Efficiently listing bounded length st-paths

TL;DR

This paper introduces new algorithms for listing all simple $st$-paths in a graph with bounded length, achieving similar time complexity to classic methods but with significantly reduced space requirements.

Contribution

It presents a novel length-bounded $st$-path listing algorithm with optimal time complexity and a unified framework connecting classic and new path enumeration methods.

Findings

Algorithms match classic time complexities

Reduced space complexity to O(m+n)

Unified traversal framework for different parameterizations

Abstract

The problem of listing the $K$ shortest simple (loopless) $st$-paths in a graph has been studied since the early 1960s. For a non-negatively weighted graph with $n$ vertices and $m$ edges, the most efficient solution is an $O(K(mn + n^2 \log n))$ algorithm for directed graphs by Yen and Lawler [Management Science, 1971 and 1972], and an $O(K(m+n \log n))$ algorithm for the undirected version by Katoh et al. [Networks, 1982], both using $O(Kn + m)$ space. In this work, we consider a different parameterization for this problem: instead of bounding the number of $st$-paths output, we bound their length. For the bounded length parameterization, we propose new non-trivial algorithms matching the time complexity of the classic algorithms but using only $O(m+n)$ space. Moreover, we provide a unified framework such that the solutions to both parameterizations -- the classic $K$-shortest and the new length-bounded paths -- can be seen as two different traversals of a same tree, a Dijkstra-like and a DFS-like traversal, respectively.