The K Shortest Paths Problem with Application to Routing

Abstract

Due to the computational complexity of finding almost shortest simple paths, we propose that identifying a larger collection of (nonbacktracking) paths is more efficient than finding almost shortest simple paths on positively weighted real-world networks. First, we present an easy to implement $O(m\log m+kL)$ solution for finding all (nonbacktracking) paths with bounded length $D$ between two arbitrary nodes on a positively weighted graph, where $L$ is an upperbound for the number of nodes in any of the $k$ outputted paths. Subsequently, we illustrate that for undirected Chung-Lu random graphs, the ratio between the number of nonbacktracking and simple paths asymptotically approaches $1$ with high probability for a wide range of parameters. We then consider an application to the almost shortest paths algorithm to measure path diversity for internet routing in a snapshot of the Autonomous System graph subject to an edge deletion process.