Asymptotic Analysis and Spatial Coupling of Counter Braids

A counter braid (CB) is a novel counter architecture introduced by Lu et al. in 2007 for per-flow measurements on high-speed links which can be decoded with low complexity using message passing (MP). CBs achieve an asymptotic compression rate (under optimal decoding) that matches the entropy lower bound of the flow size distribution. In this paper, we apply the concept of spatial coupling to CBs to improve the performance of the original CBs and analyze the performance of the resulting spatially-coupled CBs (SC-CBs). We introduce an equivalent bipartite graph representation of CBs with identical iteration-by-iteration finite-length and asymptotic performance. Based on this equivalent representation, we then analyze the asymptotic performance of single-layer CBs and SC-CBs under the MP decoding algorithm proposed by Lu et al.. In particular, we derive the potential threshold of the uncoupled system and show that it is equal to the area threshold. We also derive the Maxwell decoder for CBs and prove that the potential threshold is an upper bound on the Maxwell decoding threshold, which, in turn, is a lower bound on the maximum a posteriori (MAP) decoding threshold. We then show that the area under the extended MP extrinsic information transfer curve (defined for the equivalent graph), computed for the expected residual CB graph when a peeling decoder equivalent to the MP decoder stops, is equal to zero precisely at the area threshold. This, combined with the analysis of the Maxwell decoder and simulation results, leads us to the conjecture that the potential threshold is, in fact, equal to the Maxwell decoding threshold and hence a lower bound on the MAP decoding threshold. Interestingly, SC-CBs do not show the well-known phenomenon of threshold saturation of the MP decoding threshold to the potential threshold characteristic of spatially-coupled low-density parity-check codes and other coupled systems. However, SC-CBs yield better MP decoding thresholds than their uncoupled counterparts. Finally, we also consider SC-CBs as a compressed sensing scheme and show that low undersampling factors can be achieved.