Metraipolis › archive › dc_ef26d8780fa2

Fast Sparse $α$-Navigable Graph Construction with Improved Resource Augmentation — arXiv:2610.38205

Source: arXiv (en)
Original: https://arxiv.org/abs/2610.38205
License (as archived): CC0-1.0
Archived: 2026-11-01T02:46:26.651Z
SHA-256 of the archived text: 32ac41f5ad0cb84706e637681f15daf059cc095b03278a6fae187a6ae56afd65
Stable URL of this page: https://metraipolis.com/documento/dc_ef26d8780fa2

How to cite

Fast Sparse $α$-Navigable Graph Construction with Improved Resource Augmentation — arXiv:2610.38205 — arXiv (en), https://arxiv.org/abs/2610.38205 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_ef26d8780fa2 (Metraipolis archive dc_ef26d8780fa2, sha256 32ac41f5ad0cb84706e637681f15daf059cc095b03278a6fae187a6ae56afd65)

BibTeX

@misc{dcef26d8780fa2,
  title = {Fast Sparse $α$-Navigable Graph Construction with Improved Resource Augmentation — arXiv:2610.38205},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2610.38205},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_ef26d8780fa2; sha256 32ac41f5ad0cb84706e637681f15daf059cc095b03278a6fae187a6ae56afd65},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2610.38205. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

Graph-based nearest neighbor search methods, which run greedy search on a directed graph over the data points, have surged in popularity in recent years. A common goal is to construct a search graph with the $\alpha$-navigability property, which ensures that greedy search returns provably good approximate nearest neighbors and converges in few iterations. Increasing $\alpha \geq 1$ improves accuracy and convergence rate, but requires more edges, which increases both space and search time. This tradeoff has motivated the study of algorithms for constructing a sparsest $\alpha$-navigable graph for a given set of points. For any point set and any distance function, we show how to compute an $\alpha$-navigable graph with at most $O(\log n)$ times as many edges as the sparsest $(1+\varepsilon)\alpha$-navigable graph, in $O(n^\omega \log(\Delta)/\varepsilon)$ time, where $\omega < 2.372$ is the matrix multiplication exponent and $\Delta$ is the aspect ratio of the input. This improves on Khanna, Padaki, and Waingarten (SODA 2026), who achieve the same runtime but are only competitive with the sparsest $2(1+\varepsilon)\alpha$-navigable graph. Since sparsity often increases rapidly with $\alpha$ (typically $\alpha \leq 1.1$), their bound can be significantly weaker. Previously, the best algorithm competing with the sparsest $(1+\varepsilon)\alpha$-navigable graph required $\widetilde{O}(n^{2.5})$ time. Our techniques also extend to constructing sparse $\tau$-monotonic graphs, another popular target in nearest neighbor search. Finally, we give a fine-grained complexity analysis suggesting that improving the $O(n^\omega)$ running time is unlikely, or would require a very different approach. It is based on hardness results for problems related to verifying whether a graph is navigable, a topic of independent interest.

Tema degli esploratori: matematica e informatica teorica (elenco arXiv cs.DS, nuovi lavori del giorno).
Categorie arXiv: Data Structures and Algorithms (cs.DS).
Pagina del lavoro: https://arxiv.org/abs/2610.38205 — 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 2026-11-01T02:17:58.590Z dalla pagina https://arxiv.org/list/cs.DS/new (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).