Stopping Sets Research Articles

Stopping Sets of Algebraic Geometry Codes over Hyperelliptic Curves of Genus Two

Stopping sets are useful for analyzing the performance of a linear code under an iterative decoding algorithm over an erasure channel. In this paper, we consider stopping sets of one-point algebraic geometry codes defined by a hyperelliptic curve of genus g=2 defined by the plane model y2=f(x), where the degree of f(x) was 5. We completely classify the stopping sets of the one-point algebraic geometric codes C=CΩ(D,mP∞) defined by a hyperelliptic curve of genus 2 with m≤4. For m=3, we proved in detail that all sets S⊆{1,2,⋯,n} of a size greater than 3 are stopping sets and we give an example of sets of size 2,3 that are not.

Invertible Bloom Lookup Tables with Listing Guarantees

The Invertible Bloom Lookup Table (IBLT) is a probabilistic concise data structure for set representation that supports a listing operation as the recovery of the elements in the represented set. Its applications can be found in network synchronization and traffic monitoring as well as in error-correction codes. IBLT can list its elements with probability affected by the size of the allocated memory and the size of the represented set, such that it can fail with small probability even for relatively small sets. While previous works only studied the failure probability of IBLT, this work initiates the worst case analysis of IBLT that guarantees successful listing for all sets of a certain size. The worst case study is important since the failure of IBLT imposes high overhead. We describe a novel approach that guarantees successful listing when the set satisfies a tunable upper bound on its size. To allow that, we develop multiple constructions that are based on various coding techniques such as stopping sets and the stopping redundancy of error-correcting codes, and Steiner systems as well as new methodologies we develop. We analyze the sizes of IBLTs with listing guarantees obtained by the various methods as well as their mapping memory and runtime overheads. Lastly, we study lower bounds on the achievable sizes of IBLT with listing guarantees and verify the results in the paper by simulations.

Invertible Bloom Lookup Tables with Listing Guarantees

The Invertible Bloom Lookup Table (IBLT) is a probabilistic concise data structure for set representation that supports a listing operation as the recovery of the elements in the represented set. Its applications can be found in network synchronization and traffic monitoring as well as in error-correction codes. IBLT can list its elements with probability affected by the size of the allocated memory and the size of the represented set, such that it can fail with small probability even for relatively small sets. While previous works only studied the failure probability of IBLT, this work initiates the worst case analysis of IBLT that guarantees successful listing for all sets of a certain size. The worst case study is important since the failure of IBLT imposes high overhead. We describe a novel approach that guarantees successful listing when the set satisfies a tunable upper bound on its size. To allow that, we develop multiple constructions that are based on various coding techniques such as stopping sets and the stopping redundancy of error-correcting codes, and Steiner systems as well as new methodologies we develop. We analyze the sizes of IBLTs with listing guarantees obtained by the various methods as well as their mapping memory and runtime overheads. Lastly, we study lower bounds on the achievable sizes of IBLT with listing guarantees and verify the results in the paper by simulations.

Optimizing quasi-cyclic spatially coupled LDPC codes by eliminating harmful objects

It is well known that some harmful objects in the Tanner graph of low-density parity-check (LDPC) codes have a negative impact on their error correction performance under iterative message-passing decoding. Depending on the channel and the decoding algorithm, these harmful objects are different in nature and can be stopping sets, trapping sets, absorbing sets, or pseudocodewords. Differently from LDPC block codes, the design of spatially coupled LDPC codes must take into account the semi-infinite nature of the code, while still reducing the number of harmful objects as much as possible. We propose a general procedure, based on edge spreading, enabling the design of good quasi-cyclic spatially coupled LDPC (QC-SC-LDPC) codes. These codes are derived from quasi-cyclic LDPC (QC-LDPC) block codes and contain a considerably reduced number of harmful objects with respect to the original QC-LDPC block codes. We use an efficient way of enumerating harmful objects in QC-SC-LDPCCs to obtain a fast algorithm that spans the search space of potential candidates to select those minimizing the multiplicity of the target harmful objects. We validate the effectiveness of our method via numerical simulations, showing that the newly designed codes achieve better error rate performance than codes presented in previous literature.

Collaborative Optimization of Company Commuter Bus Stop Selection and Route Programming Considering Stochastic Travel Time and Demand

Company commuter bus service (CCBS) is a convenient commuting mode provided for employees. which has always been popular because of its flexible route planning and low cost. Some companies offer CCBS to their employees for free. However, there are also some companies that need employees to pay the fare for CCBS. For fee-based CCBS, profit is an important consideration. Appropriate stop setting and route programming can attract more commuters and generate greater profits. This paper studies the stop selection and route planning for CCBS considering uncertain demand and travel time. In the data preparation phase, we propose an improved fuzzy c-means clustering to obtain appropriate clusters of commuters’ addresses. In the solution phase, we designed a collaborative framework for stop selection and route programming with the objective to maximize the profit of CCBS. A novel heuristic stochastic dynamic programming (H-SDP) method is then designed for the stop selection sub-problem considering the uncertainties of both traveling time and commuting demand. Meanwhile, we employ a variable neighborhood search algorithm with a novel shaking operation suitable for the routing problem. Finally, we conduct a series of computational experiments to demonstrate the effectiveness and efficiency of the developed algorithms.

A new integral equation for Brownian stopping problems with finite time horizon

For classical finite time horizon stopping problems driven by a Brownian motion V(t,x)=supt≤τ≤0E(t,x)[g(τ,Wτ)],we derive a new class of Fredholm type integral equations for the stopping set. For a large class of discounted problems, we show by analytical arguments that the equation uniquely characterizes the stopping boundary of the problem. Regardless of uniqueness, we use the representation to rigorously find the limit behavior of the stopping boundary close to the terminal time. Interestingly, it turns out that the leading-order coefficient is universal for wide classes of problems. We also discuss how the representation can be used for numerical purposes.

Phase transitions and noise sensitivity on the Poisson space via stopping sets and decision trees

AbstractProofs of sharp phase transition and noise sensitivity in percolation have been significantly simplified by the use of randomized algorithms, via the OSSS inequality (proved by O'Donnell et al. and the Schramm–Steif inequality for the Fourier‐Walsh coefficients of functions defined on the Boolean hypercube. In this article, we prove intrinsic versions of the OSSS and Schramm–Steif inequalities for functionals of a general Poisson process, and apply these new estimates to deduce sufficient conditions—expressed in terms of randomized stopping sets—yielding sharp phase transitions, quantitative noise sensitivity, exceptional times and bounds on critical windows for monotonic Boolean Poisson functions. Our analysis is based on a new general definition of “stopping set”, not requiring any topological property for the underlying measurable space, as well as on the new concept of a “continuous‐time decision tree”, for which we establish several fundamental properties. We apply our findings to the ‐percolation of the Poisson Boolean model and to the Poisson‐based confetti percolation with bounded random grains. In these two models, we reduce the proof of sharp phase transitions for percolation, and of noise sensitivity for crossing events, to the construction of suitable randomized stopping sets and the computation of one‐arm probabilities at criticality. This enables us to settle an open problem suggested by Ahlberg et al. (a special case of which was conjectured earlier by Ahlberg et al. on noise sensitivity of crossing events for the planar Poisson Boolean model with random balls whose radius distribution has finite ‐moments and also show the same for planar confetti percolation model with bounded random balls. We also prove that critical probability is for the planar confetti percolation model with bounded, ‐rotation invariant and reflection invariant random grains. Such a result was conjectured by Benjamini and Schramm in the case of fixed balls and proved by Müller, Hirsch and Ghosh and Roy in the case of balls, boxes and random boxes, respectively; our results contain all previous findings as special cases.

Overcoming Data Availability Attacks in Blockchain Systems: Short Code-Length LDPC Code Design for Coded Merkle Tree

Light nodes are clients in blockchain systems that only store a small portion of the blockchain ledger. In certain blockchains, light nodes are vulnerable to a data availability (DA) attack where a malicious node makes the light nodes accept an invalid block by hiding the invalid portion of the block from the nodes in the system. Recently, a technique based on LDPC codes called Coded Merkle Tree was proposed by Yu et al. that enables light nodes to detect a DA attack by randomly requesting/sampling portions of the block from the malicious node. However, light nodes fail to detect a DA attack with high probability if a malicious node hides a small stopping set of the LDPC code. In this paper, we demonstrate that a suitable co-design of specialized LDPC codes and the light node sampling strategy leads to a high probability of detection of DA attacks. We consider different adversary models based on their computational capabilities of finding stopping sets. For the different adversary models, we provide new specialized LDPC code constructions and coupled light node sampling strategies and demonstrate that they lead to a higher probability of detection of DA attacks compared to approaches proposed in earlier literature.

On strongly rigid hyperfluctuating random measures

AbstractIn contrast to previous belief, we provide examples of stationary ergodic random measures that are both hyperfluctuating and strongly rigid. Therefore we study hyperplane intersection processes (HIPs) that are formed by the vertices of Poisson hyperplane tessellations. These HIPs are known to be hyperfluctuating, that is, the variance of the number of points in a bounded observation window grows faster than the size of the window. Here we show that the HIPs exhibit a particularly strong rigidity property. For any bounded Borel set B, an exponentially small (bounded) stopping set suffices to reconstruct the position of all points in B and, in fact, all hyperplanes intersecting B. Therefore the random measures supported by the hyperplane intersections of arbitrary (but fixed) dimension, are also hyperfluctuating. Our examples aid the search for relations between correlations, density fluctuations, and rigidity properties.

On the Sn/n problem

AbstractThe Chow–Robbins game is a classical, still partly unsolved, stopping problem introduced by Chow and Robbins in 1965. You repeatedly toss a fair coin. After each toss, you decide whether you take the fraction of heads up to now as a payoff, otherwise you continue. As a more general stopping problem this reads $V(n,x) = \sup_{\tau }\mathbb{E} \left [ \frac{x + S_\tau}{n+\tau}\right]$ , where S is a random walk. We give a tight upper bound for V when S has sub-Gaussian increments by using the analogous time-continuous problem with a standard Brownian motion as the driving process. For the Chow–Robbins game we also give a tight lower bound and use these to calculate, on the integers, the complete continuation and the stopping set of the problem for $n\leq 489\,241$ .

On the complexity of and solutions to the minimum stopping and trapping set problems

In this paper, we discuss the problems of finding minimum stopping sets and trapping sets in Tanner graphs, using integer linear programming. These problems are important for establishing reliable communication across noisy channels. Indeed, stopping sets and trapping sets correspond to combinatorial structures in information-theoretic codes, which lead to errors in decoding once a message is received. In particular, these sets correspond to variables that are not eventually corrected by the decoder, thus causing failures in decoding when using iterative algorithms. We present integer linear programs (ILPs) for finding stopping sets and several trapping set variants. Furthermore, we prove that two of these trapping set problem variants are NP-hard for the first time. Additionally, we analyze these variants from the parameterized perspective. Finally, we model stopping set and trapping set problems as Integer Linear Programs (ILPs). The effectiveness of our approach is demonstrated by finding stopping sets of size up to 48 in the (4896,2474) Margulis code. This compares favorably to the current state-of-the-art, which finds stopping sets of size up to 26. For the trapping set problems, we show for which cases an ILP produces solutions efficiently and for which cases it performs poorly. The proposed approach is applicable to codes represented by regular and irregular graphs alike.1

Stopping-Set Assisted Reinforced Belief Propagation for Decoding Short LDPC Codes

The low-density parity-check (LDPC) code is chosen to be a channel coding scheme for ultra-reliable and low-latency communications (URLLC) in 5G. However, short LDPC compromises the belief propagation (BP) decoder because of the stopping set (SS) in the decoding graph. We propose to perform different node operations according to the SS. A dynamic SS selection according to the received sequence is further applied. In the length 308 5G LDPC, our decoders outperform prior work, improving BP and collaborative BP at block error probability (BLER) of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$3 \times 10^{-5}$ </tex-math></inline-formula> by around 0.176 and 0.092 dB respectively, with almost the same number of iterations.

The choice of the entrance pupil location in optical measurement of eyewear displays: Theoretical and experimental analyses

AbstractThe location of the entrance pupil, field of view, and motion of the viewer's eye is frequently discussed for virtual reality/augmented reality (VR/AR) displays. In order to properly measure optical performance of these displays, the light measuring device (LMD) must simulate the characteristics of the human eye. An important factor to consider is the location of the entrance pupil (aperture stop setting) of the LMD. We compared field of view (FOV) and luminance deviations with various pupil entrance locations at the LMD in this paper. As a result of the experiment, the entrance pupil at the front focal point of the LMD was the best choice.

On the forward algorithm for stopping problems on continuous-time Markov chains

AbstractWe revisit the forward algorithm, developed by Irle, to characterize both the value function and the stopping set for a large class of optimal stopping problems on continuous-time Markov chains. Our objective is to renew interest in this constructive method by showing its usefulness in solving some constrained optimal stopping problems that have emerged recently.

19.3: Evaluation of Entrance Pupil Location in Measuring VR\AR Eyewear Displays: Theoretical and Experimental Analyses in Field of View

The conventional display measurements are no longer satisfying the need for eyewear displays. The measurement of VR/AR type display requires light measuring devices (LMDs) to simulate the characteristics of the human eye. Factors like the position of the pupil, field of view, and the motion of the eye are often discussed. One of the factors to be considered is the choice of entrance pupil location (aperture stop setting) of the LMD. In this paper, a preliminary study was carried out to compare the deviations of FOV and luminance with the choice of entrance pupil location. The experiment demonstrated that it exists as the most accurate eye‐viewing when the entrance pupil is located at the front focal point.

On Stopping Sets of AG Codes Over Certain Curves With Separated Variables

In this paper we study stopping sets of AG codes over a family of curves, denoted by $\mathcal {X}_{f,g}$ , that includes several important curves with applications in coding theory. We present results concerning stopping sets of one-point and $m$ -point codes over $\mathcal {X}_{f,g}$ , generalizing some results for Hermitian codes presented by Anderson and Matthews.

Cardinality estimation for random stopping sets based on Poisson point processes

We construct unbiased estimators for the distribution of the number of points inside random stopping sets based on a Poisson point process. Our approach is based on moment identities for stopping sets, showing that the random count of points inside the complement S̅ of a stopping set S has a Poisson distribution conditionally to S. The proofs do not require the use of set-indexed martingales, and our estimators have a lower variance when compared to standard sampling. Numerical simulations are presented for examples such as the convex hull and the Voronoi flower of a Poisson point process, and their complements.

Exploring Influencing Factors of Passenger Satisfaction toward Bus Transit in Small-Medium City in China

This paper aims to explore the factors influencing passengers’ satisfaction toward bus transit and develop the improvement strategy for the small-medium city. Data including individual attributes, travel activity attributes, and perceived service attributes were collected in the city of Weinan, China. The ordered logit (OL) model and ordered Probit (OP) model were employed to explore the significant factors associated with passengers’ satisfaction toward bus transit. The odds ratio (OR) was applied to quantitatively measure the effects of the significant factors. Improving strategies of bus transit service were proposed based on the model results using the three-factor theory. Results show that the OL model outperforms the OP model. The age, daily average waiting time, perceived waiting time, transferring convenience, the attitude of the driver, intelligent travel information service, hygienic environment inside the bus, ticket price, bus route setting, and bus stop setting significantly affect the passengers’ satisfaction. Among them, the ticket price, perceived waiting time, bus stop setting, intelligent travel information service, transferring convenience, and bus route setting were identified as exciting factors. It is recommended that optimization of bus route and bus stop setting, building bus dedicated lanes, optimizing dynamic charging system, and providing intelligent travel information service could be effective strategies to improve passengers’ satisfaction toward bus transit in Chinese small-medium cities.

How to Optimize Train Stops under Diverse Passenger Demand: a New Line Planning Method for Large-Scale High-Speed Rail Networks

Optimizing stop plans of train lines greatly contributes to improving the quality of rail passenger service. Traditionally, stop plans are pre-specified according to the classification of stations in the line planning process. However, with the expansion of railway networks and the great changes of travel demand among different origin–destination pairs (ODs), it becomes more difficult to generate stops using this simple method. To meet the new challenges of line planning in railway management, a linear integer programming model based on a new definition of lines is presented in this paper. In the proposed model, proper lines with reasonable train ODs and frequencies can be selected from a series of potential lines in the given line pool. The stops of selected lines can be chosen from candidate stop sets that are generated based on both station classification and types of trains. More practical considerations of stop planning are integrally considered, i.e., the balance of the stop distribution and restrictions on the stop numbers. In experiments involving small-scale cases, the stop numbers decrease under different types of demand scenarios when using the proposed method compared to the traditional methods. We also conduct a series of sensitivity analyses using large-scale random cases to reveal the impact of different stopping rules on the quality of passenger services. Finally, two real-world case studies based on the Beijing-Shanghai high-speed rail network are utilized to demonstrate the applicability of the proposed method.

A Dynkin Game on Assets with Incomplete Information on the Return

This paper studies a two-player zero-sum Dynkin game arising from pricing an option on an asset whose rate of return is unknown to both players. Using filtering techniques, we first reduce the problem to a zero-sum Dynkin game on a bidimensional diffusion (X,Y). Then we characterize the existence of a Nash equilibrium in pure strategies in which each player stops at the hitting time of (X,Y) to a set with a moving boundary. A detailed description of the stopping sets for the two players is provided along with global C1 regularity of the value function.