Optimal Fault-Tolerant Spanners in Euclidean and Doubling Metrics: Breaking the $Ω(\log n)$ Lightness Barrier
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| Natura: | Preprint |
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2023
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| _version_ | 1866929237420670976 |
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| author | Le, Hung Solomon, Shay Than, Cuong |
| author_facet | Le, Hung Solomon, Shay Than, Cuong |
| contents | An essential requirement of spanners in many applications is to be fault-tolerant: a $(1+ε)$-spanner of a metric space is called (vertex) $f$-fault-tolerant ($f$-FT) if it remains a $(1+ε)$-spanner (for the non-faulty points) when up to $f$ faulty points are removed from the spanner. Fault-tolerant (FT) spanners for Euclidean and doubling metrics have been extensively studied since the 90s.
For low-dimensional Euclidean metrics, Czumaj and Zhao in SoCG'03 [CZ03] showed that the optimal guarantees $O(f n)$, $O(f)$ and $O(f^2)$ on the size, degree and lightness of $f$-FT spanners can be achieved via a greedy algorithm, which na\"ıvely runs in $O(n^3) \cdot 2^{O(f)}$ time. The question of whether the optimal bounds of [CZ03] can be achieved via a fast construction has remained elusive, with the lightness parameter being the bottleneck. Moreover, in the wider family of doubling metrics, it is not even clear whether there exists an $f$-FT spanner with lightness that depends solely on $f$ (even exponentially): all existing constructions have lightness $Ω(\log n)$ since they are built on the net-tree spanner, which is induced by a hierarchical net-tree of lightness $Ω(\log n)$.
In this paper we settle in the affirmative these longstanding open questions. Specifically, we design a construction of $f$-FT spanners that is optimal with respect to all the involved parameters (size, degree, lightness and running time): For any $n$-point doubling metric, any $ε> 0$, and any integer $1 \le f \le n-2$, our construction provides, within time $O(n \log n + f n)$, an $f$-FT $(1+ε)$-spanner with size $O(f n)$, degree $O(f)$ and lightness $O(f^2)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_11226 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Optimal Fault-Tolerant Spanners in Euclidean and Doubling Metrics: Breaking the $Ω(\log n)$ Lightness Barrier Le, Hung Solomon, Shay Than, Cuong Computational Geometry Data Structures and Algorithms F.2.2; G.2.2 An essential requirement of spanners in many applications is to be fault-tolerant: a $(1+ε)$-spanner of a metric space is called (vertex) $f$-fault-tolerant ($f$-FT) if it remains a $(1+ε)$-spanner (for the non-faulty points) when up to $f$ faulty points are removed from the spanner. Fault-tolerant (FT) spanners for Euclidean and doubling metrics have been extensively studied since the 90s. For low-dimensional Euclidean metrics, Czumaj and Zhao in SoCG'03 [CZ03] showed that the optimal guarantees $O(f n)$, $O(f)$ and $O(f^2)$ on the size, degree and lightness of $f$-FT spanners can be achieved via a greedy algorithm, which na\"ıvely runs in $O(n^3) \cdot 2^{O(f)}$ time. The question of whether the optimal bounds of [CZ03] can be achieved via a fast construction has remained elusive, with the lightness parameter being the bottleneck. Moreover, in the wider family of doubling metrics, it is not even clear whether there exists an $f$-FT spanner with lightness that depends solely on $f$ (even exponentially): all existing constructions have lightness $Ω(\log n)$ since they are built on the net-tree spanner, which is induced by a hierarchical net-tree of lightness $Ω(\log n)$. In this paper we settle in the affirmative these longstanding open questions. Specifically, we design a construction of $f$-FT spanners that is optimal with respect to all the involved parameters (size, degree, lightness and running time): For any $n$-point doubling metric, any $ε> 0$, and any integer $1 \le f \le n-2$, our construction provides, within time $O(n \log n + f n)$, an $f$-FT $(1+ε)$-spanner with size $O(f n)$, degree $O(f)$ and lightness $O(f^2)$. |
| title | Optimal Fault-Tolerant Spanners in Euclidean and Doubling Metrics: Breaking the $Ω(\log n)$ Lightness Barrier |
| topic | Computational Geometry Data Structures and Algorithms F.2.2; G.2.2 |
| url | https://arxiv.org/abs/2306.11226 |