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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| Online Access: | https://arxiv.org/abs/2407.04976 |
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| _version_ | 1866912184864342016 |
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| author | Li, Jason Rao, Satish Wang, Di |
| author_facet | Li, Jason Rao, Satish Wang, Di |
| contents | We develop a novel algorithm to construct a congestion-approximator with polylogarithmic quality on a capacitated, undirected graph in nearly-linear time. Our approach is the first *bottom-up* hierarchical construction, in contrast to previous *top-down* approaches including that of Racke, Shah, and Taubig (SODA 2014), the only other construction achieving polylogarithmic quality that is implementable in nearly-linear time (Peng, SODA 2016). Similar to Racke, Shah, and Taubig, our construction at each hierarchical level requires calls to an approximate max-flow/min-cut subroutine. However, the main advantage to our bottom-up approach is that these max-flow calls can be implemented directly *without recursion*. More precisely, the previously computed levels of the hierarchy can be converted into a *pseudo-congestion-approximator*, which then translates to a max-flow algorithm that is sufficient for the particular max-flow calls used in the construction of the next hierarchical level. As a result, we obtain the first non-recursive algorithms for congestion-approximator and approximate max-flow that run in nearly-linear time, a conceptual improvement to the aforementioned algorithms that recursively alternate between the two problems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_04976 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Congestion-Approximators from the Bottom Up Li, Jason Rao, Satish Wang, Di Data Structures and Algorithms We develop a novel algorithm to construct a congestion-approximator with polylogarithmic quality on a capacitated, undirected graph in nearly-linear time. Our approach is the first *bottom-up* hierarchical construction, in contrast to previous *top-down* approaches including that of Racke, Shah, and Taubig (SODA 2014), the only other construction achieving polylogarithmic quality that is implementable in nearly-linear time (Peng, SODA 2016). Similar to Racke, Shah, and Taubig, our construction at each hierarchical level requires calls to an approximate max-flow/min-cut subroutine. However, the main advantage to our bottom-up approach is that these max-flow calls can be implemented directly *without recursion*. More precisely, the previously computed levels of the hierarchy can be converted into a *pseudo-congestion-approximator*, which then translates to a max-flow algorithm that is sufficient for the particular max-flow calls used in the construction of the next hierarchical level. As a result, we obtain the first non-recursive algorithms for congestion-approximator and approximate max-flow that run in nearly-linear time, a conceptual improvement to the aforementioned algorithms that recursively alternate between the two problems. |
| title | Congestion-Approximators from the Bottom Up |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2407.04976 |