Embeddings and near-neighbor searching with constant additive error for hyperbolic spaces
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arXiv
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| Format: | Preprint |
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2024
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| author | Park, Eunku Vigneron, Antoine |
| author_facet | Park, Eunku Vigneron, Antoine |
| contents | We give an embedding of the Poincaré halfspace $H^D$ into a discrete metric space based on a binary tiling of $H^D$, with additive distortion $O(\log D)$. It yields the following results. We show that any subset $P$ of $n$ points in $H^D$ can be embedded into a graph-metric with $2^{O(D)}n$ vertices and edges, and with additive distortion $O(\log D)$. We also show how to construct, for any $k$, an $O(k\log D)$-purely additive spanner of $P$ with $2^{O(D)}n$ Steiner vertices and $2^{O(D)}n \cdot λ_k(n)$ edges, where $λ_k(n)$ is the $k$th-row inverse Ackermann function. Finally, we show how to construct an approximate Voronoi diagram for $P$ of size $2^{O(D)}n$. It allows us to answer approximate near-neighbor queries in $2^{O(D)}+O(\log n)$ time, with additive error $O(\log D)$. These constructions can be done in $2^{O(D)}n \log n$ time. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2402_14604 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Embeddings and near-neighbor searching with constant additive error for hyperbolic spaces Park, Eunku Vigneron, Antoine Computational Geometry We give an embedding of the Poincaré halfspace $H^D$ into a discrete metric space based on a binary tiling of $H^D$, with additive distortion $O(\log D)$. It yields the following results. We show that any subset $P$ of $n$ points in $H^D$ can be embedded into a graph-metric with $2^{O(D)}n$ vertices and edges, and with additive distortion $O(\log D)$. We also show how to construct, for any $k$, an $O(k\log D)$-purely additive spanner of $P$ with $2^{O(D)}n$ Steiner vertices and $2^{O(D)}n \cdot λ_k(n)$ edges, where $λ_k(n)$ is the $k$th-row inverse Ackermann function. Finally, we show how to construct an approximate Voronoi diagram for $P$ of size $2^{O(D)}n$. It allows us to answer approximate near-neighbor queries in $2^{O(D)}+O(\log n)$ time, with additive error $O(\log D)$. These constructions can be done in $2^{O(D)}n \log n$ time. |
| title | Embeddings and near-neighbor searching with constant additive error for hyperbolic spaces |
| topic | Computational Geometry |
| url | https://arxiv.org/abs/2402.14604 |