Abstract
The k-shortest path problem is a generalization of the fundamental shortest path problem, where the goal is to compute k simple paths from a given source to a target node, in non-decreasing order of their weight. With numerous applications modeling various optimization problems and as a feature in some learning systems, there is a need for efficient algorithms for this problem. Unfortunately, despite many decades of research, the best directed graph algorithm still has a worst-case asymptotic complexity of \(\tilde{O}(k\, n (n+m))\). In contrast to the worst-case complexity, many algorithms have been shown to perform well on small diameter directed graphs in practice. In this paper, we prove that the average-case complexity of the popular Yen’s algorithm on directed random graphs with edge probability \(p = \varOmega (\log {n})/n\) in the unweighted and uniformly distributed weight setting is \(O(k\, m \log {n})\), thus explaining the gap between the worst-case complexity and observed empirical performance. While we also provide a weaker bound of \(O(k\, m \log ^4{n})\) for sparser graphs with \(p \ge 4/n\), we show empirical evidence that the stronger bound should also hold in the sparser setting. We then prove that Feng’s directed k-shortest path algorithm computes the second shortest path in expected O(m) time on random graphs with edge probability \(p = \varOmega (\log {n})/n\). Empirical evidence suggests that the average-case result for the Feng’s algorithm holds even for \(k>2\) and sparser graphs.
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Notes
- 1.
In fact, Priebe showed (P1) in the form of his Lemma 3.4 and (P2) in the form of his Lemma 3.10, for any edge weight distribution function F that satisfies the following requirements: F is concentrated on \([0,\infty )\), \(F(0) = 0\) and that \(F'(0)\) exists and is strictly positive. Since the uniform edge weight distribution between 0 and 1 is compatible with these requirements, (P1) and (P2) hold for \(\mathcal{D}(n,p,[0,1])\), too.
- 2.
For \(p\ge 4/n\) the size of the giant strong component is \(\varOmega (n)\) w.h.p. [20].
- 3.
We are aware that there exist improved SSSP algorithms with linear average-case time (e.g., [16, 26]) for initially uniformly distributed edge weights. Unfortunately, for the particular subgraphs on which we would like to apply these better SSSP algorithms it seems difficult to prove that their overall edge weight distribution remains sufficiently uniform
- 4.
We note that for convex properties, the G(n, p) and G(n, m) random graph models are equivalent up to lower order terms, provided \(m \approx p \cdot N\) (where N is the maximum number of edges insertable in a graph, \(N=n(n-1)\) for directed graphs) [7]. Thus, the empirical results shown for the G(n, m) model should also hold for the G(n, p) model.
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Acknowledgements
We are grateful to Erika Duriakova for providing us the code for the implementation of Yen’s algorithm, the SSSP subroutines and her generous help with debugging our usage of her code.
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Schickedanz, A., Ajwani, D., Meyer, U., Gawrychowski, P. (2019). Average-Case Behavior of k-Shortest Path Algorithms. In: Aiello, L., Cherifi, C., Cherifi, H., Lambiotte, R., Lió, P., Rocha, L. (eds) Complex Networks and Their Applications VII. COMPLEX NETWORKS 2018. Studies in Computational Intelligence, vol 812. Springer, Cham. https://doi.org/10.1007/978-3-030-05411-3_3
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