LDPC codes are typically decoded by running a synchronous message passing algorithm over the corresponding bipartite factor graph (made of variable and check nodes). More specifically, each synchronous round consists of 1) updating all variable nodes based on the information received from the check nodes in the previous round, and then 2) updating all the check nodes based on the information sent from variable nodes in the current round. However, in many applications, ranging from message passing in neural networks to hardware implementation of LDPC codes, assuming that all messages are sent and received at the same time is far from realistic. In this paper, we investigate the effect of asynchronous message passing on the decoding of LDPC codes over a Binary Erasure Channel (BEC). We effectively assume that there is a random delay assigned to each edge of the factor graph that models the random propagation delay of a message along the edge. As a result, the output messages of a check/variable node are also asynchronously updated upon arrival of a new message in its input. We show, for the first time for BEC, that the asymptotic performance of the asynchronous message passing is fully characterized by a fixed point integral equation that takes into account both the temporal and the spatial features of the factor graph. Our theoretical result is reminiscent of the fixed point equation in traditional BP decoding. Also, our simulation results show that asynchronous scheduling reduces decoding time compared to the traditional BP in certain cases in the finite block-length regime.
LDPC codes are typically decoded by running a synchronous message passing algorithm over the corresponding bipartite factor graph (made of variable and check nodes). More specifically, each synchronous round consists of 1) updating all variable nodes based on the information received from the check nodes in the previous round, and then 2) updating all the check nodes based on the information sent from variable nodes in the current round. However, in many applications, ranging from message passing in neural networks to hardware implementation of LDPC codes, assuming that all messages are sent and received at the same time is far from realistic. In this paper, we investigate the effect of asynchronous message passing on the decoding of LDPC codes over a Binary Erasure Channel (BEC). We effectively assume that there is a random delay assigned to each edge of the factor graph that models the random propagation delay of a message along the edge. As a result, the output messages of a check/variable node are also asynchronously updated upon arrival of a new message in its input. We show, for the first time for BEC, that the asymptotic performance of the asynchronous message passing is fully characterized by a fixed point integral equation that takes into account both the temporal and the spatial features of the factor graph. Our theoretical result is reminiscent of the fixed point equation in traditional BP decoding. Also, our simulation results show that asynchronous scheduling reduces decoding time compared to the traditional BP in certain cases in the finite block-length regime.
Conventionally, in each low-density parity-check (LDPC) decoding iteration all the variable nodes and subsequently all the check nodes send messages to their neighbors (flooding schedule). An alternative, more efficient, approach is to update the nodes' messages serially (serial schedule). A theoretical analysis of serial message passing decoding schedules is presented. In particular, the evolution of the computation tree under serial scheduling is analyzed. It shows that the tree grows twice as fast in comparison to the flooding schedule's one, indicating that the serial schedule propagates information twice as fast in the code's underlying graph. Furthermore, an asymptotic analysis of the serial schedule's convergence rate is done using the density evolution (DE) algorithm. Applied to various ensembles of LDPC codes, it shows that for long codes the serial schedule is expected to converge in half the number of iterations compared to the standard flooding schedule, when working near the ensemble's threshold. This observation is generally proved for the binary erasure channel (BEC) under some natural assumptions. Finally, an accompanying concentration theorem is proved.
Abstract — Low-Density Parity-Check (LDPC) codes are usually decoded by running an iterative belief-propagation, or message-passing, algorithm over the factor graph of the code. The traditional message-passing schedule consists of updating all the variable nodes in the graph, using the same pre-update information, followed by updating all the check nodes of the graph, again, using the same pre-update information. Recently several studies show that sequential scheduling, in which messages are generated using the latest available information, significantly improves the convergence speed in terms of number of iterations. Sequential scheduling raises the problem of finding the best sequence of message updates. This paper presents practical scheduling strategies that use the value of the messages in the graph to find the next message to be updated. Simulation results show that these informed update sequences require significantly fewer iterations than standard sequential schedules. Complexity and implementability issues are also addressed. Index Terms — Belief propagation-algorithm, messagepassing schedule, error-control codes, low-density paritycheck codes. I.
Serial decoding schedules for low-density parity-check (LDPC) codes are described and analyzed. Conventionally, in each iteration all the variable nodes and subsequently all the check nodes send messages to their neighbors (“flooding schedule”). In contrast, in the considered methods, the updating of the nodes is implemented according to a serial schedule. The evolution of the decoding algorithm’s computation tree under serial scheduling is analyzed. The analysis shows that it grows twice as fast in comparison to the flooding schedule’s computation tree, indicating that the serial schedule propagates information twice as fast in the code’s underlying graph. Furthermore, asymptotic analysis of the serial schedule’s convergence rate is done using the Density Evolution (DE) algorithm. Applied to various ensembles of LDPC codes, it shows that when working near the ensemble’s threshold, for long enough codes the serial schedule is expected to converge in half the number of iterations compared to the standard flooding schedule. This observation is generally proved for the Binary Erasure Channel (BEC) under some likely assumptions. Finally, an accompanying concentration theorem is proved, justifying the asymptotic DE analysis assumptions. 1
Low-density parity-check (LDPC) codes are usually decoded by running an iterative belief-propagation, or message-passing, algorithm over the factor graph of the code. The traditional message-passing schedule consists of updating all the variable nodes in the graph, using the same pre-update information, followed by updating all the check nodes of the graph, again, using the same pre-update information. Recently several studies show that sequential scheduling, in which messages are generated using the latest available information, significantly improves the convergence speed in terms of number of iterations. Sequential scheduling raises the problem of finding the best sequence of message updates. This paper presents practical scheduling strategies that use the value of the messages in the graph to find the next message to be updated. Simulation results show that these informed update sequences require significantly fewer iterations than standard sequential schedules. Furthermore, the paper shows that informed scheduling solves some standard trapping set errors. Therefore, it also outperforms traditional scheduling for a large numbers of iterations. Complexity and implementability issues are also addressed.
Low-Density Parity-Check (LDPC) codes are usually decoded by running an iterative belief-propagation (BP), or message-passing, algorithm over the factor graph of the code. The traditional message-passing scheduling, called flooding, consists of updating all the variable nodes in the graph, using the same pre-update information, followed by updating all the check nodes of the graph, again, using the same pre-update information. Recently, several studies show that sequential scheduling, in which messages are generated using the latest available information, significantly improves the convergence speed in terms of number of iterations. Sequential scheduling introduces the problem of finding the best sequence of message updates. We propose Informed Dynamic Scheduling (IDS) strategies that select the message-passing schedule according to the observed rate of change of the messages. In general, IDS strategies require computation to select the message to update but converge in fewer message updates because they focus on the part of the graph that has not converged. Moreover, IDS yields a lower error-rate performance than either flooding or sequential scheduling because IDS strategies overcome traditional trapping-set errors. This paper presents IDS strategies that address several issues including performance for short-blocklength codes, complexity, and implementability.
We study a wide range of graph-based message-passing schedules for iterative decoding of low-density parity-check (LDPC) codes. Using the Tanner graph (TG) of the code and for different nodes and edges of the graph, we relate the first iteration in which the corresponding messages deviate from their optimal value (corresponding to a cycle-free graph) to the girths and the lengths of the shortest closed walks in the graph. Using this result, we propose schedules, which are designed based on the distribution of girths and closed walks in the TG of the code, and categorize them as node based versus edge based, unidirectional versus bidirectional, and deterministic versus probabilistic. These schedules, in some cases, outperform the previously known schedules, and in other cases, provide less complex alternatives with more or less the same performance. The performance/complexity tradeoff and the best choice of schedule appear to depend not only on the girth and closed-walk distributions of the TG, but also on the iterative decoding algorithm and channel characteristics. We examine the application of schedules to belief propagation (sum-product) over additive white Gaussian noise (AWGN) and Rayleigh fading channels, min-sum (max-sum) over an AWGN channel, and Gallager's algorithm A over a binary symmetric channel.