A message-passing algorithm for counting short cycles in a graph

Abstract

A message-passing algorithm for counting short cycles in a graph is presented. For bipartite graphs, which are of particular interest in coding, the algorithm is capable of counting cycles of length g, g + 2, . . . , 2g − 2, where g is the girth of the graph. For a general (non-bipartite) graph, cycles of length g, g + 1, . . . , 2g − 1 can be counted.The algorithm is based on performing integer additions and subtractions in the nodes of the graph and passing extrinsic messages to adjacent nodes. The complexity of the proposed algorithm grows as O(g|E| 2 ), where |E| is the number of edges in the graph. For sparse graphs, the proposed algorithm significantly outperforms the existing algorithms in terms of computational complexity and memory requirements.

Efficient Algorithm for Finding Dominant Trapping Sets of LDPC Codes

“…9, the importance sampling estimation closely matches the Monte Carlo simulation, further verifying the dominance of the trapping sets found by the proposed algorithm. Example 20: As the last example, we use the following degree distribution optimized for the min-sum algorithm in [6] and construct a (1000, 499) LDPC code using the PEG algorithm: 19 and ρ(x) = .0160x 5 + .9840x 6 . The girth of the resultant graph is 6, and we use the short cycles of length 6 and 8, and cycles of length up to 20 with ACE less than 4 as the initial input set of Algorithm 2.…”

High Rate QC-LDPC Codes with Optimization of Maximum Average Girth

“…The number of the edges in a loop is called loop length. And the shortest loop length in a Tanner graph is defined as the girth of the code [17], which usually decides the code performance to some extent. And the girth should be optimized to reduce the performance degradation caused by false message selffeedback propagation in the short cycles.…”

Rate-compatible LDPC codes with short block lengths based on puncturing and extension techniques

“…The proposed counting cycle (CC)-based puncturing technique is developed based on the counting cycle algorithms [26] and [30]. The former algorithm employs matrix multiplications while the latter takes advantage of the message-passing nature of BP decoding.…”

“…Given the same TG, we have verified that both algorithms produce similar results for counting cycles of length g and g + 2, where g is the girth. But the algorithm in [30] has much lower complexity (O(g|E| 2 )) than its counterpart [26] (O(gN 3 )), especially for graphs with large sizes. Provided with the cycle distribution, the objective is to select an ideal puncturing pattern that can break as many girthlength cycles as possible, which may reduce the performance degradation introduced by puncturing.…”

“…, S). After running the counting cycle algorithm [30], we select the candidate submatrix with the largest g h and the smallest N g as h ext . As a result, the CCbased extension scheme maximizes the local girth g h of h ext , and the selected h ext has the smallest number of lengthg h cycles.…”