Single-Source Shortest Paths with Negative Real Weights in Õ(𝑚𝑛8/9) Time
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Abstract
This paper presents a randomized algorithm for single-source shortest paths on directed graphs with real (both positive and negative) edge weights. Given an input graph with n vertices and m edges, the algorithm completes in Õ(mn8/9) time with high probability. For real-weighted graphs, this result constitutes the first asymptotic improvement over the classic O(mn)-time algorithm variously attributed to Shimbel, Bellman, Ford, and Moore.
References
[1]
Kyriakos Axiotis, Aleksander Madry, and Adrian Vladu. 2020. Circulation Control for Faster Minimum Cost Flow in Unit-Capacity Graphs. In 61st IEEE Annual Symposium on Foundations of Computer Science. 93–104. https://doi.org/10.1109/FOCS46700.2020.00018
[2]
Richard Bellman. 1958. On a Routing Problem. Quart. Appl. Math., 16, 1 (1958).
[3]
Aaron Bernstein, Danupon Nanongkai, and Christian Wulff-Nilsen. 2022. Negative-Weight Single-Source Shortest Paths in Near-linear Time. In 63rd IEEE Annual Symposium on Foundations of Computer Science. 600–611. https://doi.org/10.1109/FOCS54457.2022.00063
[4]
Karl Bringmann, Alejandro Cassis, and Nick Fischer. 2023. Negative-Weight Single-Source Shortest Paths in Near-Linear Time: Now Faster!. In 64th IEEE Annual Symposium on Foundations of Computer Science. 515–538. https://doi.org/10.1109/FOCS57990.2023.00038
[5]
Li Chen, Rasmus Kyng, Yang P. Liu, Richard Peng, Maximilian Probst Gutenberg, and Sushant Sachdeva. 2022. Maximum Flow and Minimum-Cost Flow in Almost-Linear Time. In 63rd IEEE Annual Symposium on Foundations of Computer Science. 612–623. https://doi.org/10.1109/FOCS54457.2022.00064
[6]
Michael B. Cohen, Aleksander Madry, Piotr Sankowski, and Adrian Vladu. 2017. Negative-Weight Shortest Paths and Unit Capacity Minimum Cost Flow in O(m^10/7 ologW) time. In Proceedings of the 28th ACM-SIAM Symposium on Discrete Algorithms. 752–771. https://doi.org/10.1137/1.9781611974782.48
[7]
Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. 2009. Introduction to Algorithms, 3rd Edition. MIT Press. isbn:978-0-262-03384-8
[8]
Yefim Dinitz and Rotem Itzhak. 2017. Hybrid Bellman-Ford-Dijkstra Algorithm. J. of Discrete Algorithms, 42, C (2017), jan, 35–44. issn:1570–8667
[9]
Ran Duan, Jiayi Mao, Xinkai Shu, and Longhui Yin. 2023. A Randomized Algorithm for Single-Source Shortest Path on Undirected Real-Weighted Graphs. In 64th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2023, Santa Cruz, CA, USA, November 6-9, 2023. 484–492. https://doi.org/10.1109/FOCS57990.2023.00035
[10]
Jeff Erickson, Ivor van der Hoog, and Tillmann Miltzow. 2020. Smoothing the gap between NP and ER. In 61st IEEE Annual Symposium on Foundations of Computer Science, FOCS 2020, Durham, NC, USA, November 16-19, 2020, Sandy Irani (Ed.). 1022–1033. https://doi.org/10.1109/FOCS46700.2020.00099
[11]
Lester R. Ford. 1956. Paper P-923. Network Flow Theory.
[12]
Michael L. Fredman and Robert Endre Tarjan. 1987. Fibonacci heaps and their uses in improved network optimization algorithms. J. ACM, 34, 3 (1987), 596–615. https://doi.org/10.1145/28869.28874
[13]
Harold N. Gabow. 1983. Scaling Algorithms for Network Problems. In 24th Annual Symposium on Foundations of Computer Science. 248–257. https://doi.org/10.1109/SFCS.1983.68
[14]
Harold N. Gabow and Robert Endre Tarjan. 1989. Faster Scaling Algorithms for Network Problems. SIAM J. Comput., 18, 5 (1989), 1013–1036.
[15]
Andrew V. Goldberg. 1995. Scaling Algorithms for the Shortest Path Problem. SIAM J. Comput., 24, 3 (1995), 494–504.
[16]
Donald B. Johnson. 1977. Efficient Algorithms for Shortest Paths in Sparse Networks. J. ACM, 24, 1 (1977), jan, 1–13. issn:0004-5411
[17]
Adam Karczmarz, Wojciech Nadara, and Marek Sokołowski. 2024. Exact Shortest Paths with Rational Weights on the Word RAM. In Proceedings of the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms. 2597–2608. https://doi.org/10.1137/1.9781611977912.92
[18]
Edward F. Moore. 1959. The Shortest Path Through a Maze. In Proceedings of the International Symposium on the Theory of Switching. 285–292.
[19]
Seth Pettie and Vijaya Ramachandran. 2005. A Shortest Path Algorithm for Real-Weighted Undirected Graphs. SIAM J. Comput., 34, 6 (2005), 1398–1431. https://doi.org/10.1137/S0097539702419650
[20]
Alfonso Shimbel. 1955. Structure in Communication Nets. In Proceedings of the Symposium on Information Networks. 199–203.
[21]
Jan van den Brand, Yin Tat Lee, Danupon Nanongkai, Richard Peng, Thatchaphol Saranurak, Aaron Sidford, Zhao Song, and Di Wang. 2020. Bipartite Matching in Nearly-linear Time on Moderately Dense Graphs. In 61st IEEE Annual Symposium on Foundations of Computer Science. 919–930. https://doi.org/10.1109/FOCS46700.2020.00090
Index Terms
- Single-Source Shortest Paths with Negative Real Weights in Õ(𝑚𝑛8/9) Time
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Published In
June 2024
2049 pages
ISBN:9798400703836
DOI:10.1145/3618260
- General Chairs:
- Bojan Mohar,
- Igor Shinkar,
- Program Chair:
- Ryan O'Donnell
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Published: 11 June 2024
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