Stopping Sets Research Articles

On the threshold strategies in optimal stopping problems for diffusion processes

Abstract We study a problem when the optimal stopping for a one-dimensional diffusion process is generated by a threshold strategy. Namely, we give necessary and sufficient conditions (on the diffusion process and the payoff function) under which a stopping set has a threshold structure.

Asymmetric Spatially Coupled LDPC Codes for Multiple-Burst Erasure Channels

The outstanding performances of spatially coupled low-density parity-check (SC-LDPC) codes deteriorate over the channels with memory due to burst erasures. In order to improve the performances of SC-LDPC codes over multiple-burst erasure channels, an asymmetric spatial coupling (ASC) structure is proposed, where each variable node at the same position of the coupled protograph can flexibly choose its coupling width. It is shown that the base matrices of the proposed ASC-LDPC codes possess larger minimal cardinality of stopping sets, which is beneficial to the multiple-burst erasure correcting capability of SC-LDPC codes. Simulation results further demonstrate that, aided by the ASC structure, the performances of SC-LDPC codes can be greatly improved over multiple-burst erasure channels.

A Recursive Algorithm for Selling at the Ultimate Maximum in Regime-Switching Models

We propose a recursive algorithm for the numerical computation of the optimal value function $\inf_{t\le\tau\le T} E \Big[\sup_{0\le s\le T } Y_s / Y_{\tau} \big| {\cal F}_t\Big]$ over the stopping times $\tau$ with respect to the filtration of a geometric Brownian motion $Y_t$ with Markovian regime switching. This method allows us to determine the boundary functions of the optimal stopping set when no associated Volterra integral equation is available. It applies in particular when regime-switching drifts have mixed signs, in which case the boundary functions may not be monotone.

Edge Coloring and Stopping Sets Analysis in Product Codes With MDS Components

We consider non-binary product codes with MDS components and their iterative row-column algebraic decoding on the erasure channel. Both independent and block erasures are considered in this paper. A compact graph representation is introduced on which we define double-diversity edge colorings via the rootcheck concept. An upper bound of the number of decoding iterations is given as a function of the graph size and the color palette size $M$ . Then, we propose a differential evolution edge coloring algorithm that produces colorings with a large population of minimal rootcheck order symbols. The complexity of this algorithm per iteration is $o(M^{\aleph })$ , for a given differential evolution parameter $\aleph $ , where $M^{\aleph }$ itself is small with respect to the huge cardinality of the coloring ensemble. Stopping sets of a product code are defined in the context of MDS components and a relationship is established with the graph representation. A full characterization of these stopping sets is given up to a size $(d+1)^{2}$ , where $d$ is the minimum Hamming distance of the MDS component code. The performance of MDS-based product codes with and without double-diversity coloring is analyzed in presence of both the block and the independent erasures. In the latter case, ML and iterative decoding are proven to coincide at small channel erasure probability. Furthermore, numerical results show excellent performance in presence of unequal erasure probability due to double-diversity colorings.

Weight Distributions of Non-Binary Multi-Edge Type LDPC Code Ensembles: Analysis and Efficient Evaluation

Non-binary multi-edge type ensembles of low-density parity-check codes are analyzed in terms of non-binary codeword weight distribution and its growth rate. In particular, an exact expression of the growth rate for small weights is developed. As a side result, the stopping set distributions of these ensembles are developed. Examples of weight distributions are provided, showing that the derived closed-form expressions can be easily evaluated. The obtained results can thus be exploited to analyze and design non-binary low-density parity-check codes that fall within the multi-edge type framework such as, but not limited to, protograph-based codes.

Broadcast Coded Slotted ALOHA: A Finite Frame Length Analysis

We propose an uncoordinated medium access control (MAC) protocol, called all-to-all broadcast coded slotted ALOHA (B-CSA) for reliable all-to-all broadcast with strict latency constraints. In B-CSA, each user acts as both transmitter and receiver in a half-duplex mode. The half-duplex mode gives rise to a double unequal error protection (DUEP) phenomenon: the more a user repeats its packet, the higher the probability that this packet is decoded by other users, but the lower the probability for this user to decode packets from others. We analyze the performance of B-CSA over the packet erasure channel for a finite frame length. In particular, we provide a general analysis of stopping sets for B-CSA and derive an analytical approximation of the performance in the error floor (EF) region, which captures the DUEP feature of B-CSA. Simulation results reveal that the proposed approximation predicts very well the performance of B-CSA in the EF region. Finally, we consider the application of B-CSA to vehicular communications and compare its performance with that of carrier sense multiple access (CSMA), the current MAC protocol in vehicular networks. The results show that B-CSA is able to support a much larger number of users than CSMA with the same reliability.

On Frame Asynchronous Coded Slotted ALOHA: Asymptotic, Finite Length, and Delay Analysis

We consider a frame asynchronous coded slotted ALOHA (FA-CSA) system for uncoordinated multiple access, where users join the system on a slot-by-slot basis according to a Poisson random process, and in contrast to standard frame synchronous CSA (FS-CSA), users are not frame-synchronized. We analyze the performance of FA-CSA in terms of packet loss rate and delay. In particular, we derive the (approximate) density evolution that characterizes the asymptotic performance of FA-CSA when the frame length goes to infinity. We show that, if the receiver can monitor the system before anyone starts transmitting, a boundary effect similar to that of spatially coupled codes occurs, which greatly improves the iterative decoding threshold. Furthermore, we derive tight approximations of the error floor (EF) for the finite frame length regime, based on the probability of occurrence of the most frequent stopping sets. We show that, in general, FA-CSA provides better performance in both the EF and waterfall regions as compared to FS-CSA. Moreover, FA-CSA exhibits better delay properties than FS-CSA.

Stopping Sets of Hermitian Codes

Combinatorial structures called stopping sets are useful in analyzing the performance of a linear code when coupled with an iterative decoding algorithm over an erasure channel. In this paper, we consider stopping sets of Hermitian codes.

Sequential Detection of Market Shocks With Risk-Averse CVaR Social Sensors

This paper considers a statistical signal processing problem involving agent-based models of financial markets, which at a microlevel are driven by socially aware and risk-averse agents. These agents trade (buy or sell) stocks at each trading instant by using the decisions of all previous agents (social learning) in addition to a private (noisy) signal they receive on the value of the stock. We are interested in the following: 1) modelling the dynamics of these risk averse agents and 2) sequential detection of a market shock based on the behaviour of these agents. Structural results that characterize social learning under a risk measure, conditional value-at-risk (CVaR), are presented and formulation of the Bayesian change point detection problem is provided. The structural results exhibit two interesting properties: 1) risk averse agents herd more often than risk neutral agents and 2) the stopping set in the sequential detection problem is nonconvex. The framework is validated on data from the Yahoo! Tech Buzz game dataset and it is revealed that 1) the model identifies the value changes based on agent’s trading decisions. 2) Reasonable quickest detection performance is achieved when the agents are risk-averse.

Novel joint encoding/decoding algorithms of fountain codes for underwater acoustic communication

Fountain codes are considered to be a promising coding technique in underwater acoustic communication (UAC) which is challenged with the unique propagation features of the underwater acoustic channel and the harsh marine environment. And Luby transform (LT) codes are the first codes fully realizing the digital fountain concept. However, in conventional LT encoding/decoding algorithms, due to the imperfect coverage (IC) of input symbols and short cycles in the generator matrix, stopping sets would occur and terminate the decoding. Thus, the recovery probability is reduced, high coding overhead is required and decoding delay is increased. These issues would be disadvantages while applying LT codes in underwater acoustic communication. Aimed at solving those issues, novel encoding/decoding algorithms are proposed. First, a doping and non-uniform selecting (DNS) encoding algorithm is proposed to solve the IC and the generation of short cycles problems. And this can reduce the probability of stopping sets occur during decoding. Second, a hybrid on the fly Gaussian elimination and belief propagation (OFG-BP) decoding algorithm is designed to reduce the decoding delay and efficiently utilize the information of stopping sets. Comparisons via Monte Carlo simulation confirm that the proposed schemes could achieve better overall decoding performances in comparison with conventional schemes.

An Efficient Algorithm to Improve the Success Threshold of Node-Based Verification-Based Algorithms in Compressed Sensing

In this letter, we introduce an algorithm to enhance the success threshold of node-based verification-based (NB-VB) algorithms in compressed sensing. The NB-VB algorithms have low computational complexity, and are generally classified as iterative message passing algorithms employed for signal recovery. However, similar to the standard iterative decoding of low-density parity-check codes over the binary erasure channel (BEC) in the context of channel coding, these algorithms become inefficient in the stopping sets. The proposed method, with the inspiration of improved decoding algorithms over the BEC, enhances the performance of NB-VB algorithms by guessing values of some unverified signal elements. Our simulation results indicate that although the proposed method improves the success threshold significantly, it does not cause any considerable increase in the complexity of standard NB-VB algorithms.

Predictable Policing: Measuring the Crime Control Benefits of Hotspots Policing at Bus Stops

ObjectivesA fairly robust body of evidence suggests that hotspots policing is an effective crime prevention strategy. In this paper, we present contradictory evidence of a backfiring effect.MethodsIn a randomized controlled trial, aimed at reducing crime and disorder, London’s ‘hottest’ 102 bus-stops were targeted. Double patrol teams of Metropolitan Police Service uniformed officers visited the stops three times per shift (12:00–20:00), 5-times per week, for a duration of 15 min, over a 6 month period. Crucially, officers arrived and departed the bus stop on a bus, with significantly less time spent outside the bus stop setting. Outcomes were measured in terms of victim-generated crimes reported to the police and bus driver incident reports (DIRs), within targeted and catchment areas. We used adjusted Poisson-regression models to compare differences in pre- and post-treatment measures of outcomes and estimated-marginal-means to illustrate the treatment effect.ResultsDIRs went down significantly by 37 % (p = 0.07) in the near vicinity of the bus stops (50 m), by 40 % in the 100 m catchment area (p = 0.04) and marginally and non-significantly in the farthest catchment (10 %; p = 0.66), compared to control conditions. However, victim-generated crimes—the primary outcome measured in previous experiments—increased by 25 % (p = 0.10) in the near vicinity, by 23 % (p = 0.08) and 11 % (p ≤ 0.001) within the 100–150 m catchment areas, respectively.ConclusionsThese findings illustrate the role of bounded-rationality in everyday policing: reductions in crime are predicated on an elevated perceived risk-of-apprehension. Previous studies focused on clusters of addresses or public facilities, with police moving freely and unpredictably within the boundaries of the hotspot, but the patrol areas of officers in this experiment were limited to bus stops so offenders could anticipate their movements. Hotspots policing therefore backfires when offenders can systematically and accurately predict the temporal and spatial pattern of long-term targeting at a single location.

Laplace transform identities for the volume of stopping sets based on Poisson point processes

We derive Laplace transform identities for the volume content of random stopping sets based on Poisson point processes. Our results are based on anticipating Girsanov identities for Poisson point processes under a cyclic vanishing condition for a finite difference gradient. This approach does not require classical assumptions based on set-indexed martingales and the (partial) ordering of index sets. The examples treated focus on stopping sets in finite volume, and include the random missed volume of Poisson convex hulls.

Laplace transform identities for the volume of stopping sets based on Poisson point processes

We derive Laplace transform identities for the volume content of random stopping sets based on Poisson point processes. Our results are based on anticipating Girsanov identities for Poisson point processes under a cyclic vanishing condition for a finite difference gradient. This approach does not require classical assumptions based on set-indexed martingales and the (partial) ordering of index sets. The examples treated focus on stopping sets in finite volume, and include the random missed volume of Poisson convex hulls.

LDPC code optimization techniques to improve the error correction threshold

Non-empty stopping sets, which are the main reason for achieving a threshold of errors in data transmission channels, are studied. New algorithm of transfer smallest stopping sets and stop distance of any LDPC code is proposed. More functional and flexible technique of splitting-and-filling is proposed. Time for which will be transferred the smallest stopping sets and founded stop distance of any LDPC code is calculated

Relay Selection with Channel Probing in Sleep-Wake Cycling Wireless Sensor Networks

In geographical forwarding of packets in a large wireless sensor network (WSN) with sleep-wake cycling nodes, we are interested in the local decision problem faced by a node that has “custody” of a packet and has to choose one among a set of next-hop relay nodes to forward the packet toward the sink. Each relay is associated with a “reward” that summarizes the benefit of forwarding the packet through that relay. We seek a solution to this local problem, the idea being that such a solution, if adopted by every node, could provide a reasonable heuristic for the end-to-end forwarding problem. Toward this end, we propose a local <i>relay selection problem</i> consisting of a forwarding node and a collection of relay nodes, with the relays waking up sequentially at random times. At each relay wake-up instant, the forwarder can choose to <i>probe</i> a relay to learn its reward value, based on which the forwarder can then decide whether to <i>stop</i> (and forward its packet to the chosen relay) or to <i>continue</i> to wait for further relays to wake up. The forwarder’s objective is to select a relay so as to minimize a combination of waiting delay, reward, and probing cost. The local decision problem can be considered as a variant of the asset selling problem studied in the operations research literature. We formulate the local problem as a Markov decision process (MDP) and characterize the solution in terms of <i>stopping sets</i> and <i>probing sets</i>. We provide results illustrating the structure of the stopping sets, namely, the (lower bound) threshold and the stage independence properties. Regarding the probing sets, we make an interesting conjecture that these sets are characterized by upper bounds. Through simulation experiments, we provide valuable insights into the performance of the optimal local forwarding and its use as an end-to-end forwarding heuristic.

Stopping Set Elimination by Parity-Check Matrix Extension via Integer Linear Programming

Error-rate floor phenomenon is known to be a serious impediment to the use of low-density parity-check (LDPC) codes for some practical applications that demand high data reliability. In the case of binary erasure channels (BECs), certain error-prone patterns, known as stopping sets, are proven to cause this performance degradation. A possible approach to diminish this drawback over BECs is to eliminate stopping sets by parity-check matrix extension. Given a parity-check matrix $H$ , and a list $\mathcal{L} $ of its stopping sets, we present an integer linear programming (ILP) formulation to find a parity-check equation which eliminates the maximum number of stopping sets in $\mathcal{L} $ . One of the distinguishing advantages of the proposed scheme is its flexibility for modifications such as: limiting the weight of the new parity-check row, making the new row redundant or linearly independent, 4-cycle avoidance, and taking into account the sizes of stopping sets. Armed with these adjustments, the method can provide good performance improvements, as evidenced by simulation results. Furthermore, for a given $\varrho\in\mathbb{N} $ , by extending the basic formulation, we provide an ILP formulation for finding a set of size $\varrho$ of parity-check equations which can best eliminate the stopping sets in $\mathcal{L}$ , among all such sets.

Error Floor Analysis of Coded Slotted ALOHA Over Packet Erasure Channels

We present a framework for the analysis of the error floor of coded slotted ALOHA (CSA) for finite frame lengths over the packet erasure channel. The error floor is caused by stopping sets in the corresponding bipartite graph, whose enumeration is, in general, not a trivial problem. We therefore identify the most dominant stopping sets for the distributions of practical interest. The derived analytical expressions allow us to accurately predict the error floor at low to moderate channel loads and characterize the unequal error protection inherent in CSA.

Study of optical design of Blu-ray pickup head system with a liquid crystal element.

This paper proposes a newly developed optical design and an active compensation method for a Blu-ray pickup head system with a liquid crystal (LC) element. Different from traditional pickup lens design, this new optical design delivers performance as good as the conventional one but has more room for tolerance control, which plays a role in antishaking devices, such as portable Blu-ray players. A hole-pattern electrode and LC optics with external voltage input were employed to generate a symmetric nonuniform electrical field in the LC layer that directs LC molecules into the appropriate gradient refractive index distribution, resulting in the convergence or divergence of specific light beams. LC optics deliver fast and, most importantly, active compensation through optical design when errors occur. Simulations and tolerance analysis were conducted using Code V software, including various tolerance analyses, such as defocus, tilt, and decenter, and their related compensations. Two distinct Blu-ray pickup head system designs were examined in this study. In traditional Blu-ray pickup head system designs, the aperture stop is always set on objective lenses. In the study, the aperture stop is on the LC lens as a newly developed lens. The results revealed that an optical design with aperture stop set on the LC lens as an active compensation device successfully eliminated up to 57% of coma aberration compared with traditional optical designs so that this pickup head lens design will have more space for tolerance control.

Non-Binary Protograph-Based LDPC Codes: Enumerators, Analysis, and Designs

This paper provides a comprehensive analysis of nonbinary low-density parity check (LDPC) codes built out of protographs. We consider both random and constrained edge-weight labeling, and refer to the former as the unconstrained nonbinary protograph-based LDPC codes (U-NBPB codes) and to the latter as the constrained nonbinary protograph-based LDPC codes (C-NBPB codes). Equipped with combinatorial definitions extended to the nonbinary domain, ensemble enumerators of codewords, trapping sets, stopping sets, and pseudocodewords are calculated. The exact enumerators are presented in the finite-length regime, and the corresponding growth rates are calculated in the asymptotic regime. An EXIT chart tool for computing the iterative decoding thresholds of protograph-based LDPC codes is presented, followed by several examples of finite-length U-NBPB and C-NBPB codes with high performance. Throughout this paper, we provide accompanying examples, which demonstrate the advantage of nonbinary protograph-based LDPC codes over their binary counterparts and over random constructions. The results presented in this paper advance the analytical toolbox of nonbinary graph-based codes.