Space Complexity of Vertex Connectivity Oracles
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866911286725443584 |
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| author | Pettie, Seth Saranurak, Thatchaphol Yin, Longhui |
| author_facet | Pettie, Seth Saranurak, Thatchaphol Yin, Longhui |
| contents | A $k$-vertex connectivity oracle for undirected $G$ is a data structure that, given $u,v\in V(G)$, reports $\min\{k,κ(u,v)\}$, where $κ(u,v)$ is the pairwise vertex connectivity between $u,v$. There are three main measures of efficiency: construction time, query time, and space. Prior work of Izsak and Nutov shows that a data structure of total size $\tilde{O}(kn)$ can even be encoded as a $\tilde{O}(k)$-bit labeling scheme so that vertex-connectivity queries can be answered in $\tilde{O}(k)$ time. The construction time is polynomial, but unspecified.
In this paper we address the top three complexity measures: Space, Query Time, and Construction Time. We give an $Ω(kn)$-bit lower bound on any vertex connectivity oracle. We construct an optimal-space connectivity oracle in max-flow time that answers queries in $O(\log n)$ time, independent of $k$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2201_00408 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Space Complexity of Vertex Connectivity Oracles Pettie, Seth Saranurak, Thatchaphol Yin, Longhui Data Structures and Algorithms Combinatorics A $k$-vertex connectivity oracle for undirected $G$ is a data structure that, given $u,v\in V(G)$, reports $\min\{k,κ(u,v)\}$, where $κ(u,v)$ is the pairwise vertex connectivity between $u,v$. There are three main measures of efficiency: construction time, query time, and space. Prior work of Izsak and Nutov shows that a data structure of total size $\tilde{O}(kn)$ can even be encoded as a $\tilde{O}(k)$-bit labeling scheme so that vertex-connectivity queries can be answered in $\tilde{O}(k)$ time. The construction time is polynomial, but unspecified. In this paper we address the top three complexity measures: Space, Query Time, and Construction Time. We give an $Ω(kn)$-bit lower bound on any vertex connectivity oracle. We construct an optimal-space connectivity oracle in max-flow time that answers queries in $O(\log n)$ time, independent of $k$. |
| title | Space Complexity of Vertex Connectivity Oracles |
| topic | Data Structures and Algorithms Combinatorics |
| url | https://arxiv.org/abs/2201.00408 |