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. 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 their belief propagation (BP) threshold, and analyze the performance of the resulting spatially-coupled CBs (SC-CBs). We introduce an equivalent bipartite graph representation of CBs with identical iterationby-iteration finite-length and asymptotic performance. Based on this equivalent representation, we then analyze the asymptotic performance of single-layer CBs and SC-CBs. In particular, we show that the potential threshold and the area threshold are equal. We also derive the Maxwell decoder for CBs and prove that the area 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 BP extrinsic information transfer curve (defined for the equivalent graph), computed for the expected residual CB graph when a peeling decoder equivalent to the BP 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 area 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 BP decoding threshold to the MAP decoding threshold characteristic of spatially-coupled low-density parity-check codes and other coupled systems. However, SCCBs yield better BP 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.