An $O(n^2)$ Self-Stabilizing Algorithm for Computing Bridge-Connected Components

Abstract

The basic idea of our new approach is to determine in a first step for each node those pairs of nodes which allow a good interpolation of the unknowns located at this node. These pairs of neighbor nodes (in some cases only one node) are called parent nodes. This is done by solving a local minimization problem which, in addition, yields the interpolation and restriction coefficients. The construction scheme has been generalized to systems of convection-diffusion-reaction equations using a point-block approach. After these suitable pairs of parent nodes have been determined, the nodes are labeled as C- and F-nodes such that each F-node can be interpolated using one of these suitable pairs of parent nodes and the already computed coefficients. Additionally, a simple heuristic algorithm tries to minimize the number of C-nodes and the number of non-zero entries in the coarse grid matrix. The algorithm has been parallelized and shows mesh size independent convergence for standard model problems. Realistic numerical experiments confirm the efficiency of the presented algorithm.