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In this paper, we consolidate and expand upon the current theory andpotential applications of the set of $k$ best \emph{cascading via-paths} (CVPs)and the \emph{reciprocal pointer chain} (RPC) method for identifying them. CVPsare a collection of up to $|V|$ paths between a source and a target node in agraph $G = (V,E)$, computed using two shortest path trees, that havedistinctive properties relative to other path sets. They have been shown to beparticularly useful in geospatial applications, where they are an intuitive andefficient means for identifying a set of spatially diverse alternatives to thesingle shortest path between the source and target. However, spatial diversityis not intrinsic to paths in a graph, and little theory has been developedoutside of application to describe the nature of these paths and the RPC methodin general. Here we divorce the RPC method from its typical geospatialapplications and develop a comprehensive theory of CVPs from an abstractgraph-theoretic perspective. Restricting ourselves to properties of the CVPsand 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 efficientapproaches to problems such as generating a diverse set of paths and tocomputing the $k$ shortest loopless paths between two nodes in a graph. Weconclude by demonstrating the new theory in practice, first for a typicalapplication of finding alternative routes in road networks and then for a novelapplication of identifying layer-boundaries in ground-penetrating radar (GPR)data. It is our hope that by generalizing the RPC method, providing a soundtheoretical foundation, and demonstrating novel uses, we are able to broadenits perceived applicability and stimulate new research in this area, bothapplied and theoretical.

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