Path Splitting Research Articles

A path splitting and pruning strategy on list decoder for PAC codes

AbstractRecently, Arıkan proposed a polarization‐adjusted convolutional (PAC) codes, demonstrating their superior error correction performance over polar codes at short block lengths. It was confirmed that PAC codes approached the optimal performance achievable with limited code length. This paper proposes a novel low‐complexity list decoding algorithm for PAC codes, incorporating path splitting and pruning strategies based on a set of highly reliable information bits. Simulation results reveal that the proposed algorithm significantly reduces sorting complexity and average list size, all while incurring negligible performance loss. Unlike previous pruning algorithms designed for polar codes, the proposed strategy eliminates the need to individually assess the reliability of decoding paths in each decoding process. Instead, the algorithm minimizes redundant decoding paths through a high‐reliability information bit set, constructed using Monte Carlo experiments.

Relation-Algebraic Verification of Disjoint-Set Forests

This paper studies how to use relation algebras, which are useful for high-level specification and verification, for proving the correctness of lower-level array-based implementations of algorithms. We give a simple relation-algebraic semantics of read and write operations on associative arrays. The array operations seamlessly integrate with assignments in computation models supporting while-programs. As a result, relation algebras can be used for verifying programs with associative arrays. We verify the correctness of an array-based implementation of disjoint-set forests using the union-by-rank strategy and find operations with path compression, path splitting and path halving. All results are formally proved in Isabelle/HOL. This paper is an extended version of [1].

Understanding Ionospheric and Geomagnetic Effects on Residual Biases in Radio Occultation Data for Stratospheric Climate Monitoring

AbstractHigher‐order residual ionospheric errors in Global Navigation Satellite System (GNSS) radio occultation (RO) data can induce a systematic residual ionospheric bias (RIB) in RO bending angles, which can impact stratospheric climate monitoring. The main RIB causes are the ray path splitting of dual‐frequency GNSS signals, the electron density distribution along the ray path and at the RO receiver location, and the geomagnetic field. In this study, we investigate the ionospheric and geomagnetic effects on RO‐retrieved stratospheric bending angle and temperature profiles by inspecting the kappa and bi‐local RIB correction methods, the current state‐of‐the‐art methods, using multiple RO satellite mission data in different ionospheric and geomagnetic conditions. We find that, globally, the layer mean/median RIBs of the kappa and bi‐local correction methods exhibit similar behaviors for different RO missions: the estimated bending angle RIBs reach about −0.025/−0.01 and −0.024/−0.008 μrad in the upper (40–45 km) and lower (30–35 km) stratosphere, respectively, while the temperature RIBs reach about −1.0/−0.3 and −0.2/−0.1 K in these layers. However, in the equatorial day time region, the RIB statistics of the mission results diverge. Both the kappa and bi‐local RIBs increase in magnitude with increasing ionization and geomagnetic field strength but show no increase with an increasing degree of ionospheric asymmetry. Overall, the more refined bi‐local method has higher capacity than the comparatively simple kappa method to represent the variability of the ionospheric and geomagnetic conditions that affect RO events. This is important for regional‐scale studies, where the geomagnetic term can be of key relevance.

Neural Network Aided Path Splitting Strategy for Polar Successive Cancellation List Decoding

Combined with the path splitting selecting (PSS) strategy, the successive cancellation list (SCL) decoder achieves a lower decoding complexity without any performance loss. The main purpose of the PSS strategy is to select fewer non-frozen bits to perform the path splitting and pruning operations. To make the selection more efficient, a neural network aided path splitting selecting strategy is proposed in this paper. In the proposed method, the path splitting is seen as a classification problem. With the help of the neural network, the proposed method can precisely locate the non-frozen bits which are needed to be split. Simulation results verify that the proposed method can reduce the sorting complexity (which is equal to the number of path splitting) without any performance loss.

Research on encoding and decoding of non-binary polar codes over GF(2m)

Binary Polar Codes (BPCs) have advantages of high-efficiency and capacity-achieving but suffer from large latency due to the Successive-Cancellation List (SCL) decoding. Non-Binary Polar Codes (NBPCs) have been investigated to obtain the performance gains and reduce latency under the implementation of parallel architectures for multi-bit decoding. However, most of the existing works only focus on the Reed-Solomon matrix-based NBPCs and the probability domain-based non-binary polar decoding, which lack flexible structure and have a large computation amount in the decoding process, while little attention has been paid to general non-binary kernel-based NBPCs and Log-Likelihood Ratio (LLR) based decoding methods. In this paper, we consider a scheme of NBPCs with a general structure over GF(2m). Specifically, we pursue a detailed Monte-Carlo simulation implementation to determine the construction for proposed NBPCs. For non-binary polar decoding, an SCL decoding based on LLRs is proposed for NBPCs, which can be implemented with non-binary kernels of arbitrary size. Moreover, we propose a Perfect Polarization-Based SCL (PPB-SCL) algorithm based on LLRs to reduce decoding complexity by deriving a new update function of path metric for NBPCs and eliminating the path splitting process at perfect polarized (i.e., highly reliable) positions. Simulation results show that the bit error rate of the proposed NBPCs significantly outperforms that of BPCs. In addition, the proposed PPB-SCL decoding obtains about a 40% complexity reduction of SCL decoding for NBPCs.

A highly reliable embedding algorithm for airborne tactical network virtualization

The evolution of airborne tactical networks (ATNs) is impeded by the network ossification problem. As a solution, network virtualization (NV) can provide a flexible and scalable architecture where virtual network embedding (VNE) is a key part. However, existing VNE algorithms cannot be optimally adopted in the virtualization of ATN due to the complex interference in air-combat field. In this context, a highly reliable VNE algorithm based on the transmission rate for ATN virtualization (TR-ATVNE) is proposed to adapt well to the specific electromagnetic environment of ATN. Our algorithm coordinates node and link mapping. In the node mapping, transmission-rate resource is firstly defined to effectively evaluate the ranking value of substrate nodes under the interference of both environmental noises and enemy attacks. Meanwhile, a feasible splitting rule is proposed for path splitting in the link mapping, considering the interference between wireless links. Simulation results reveal that our algorithm is able to improve the acceptance ratio of virtual network requests while maintaining a high revenue-to-cost ratio under the complex electromagnetic interference.

An Improved Path Splitting Decision-Aided SCL Decoding Algorithm for Polar Codes

Although the traditional successive cancellation list (SCL) decoding algorithm can greatly improve the performance of successive cancellation (SC) decoding algorithm, multiple path splitting and sorting operations increase the complexity. In order to reduce the complexity of the SCL algorithm, a new path splitting decision strategy is proposed to reduce the number of path splitting (PSN). This strategy is based on a new auxiliary path metric (APM) proposed in this letter, which is used to identify whether the decoding path is wrong. Wrong paths can be avoided by comparing the predicted APM with the threshold. The simulation results show that the proposed path splitting decision-aided SCL (PSD-SCL) algorithm reduces the complexity compared with other list decoding algorithms with small performance loss at high signal-noise ratio (SNR).

An improved path splitting strategy on successive cancellation list decoder for polar codes

The performance of the cyclic redundancy check aided successive cancellation list (CA-SCL) decoder for polar codes exceeds that of the Turbo codes and the Low Density Parity Check (LDPC) codes adopted in the World Interoperability for Microwave Access (WiMAX) proposal. However, CA-SCL decoder has a high computational complexity, resulting in a long time delay and high memory complexity. In order to alleviate this problem, an improved path splitting strategy on successive cancellation list (IPSS-SCL) decoder was proposed, which can significantly reduce the average list size. The influence of splitting on CA-SCL decoder was analysed at first and a new selective splitting criterion using the Gaussian approximation method was proposed according to the analysis. In addition, a path contraction mechanism based on the location estimation of the correct path was proposed to further reduce the average number of unnecessary candidate paths. Compared with existing path splitting strategies, simulation results show that the proposed IPSS-SCL decoding algorithm can reduce the average computational complexity significantly over the additive white Gaussian noise (AWGN) channel with almost no performance loss.

New Higher-Order Correction of GNSS RO Bending Angles Accounting for Ionospheric Asymmetry: Evaluation of Performance and Added Value

The residual ionospheric error (RIE) from higher-order terms in the refractive index is not negligible when using global navigation satellite system (GNSS) radio occultation (RO) data for climate and meteorology applications in the stratosphere. In this study, a new higher-order bending angle RIE correction named “Bi-local correction approach” has been implemented and evaluated, which accounts for the ray path splitting of the dual-frequency GNSS signals, the altitude of the low Earth orbit (LEO) satellite, the ionospheric inbound (GNSS to tangent point) vs. outbound (tangent point to LEO) asymmetry, and the geomagnetic field. Statistical results based on test-day ensembles of RO events show that, over the upper stratosphere and mesosphere, the order of magnitude of the mean total RIE in the bi-local correction approach is 0.01 μrad. Related to this, the so-called electron-density-squared (Ne2) and geomagnetic (BNe) terms appear to be dominant and comparable in magnitude. The BNe term takes negative or positive values, depending on the angle between the geomagnetic field vector and the direction of RO ray paths, while the Ne2 term is generally negative. We evaluated the new approach against the existing “Kappa approach” and the standard linear dual-frequency correction of bending angles and found it to perform well and in many average conditions similar to the simpler Kappa approach. On top of this, the bi-local approach can provide added value for RO missions with low LEO altitudes and for regional-scale applications, where its capacity to account for the ionospheric inbound-outbound asymmetry as well as for the geomagnetic term plays out.

DSCD: Delay Sensitive Cross-Domain Virtual Network Embedding Algorithm

In the coming 5G era, more and more applications are time-delay sensitive, and the importance of low delay in fields such as telemedicine, autonomous driving and military order transmission is self-evident. However, for different application scenarios, their delay requirements are different. How to meet different Quality of service (QoS) requirements simultaneously becomes a challenge. The emergence of network virtualization (NV) technology can overcome this problem. However, it is difficult for the existing embedding algorithms to meet the requirements of virtual networks on time delay performance. Therefore, in order to study the delay problem of virtual network, we proposed a Delay Sensitive Cross-Domain Virtual Network Embedding Algorithm (DSCD-VNE) in this paper. First of all, different from the traditional virtual network embedding (VNE) algorithm, this algorithm divides the mapping process into a “node-link-node” three-stage embedding algorithm. Second, node resource metrics and set constraints are introduced in the generation phase of candidate link matrix. Third, the algorithm takes candidate link matrix as the core, and uses Kruskal minimum spanning tree to choose the link with the shortest delay for embedding. Fourth, path splitting mechanism and $K$ -shortest path algorithm are introduced with the aim of increasing the utilization of the substrate links in link mapping stage. The simulation results show that the proposed DSCD-VNE algorithm greatly improves the performance of the algorithm compared with the traditional node-link two-stage algorithm, especially in the aspect of delay, the improvement rate is as high as 77%.

Early-Warning-Time-Based Virtual Network Live Evacuation Against Disaster Threats

Virtualization facilitates heterogeneous cloud applications to share the same physical infrastructure with admirable flexibility, while resource efficiency and survivability are critical concerns for virtual network embedding (VNE). As more and more internet applications migrate to the cloud, the resource efficiency and the survivability of VNs, such as single link failure or large-scale disaster survivability, have become crucial issues. Separating the VNE problem into node and link mapping sub-problems without coordination might cause a high embedding cost. This dissertation presents two independent approaches to solve the aforementioned challenges. First, we study two-stage coordinated survivable VNE (SVNE) problem and propose an adaptive path splitting based SVNE (APSS) scheme. We first develop a concise anchor node strategy to restrict the solution space of the candidate substrate nodes, which coordinates node mapping with link mapping to limit the distance spans of the virtual links. Then, we employ an adaptive path splitting policy to provide full protection against single-link failures with partial backup resource, and design an agile frequency slot windows choosing mechanism to mitigate the spectrum fragmentation for link resource efficiency. Simulation results demonstrate that the proposed APSS scheme can achieve satisfactory performance in terms of spectrum utilization and blocking ratio. Second, we propose a synchronous evacuation strategy for VNs with dual virtual machines (VMs) inside a disaster risk zone (DRZ), which suffer higher risks than the VNs with single. The evacuation strategy exploits post-copy technique to sustain the online service alive and enhances synchronous VM migrations to shorten the dual-VM evacuation time. Numerical results show that the proposed strategy can outperform the best-effort scheme in terms of average and total evacuation times of dual-VMs.

A survivable VON embedding algorithm based on resource mean and spectrum coherence-aware in elastic optical networks

Abstract To address the problems of severe spectrum fragmentation and high bandwidth blocking probability of virtual optical network embedding in elastic optical networks, we propose a survivable virtual optical network embedding algorithm based on resource mean and spectrum coherence-aware (SVONE-RMSC). Firstly, according to the lightpath spectrum resources state of candidate substrate nodes to the correlative substrate nodes that have been embedded, we design the spectrum resource mean formula to enhance the correlation between virtual node embedding and virtual optical link embedding. Secondly, the algorithm optimizes the spectrum allocation methods. Specifically, to weaken the disruption of spectrum allocation to the spectrum continuity of the working lightpaths, the formula of spectrum coherence is designed. When allocating the spectrum resources for the protection lightpaths, we propose a spectrum allocation method based on spectrum coherence-aware, so as to maximize the spectrum coherence while minimizing the free frequency slots consumption. Finally, when the lack of spectrum resources on the single lightpath leads to the failure of virtual optical link embedding, different path splitting methods are designed to re-embed the virtual optical link. Simulation results show that the SVONE-RMSC algorithm can effectively reduce the bandwidth blocking probability and enhance the spectrum utilization.

Adaptive path splitting based survivable virtual network embedding in elastic optical networks

Abstract Virtualization facilitates heterogeneous cloud applications to share the same physical infrastructure with admirable flexibility, while resource efficiency and survivability are critical concerns for virtual network embedding (VNE). In this paper, we study two-stage coordinated survivable VNE (SVNE) problem and propose an adaptive path splitting based SVNE (APSS) scheme for elastic optical networks (EONs). For APSS, we first develop a concise anchor node strategy to balance the node resource utilization and restrict the solution space of the candidate substrate nodes, which coordinates node mapping with link mapping to limit the distance spans of the virtual links. Then, we employ an adaptive path splitting policy to provide full protection against single-link failures with partial backup resource, and design an agile frequency slot windows choosing mechanism to mitigate the spectrum fragmentation for link resource efficiency. We evaluate the proposed APSS scheme and compare its performances with some counterpart schemes. Simulation results demonstrate that the proposed APSS scheme can achieve satisfactory performance in terms of spectrum utilization and blocking ratio, even if with higher backup redundancy ratio.

Partitioning Dynamics of Gravity‐Driven Unsaturated Flow Through Simple T‐Shaped Fracture Intersections

AbstractA fundamental understanding of the fluid movement and dynamic partitioning process at fracture intersections is important to accurately predict water infiltration and contaminant transport in networks of fractures. We present an experimental study on the flow‐splitting behavior at a T‐shaped intersection. Different combinations of apertures of the vertical (bv) and horizontal (bh) fractures are considered. Experimental results confirm that the gravity‐driven flow in the vertical fracture transitions from droplet to rivulet mode as the flow rate increases. We quantify the flow dynamics through the intersection and especially focus on the partitioning efficiency (η) defined as the percentage of flow partitioned into the horizontal fracture. We identify three regimes of flow partitioning at the intersection for the case of bv < bh: total partitioning (η → 1), splitting or partial bypass (0 < η < 1), and total bypass (η → 0). The total bypass regime is associated with the rivulet mode with a flow rate higher than ~1.5 ml/min. We find a simple relationship between η and the flow rate Q for droplet flow, η = min(1, ChQ−1), where Ch is a threshold flow rate below which droplets almost completely imbibe into the horizontal fracture, leading to η → 1. A force balance analysis links Ch to a critical droplet length for the transition from complete partitioning to path splitting. The obtained relationship is further supported by numerical simulations of droplet flow through intersections. The results and analysis from this study may provide insights and physical constraints on construction of reduced order unsaturated flow models based on simplified discrete fracture networks.

Path Splitting Selecting Strategy-Aided Successive Cancellation List Algorithm for Polar Codes

Successive cancellation list (SCL) decoder provides a good trade-off between error-correcting performance and complexity. To simplify the decoding algorithm with negligible performance loss, a path splitting selecting (PSS) strategy is proposed based on analyzing the reason why the SCL decoder fails. On the basis of the PSS strategy, two schemes are proposed to find erroneous information bits more accurately. Simulation results show that the proposed strategy based on the search set and decision function is a good solution to reduce the complexity with almost no performance loss.

A Flip-Syndrome-List Polar Decoder Architecture for Ultra-Low-Latency Communications

We consider practical hardware implementation of polar decoders. To reduce latency due to the serial nature of successive cancellation, existing optimizations improve parallelism with two approaches, i.e., multi-bit decision or reduced path splitting. In this paper, we combine the two procedures into one with an error-pattern-based architecture. It simultaneously generates a set of candidate paths for multiple bits with pre-stored patterns. For rate-1 (R1) or single parity-check nodes, we prove that a small number of deterministic patterns are required to guarantee performance preservation. For general nodes, low-weight error patterns are indexed by syndrome in a look-up table and retrieved in $O(1)$ time. The proposed flip-syndrome-list decoder fully parallelizes all constituent code blocks without sacrificing performance, and thus is suitable for ultra-low-latency applications. Meanwhile, two code construction optimizations are presented to further reduce complexity and improve performance.

Ion Permeation through a Phospholipid Membrane: Transition State, Path Splitting, and Calculation of Permeability.

We investigate the thermodynamics and kinetics of the permeation of a potassium ion through a phospholipid membrane. We illustrate that the conventional reaction coordinate (the position of the ion along the normal to the membrane plane) is insufficient to capture essential elements of the process. It is necessary to add coarse variables that measure membrane distortion. New coarse variables are suggested, and a two-dimensional coarse-space is proposed to describe the permeation. We illustrate path splitting and two transition states of comparable barrier heights. The alternative pathways differ by the extent of water solvation of the ion-phosphate pairs. The permeation process cannot be described by a local one-dimensional reaction coordinate, and a network formulation is more appropriate. We use Milestoning with Voronoi tessellation in two dimensions to quantify the equilibrium and rate of the permeation of the positively charged ion. The permeation coefficient is computed and compared favorably to experiment.

A complexity-reduced fast successive cancellation list decoder for polar codes

A multi-bit decision for polar codes based on a simplified successive cancellation (SSC) decoding algorithm can improve the throughput of polar decoding. A list algorithm is used to improve the error-correcting performance. However, list decoders are highly complex compared with decoders without a list algorithm. In this paper, a low-complexity list decoder is proposed, where path-splitting operations for a multi-bit decision can be avoided, if the decoding reliability exceeds a threshold. The threshold is determined based on the reliability of subchannels and positions of decoding nodes. Path splitting rules are designed for multi-bit decision processes, and a complexity-reduced list decoder is proposed based on this. Results show that the number of survival paths can be greatly reduced at the cost of negligible deterioration in block error performance. Thus, the computational complexity can be significantly reduced, especially for a high signal-to-noise ratio (SNR) region.

Minimizing Blocking Probability in Elastic Optical Networks by Varying the Bandwidth Granularity Based on Optical Path Fragmentation

Elastic optical networks (EONs) based on orthogonal frequency-division multiplexing (OFDM) are considered a promising solution for the next optical network’s generation. These networks make it possible to choose an adequate portion of the available spectrum to satisfy the requested capacity. In this paper, we consider the impact of spectrum fragmentation along the optical single/multipath routing transmission on the efficiency of the EONs. This involves reducing the fragmentation effects by dynamically updating and controlling the minimum bandwidth allocation granularity (ɡ). We adopt linear and nonlinear dynamic mechanisms, which are denoted as LDAɡ and NLDAɡ, respectively, to choose proper bandwidth granularities that are proportional to the optical link/path bandwidth fragmentation status. In order to avoid either splitting the capacity request over many routing paths, which would increase the management complexity, or encouraging single path transmission, the proposed schemes aim to choose a proper bandwidth allocation granularity (ɡ) for a predefined set of suggested values. Simulation results show that varying the bandwidth granularity based on the optical path fragmentation status can offer an improved performance over fixed granularity with respect to the bandwidth blocking probability, the number of path splitting actions, the throughput, and the differential delay constraint issue in terms of: the network bandwidth utilization and multipath distribution.

Minimizing Blocking Probability in Elastic Optical Networks by Varying the Bandwidth Granularity Based on Optical Path Fragmentation

Elastic optical networks (EONs) based on orthogonal frequency-division multiplexing (OFDM) are considered a promising solution for the next optical network’s generation. It basically makes it possible to choose an adequate portion of the available spectrum to satisfy the requested capacity. In this paper, we consider the impact of spectrum fragmentation along the optical single/multipath routing transmission on the efficiency of the EONs. This involves reducing the fragmentation effects by dynamically updating and controlling the minimum bandwidth allocation granularity (g).We adopt linear and nonlinear dynamic mechanisms, which are denoted as LDAEi and NLDAEi, respectively, to choose proper band width granularities that are proportional to the optical link/path bandwidth fragmentation status. In order to avoid neither splitting the capacity request over many routing paths, which would increase the management complexity, nor encouraging single path transmission, the proposed schemes aim to choose a proper bandwidth allocation granularity (g) of a predefined set of suggested values. Simulation results show that varying the bandwidth granularity based on the optical path fragmentation status can offer improved performance over fixed granularity with respect to the bandwidth blocking probability, the number of path splitting, the throughput, and differential delay constraint issue in terms of: the network bandwidth utilization and multipath distribution.