Faster negative length shortest paths by bootstrapping hop reducers
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912760665735168 |
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| author | Huang, Yufan Jin, Peter Quanrud, Kent |
| author_facet | Huang, Yufan Jin, Peter Quanrud, Kent |
| contents | The textbook algorithm for real-weighted single-source shortest paths takes $O(m n)$ time on a graph with $m$ edges and $n$ vertices. The breakthrough algorithm by Fineman [Fin24] takes $\tilde{O}(m n^{8/9})$ randomized time. The running time was subsequently improved to $\tilde{O}(mn^{4/5})$ [HJQ25].
We build on [Fin24; HJQ25] to obtain an $\tilde{O}(m n^{3/4} + m^{4/5} n)$ randomized running time. (Equivalently, $\tilde{O}(mn^{3/4})$ for $m \geq n^{5/4}$, and $\tilde{O}(m^{4/5} n)$ for $m \leq n^{5/4}$.) The main new technique replaces the hop-reducing auxiliary graph from [Fin24] with a bootstrapping process where constant-hop reducers for small subgraphs of the input graph are iteratively amplified and expanded until the desired polynomial-hop reduction is achieved over the entire graph. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_00428 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Faster negative length shortest paths by bootstrapping hop reducers Huang, Yufan Jin, Peter Quanrud, Kent Data Structures and Algorithms The textbook algorithm for real-weighted single-source shortest paths takes $O(m n)$ time on a graph with $m$ edges and $n$ vertices. The breakthrough algorithm by Fineman [Fin24] takes $\tilde{O}(m n^{8/9})$ randomized time. The running time was subsequently improved to $\tilde{O}(mn^{4/5})$ [HJQ25]. We build on [Fin24; HJQ25] to obtain an $\tilde{O}(m n^{3/4} + m^{4/5} n)$ randomized running time. (Equivalently, $\tilde{O}(mn^{3/4})$ for $m \geq n^{5/4}$, and $\tilde{O}(m^{4/5} n)$ for $m \leq n^{5/4}$.) The main new technique replaces the hop-reducing auxiliary graph from [Fin24] with a bootstrapping process where constant-hop reducers for small subgraphs of the input graph are iteratively amplified and expanded until the desired polynomial-hop reduction is achieved over the entire graph. |
| title | Faster negative length shortest paths by bootstrapping hop reducers |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2506.00428 |