A comprehensive theory of cascading via-paths and the reciprocal pointer chain method

In this paper, we consolidate and expand upon the current theory and potential applications of the set of $k$ best \emph{cascading via-paths} (CVPs) and the \emph{reciprocal pointer chain} (RPC) method for identifying them. CVPs are a collection of up to $|V|$ paths between a source and a target node in a graph $G = (V,E)$, computed using two shortest path trees, that have distinctive properties relative to other path sets. They have been shown to be particularly useful in geospatial applications, where they are an intuitive and efficient means for identifying a set of spatially diverse alternatives to the single shortest path between the source and target. However, spatial diversity is not intrinsic to paths in a graph, and little theory has been developed outside of application to describe the nature of these paths and the RPC method in general. Here we divorce the RPC method from its typical geospatial applications and develop a comprehensive theory of CVPs from an abstract graph-theoretic perspective. Restricting ourselves to properties of the CVPs and of the entire set of $k$-best CVPs that can be computed in $O(|E| + |V| \log |V|)$, we are able to then propose, among other things, new and efficient approaches to problems such as generating a diverse set of paths and to computing the $k$ shortest loopless paths between two nodes in a graph. We conclude by demonstrating the new theory in practice, first for a typical application of finding alternative routes in road networks and then for a novel application of identifying layer-boundaries in ground-penetrating radar (GPR) data. It is our hope that by generalizing the RPC method, providing a sound theoretical foundation, and demonstrating novel uses, we are able to broaden its perceived applicability and stimulate new research in this area, both applied and theoretical.