An Implementation of a Pseudoperipheral Node Finder

Many algorithms for finding orderings for sparse symmetr ic matrices operate on the corresponding undirected graph. These algorithms often require one or more "start ing nodes," and for some algorithms experience suggests tha t nodes which are at maximum or nearly maximum distance apar t are good candidates [4, 611]. In a recent paper, Gibbs et al. [8] provide a novel heuristic algorithm for finding such nodes. Our objective in this paper is to provide a well-structured flexible implementat ion of this algori thm which includes some .modifications tha t appear to improve its performance. We include some experiments demonstra t ing the effect of various modifications to the original algorithm. We now give some formal definitions and a precise s ta tement of the problem. Let G ffi (X, E) be an undirected graph with the set X of nodes and the set E of undirected edges represented as unordered pairs of nodes. A path of length k is an ordered set of distinct nodes (Xo, x l , . . . , Xk) where {x,-1, x,) E E for 1 _< i _ k. A graph is connected if for each pair of distinct nodes there is a path joining them. Throughou t this paper, graphs are assumed to be connected unless we state otherwise. Consider a connected graph G. The distance d(x, y) between two nodes x and y in G is defined to be the length of a shortest pa th connecting them. Following Berge [2], we define the eccentricity of a node x to be the quant i ty l(x) ffi max{d(x, y) [ y E X}.