Efficiently Listing Bounded Length st-Paths

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.