Optimal Parallel Basis Finding in Graphic and Related Matroids
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912692902559744 |
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| author | Khanna, Sanjeev Putterman, Aaron Song, Junkai |
| author_facet | Khanna, Sanjeev Putterman, Aaron Song, Junkai |
| contents | We study the parallel complexity of finding a basis of a graphic matroid under independence-oracle access. Karp, Upfal, and Wigderson (FOCS 1985, JCSS 1988) initiated the study of this problem and established two algorithms for finding a spanning forest: one running in $O(\log m)$ rounds with $m^{Θ(\log m)}$ queries, and another, for any $d \in \mathbb{Z}^+$, running in $O(m^{2/d})$ rounds with $Θ(m^d)$ queries. A key open question they posed was whether one could simultaneously achieve polylogarithmic rounds and polynomially many queries. We give a deterministic algorithm that uses $O(\log m)$ adaptive rounds and $\mathrm{poly}(m)$ non-adaptive queries per round to return a spanning forest on $m$ edges, and complement this result with a matching $Ω(\log m)$ lower bound for any (even randomized) algorithm with $\mathrm{poly}(m)$ queries per round. Thus, the adaptive round complexity for graphic matroids is characterized exactly, settling this long-standing problem. Beyond graphs, we show that our framework also yields an $O(\log m)$-round, $\mathrm{poly}(m)$-query algorithm for any binary matroid satisfying a smooth circuit counting property, implying, among others, an optimal $O(\log m)$-round parallel algorithms for finding bases of cographic matroids. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_04826 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Optimal Parallel Basis Finding in Graphic and Related Matroids Khanna, Sanjeev Putterman, Aaron Song, Junkai Data Structures and Algorithms Computational Complexity We study the parallel complexity of finding a basis of a graphic matroid under independence-oracle access. Karp, Upfal, and Wigderson (FOCS 1985, JCSS 1988) initiated the study of this problem and established two algorithms for finding a spanning forest: one running in $O(\log m)$ rounds with $m^{Θ(\log m)}$ queries, and another, for any $d \in \mathbb{Z}^+$, running in $O(m^{2/d})$ rounds with $Θ(m^d)$ queries. A key open question they posed was whether one could simultaneously achieve polylogarithmic rounds and polynomially many queries. We give a deterministic algorithm that uses $O(\log m)$ adaptive rounds and $\mathrm{poly}(m)$ non-adaptive queries per round to return a spanning forest on $m$ edges, and complement this result with a matching $Ω(\log m)$ lower bound for any (even randomized) algorithm with $\mathrm{poly}(m)$ queries per round. Thus, the adaptive round complexity for graphic matroids is characterized exactly, settling this long-standing problem. Beyond graphs, we show that our framework also yields an $O(\log m)$-round, $\mathrm{poly}(m)$-query algorithm for any binary matroid satisfying a smooth circuit counting property, implying, among others, an optimal $O(\log m)$-round parallel algorithms for finding bases of cographic matroids. |
| title | Optimal Parallel Basis Finding in Graphic and Related Matroids |
| topic | Data Structures and Algorithms Computational Complexity |
| url | https://arxiv.org/abs/2511.04826 |