Tight complexity bounds for diagram commutativity verification
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866908538704494592 |
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| author | Malko, Artem Spiridonov, Igor |
| author_facet | Malko, Artem Spiridonov, Igor |
| contents | A diagram $\mathcal{D} = (G, l)$ over a monoid $M$ is an oriented graph $G = (V, E)$ endowed with a labeling $l\colon E \to M$. A diagram is commutative if and only if for any two oriented paths with the same endpoints, the products in $M$ of their edge labels coincide. We propose the first asymptotically optimal algorithm for diagram commutativity verification applicable to all graph families. For graphs with $\lvert V\rvert \preceq \lvert E\rvert \preceq \lvert V\rvert^2$, which covers most practically relevant cases, our algorithm runs in $$ O\bigl(|V|\,|E|\bigr) \cdot \bigl(T_{\mathrm{equal}} + T_{\mathrm{multi}}\bigr) $$ time; here $T_{\mathrm{equal}}$ and $T_{\mathrm{multi}}$ denote the times to perform an equality check and a multiplication in $M$, respectively. We also establish new lower bounds on the numbers of equality checks and multiplications necessary for commutativity verification, which asymptotically match our algorithm's cost and thus prove its tightness. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_11331 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Tight complexity bounds for diagram commutativity verification Malko, Artem Spiridonov, Igor Combinatorics Category Theory 05C85, 68Q25, 68Q17 (Primary) 68R10, 68W05, 18A10 (Secondary) A diagram $\mathcal{D} = (G, l)$ over a monoid $M$ is an oriented graph $G = (V, E)$ endowed with a labeling $l\colon E \to M$. A diagram is commutative if and only if for any two oriented paths with the same endpoints, the products in $M$ of their edge labels coincide. We propose the first asymptotically optimal algorithm for diagram commutativity verification applicable to all graph families. For graphs with $\lvert V\rvert \preceq \lvert E\rvert \preceq \lvert V\rvert^2$, which covers most practically relevant cases, our algorithm runs in $$ O\bigl(|V|\,|E|\bigr) \cdot \bigl(T_{\mathrm{equal}} + T_{\mathrm{multi}}\bigr) $$ time; here $T_{\mathrm{equal}}$ and $T_{\mathrm{multi}}$ denote the times to perform an equality check and a multiplication in $M$, respectively. We also establish new lower bounds on the numbers of equality checks and multiplications necessary for commutativity verification, which asymptotically match our algorithm's cost and thus prove its tightness. |
| title | Tight complexity bounds for diagram commutativity verification |
| topic | Combinatorics Category Theory 05C85, 68Q25, 68Q17 (Primary) 68R10, 68W05, 18A10 (Secondary) |
| url | https://arxiv.org/abs/2509.11331 |