The K Shortest Paths Problem with Application to Routing

TL;DR

This paper introduces an efficient method to find all nonbacktracking paths within a bounded length in weighted graphs, demonstrating its effectiveness for routing applications and analyzing path diversity in real-world network models.

Contribution

It presents a simple $O(m ext{log}m + kL)$ algorithm for enumerating nonbacktracking paths and analyzes their ratio to simple paths in Chung-Lu random graphs, with applications to internet routing.

Findings

The algorithm efficiently finds all nonbacktracking paths within a length bound.

In Chung-Lu graphs, nonbacktracking and simple path counts are asymptotically similar.

Application to internet routing shows path diversity measurement under edge deletion.

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.