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Space-Optimal Algorithms for Spanning Forests in Dynamic Streams at Every Confidence Level: With and Without Deletion Constraints — arXiv:2611.04365

Source: arXiv (en)
Original: https://arxiv.org/abs/2611.04365
License (as archived): CC0-1.0
Archived: 2026-11-04T12:58:44.047Z
SHA-256 of the archived text: 559dda3a8c35ebf54d3318c733f7d3c31b596afcbf0641573705a1adb3970094
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How to cite

Space-Optimal Algorithms for Spanning Forests in Dynamic Streams at Every Confidence Level: With and Without Deletion Constraints — arXiv:2611.04365 — arXiv (en), https://arxiv.org/abs/2611.04365 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_4e5ca3df1efb (Metraipolis archive dc_4e5ca3df1efb, sha256 559dda3a8c35ebf54d3318c733f7d3c31b596afcbf0641573705a1adb3970094)

BibTeX

@misc{dc4e5ca3df1efb,
  title = {Space-Optimal Algorithms for Spanning Forests in Dynamic Streams at Every Confidence Level: With and Without Deletion Constraints — arXiv:2611.04365},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2611.04365},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_4e5ca3df1efb; sha256 559dda3a8c35ebf54d3318c733f7d3c31b596afcbf0641573705a1adb3970094},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2611.04365. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

We characterize the optimal space for one-pass spanning-forest computation in dynamic streams of $v$-vertex simple graphs with failure probability at most $\delta$. Nelson and Yu proved an $\Omega(v\log^2v\log(v/\delta))$-bit lower bound, establishing the AGM algorithm's optimality for each fixed $0<\epsilon<1$ and $2^{-v^{1-\epsilon}}<\delta\le1/4$. We resolve the remaining high-confidence regime: for every $0<\delta\le1/4$, matching lower bounds and an algorithm give \[ \Theta\!\left(\min\left\{v^2,\;v\log(v/\delta)\log^2\!\left(2+\frac{v}{\log(v/\delta)}\right)\right\}\right) \] bits, improving space by up to a factor of $\Theta((\log v/\log\log v)^2)$ in this regime. We also study streams with at most $Z$ previously inserted edges deleted and not yet reinserted at report time, where the integer $Z\ge0$ is given in advance. Total deletions, restorations, and intermediate absence counts are unrestricted. For a single report and every $0<\delta\le1/4$, matching lower bounds and a one-pass randomized algorithm yield optimal space $\Theta(\mathcal L(v,Z,\log(1/\delta)))$ bits, where $x_Z(t)=\min\{Z+1,v,\sqrt{vt}\}$ and \[ \mathcal L(v,Z,t)=v\log v\min\{\log^2v,\log^2(Z+2)\} +vt\log\!\left(1+\frac{x_Z(t)}t\right)\log\!\left(1+\frac v{x_Z(t)}\right). \] At zero error, a lower bound for all zero-error algorithms and a matching deterministic algorithm give optimal space \[ \Theta\!\left(\min\{v^2,v(Z+1)\log(1+v/(Z+1))\}\right) \] bits. Together, these results characterize the memory required across all confidence levels and report-time edge-loss bounds.

Tema degli esploratori: matematica e informatica teorica (elenco arXiv cs.DS, nuovi lavori del giorno).
Categorie arXiv: Data Structures and Algorithms (cs.DS) ; Computational Complexity (cs.CC).
Pagina del lavoro: https://arxiv.org/abs/2611.04365 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-11-04T12:35:02.303Z dalla pagina https://arxiv.org/list/cs.DS/new (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).