the number of bins of the histograms. In most of the applications, the obtained histograms are sparse; then considering only the non-empty bins drastically decreases the time consumption of the comparison. The distances and algorithms presented in this paper are experimentally validated on the comparison of images obtained from public databases and positioning of mobile robots through the recognition of indoor scenes (captured in a learning stage).

Component analysis of the responses of sensory neurons to combined sinusoidal and triangular stimulation

A method for quantitative estimation of sensory neuron sensitivity to small sinusoidal stimuli in the presence of sizable background drift (in the stimulus or response) was developed. The performance of the method was tested by analyzing the responses of 17 muscle spindle primary (Ia) afferent neurons to concomitant sinusoidal and triangular stretching of the soleus muscle. The efficacy and accuracy of several variations of the method were examined. The variations included the use of probability density (PD) and average frequency (AF) histograms as the basis for calculations and two different algorithms for the decomposition of responses to combined sinusoidal and triangular stimulation. One algorithm called the ‘inherent-drift’ method exploited the inherent half-cycle repeat property of a sine wave to extract the drift component. Another algorithm called the ‘forced-drift’ method first estimated the drift by linear regression to a response to triangular stimulation alone. The drift estimate (a slope value) was then subtracted from the response to combined sinusoidal and triangular stimulation of the same triangular (background) velocity. A comparison of the performance of the drift correction method applied either to PD or AF histograms revealed no significant differences in the estimates of sinusoidal modulation. The limitations of the AF method were manifest primarily by phase lags at low mean levels of action potential discharge. Calculation of the response parameters using the ‘inherent-drift’ correction procedure proved straightforward as long as there were at least two pairs of non-empty bins in the sine-cycle histograms on which to base the estimate of drift. The method remained effective in determining sinusoidal sensitivity in the face of distinct non-linearities (harmonic distortions) in the sine-cycle histograms. However, estimates of slope and the extraction of sinusoidal phase by the ‘inherent’ slope correction method became subject to large errors. Under such circumstances, more reliable estimates could be obtained by using the forced drift-correction method instead. The importance of extracting the drift component prior to estimating the sinusoidal response parameters was evaluated experimentally and theoretically. In general, omission of a drift correction introduced a large bias in the estimates of the phase of sinusoidal response, whereas the estimate of sinusoidal modulation was rather insensitive. Experimental findings were fully accounted for by theoretical considerations. Analytically derived relationships identified low- and high-risk regions more clearly for the estimate of sinusoidal modulation than of phase. The relationship between biased modulation estimate and underlying drift showed minima characteristics with a low-risk region, where absolute errors and dependence on slope variations were small. The precise shape of these characteristic curves depended on the phase value of the sinusoidal response component. This suggests that in situations where the phase of sinusoidal responses can be estimated independently, experimental paradigms can be optimized by adjusting the phase of sinusoidal stimuli, so as to minimize the effects of complete omission of drift correction or of inappropriate drift determination on the estimates of modulation. In contrast, for phase estimation low-risk regions are very narrow, indicating that inappropriate drift compensation almost invariably leads to significant errors in phase estimates. In its current form, the method is directly applicable to sensory responses which consist of rate-modulated trains of action potentials. Nevertheless, the same principles used to develop the method could be applied to sensory responses in the form of receptor or generator potentials. The method should therefore be applicable to the analysis of a wide variety of sensory systems.