Approximate Distance Sensitivity Oracles in Subquadratic Space
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866917741560070144 |
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| author | Bilò, Davide Chechik, Shiri Choudhary, Keerti Cohen, Sarel Friedrich, Tobias Krogmann, Simon Schirneck, Martin |
| author_facet | Bilò, Davide Chechik, Shiri Choudhary, Keerti Cohen, Sarel Friedrich, Tobias Krogmann, Simon Schirneck, Martin |
| contents | An $f$-edge fault-tolerant distance sensitive oracle ($f$-DSO) with stretch $σ\ge 1$ is a data structure that preprocesses a given undirected, unweighted graph $G$ with $n$ vertices and $m$ edges, and a positive integer $f$. When queried with a pair of vertices $s, t$ and a set $F$ of at most $f$ edges, it returns a $σ$-approximation of the $s$-$t$-distance in $G-F$.
We study $f$-DSOs that take subquadratic space. Thorup and Zwick [JACM 2005] showed that this is only possible for $σ\ge 3$. We present, for any constant $f \ge 1$ and $α\in (0, \frac{1}{2})$, and any $\varepsilon > 0$, a randomized $f$-DSO with stretch $ 3 + \varepsilon$ that w.h.p. takes $\widetilde{O}(n^{2-\fracα{f+1}}) \cdot O(\log n/\varepsilon)^{f+2}$ space and has an $O(n^α/\varepsilon^2)$ query time. The time to build the oracle is $\widetilde{O}(mn^{2-\fracα{f+1}}) \cdot O(\log n/\varepsilon)^{f+1}$. We also give an improved construction for graphs with diameter at most $D$. For any positive integer $k$, we devise an $f$-DSO with stretch $2k-1$ that w.h.p. takes $O(D^{f+o(1)} n^{1+1/k})$ space and has $\widetilde{O}(D^{o(1)})$ query time, with a preprocessing time of $O(D^{f+o(1)} mn^{1/k})$.
Chechik, Cohen, Fiat, and Kaplan [SODA 2017] devised an $f$-DSO with stretch $1{+}\varepsilon$ and preprocessing time $O(n^{5+o(1)}/\varepsilon^f)$, albeit with a super-quadratic space requirement. We show how to reduce their preprocessing time to $O(mn^{2+o(1)}/\varepsilon^f)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_11580 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Approximate Distance Sensitivity Oracles in Subquadratic Space Bilò, Davide Chechik, Shiri Choudhary, Keerti Cohen, Sarel Friedrich, Tobias Krogmann, Simon Schirneck, Martin Data Structures and Algorithms An $f$-edge fault-tolerant distance sensitive oracle ($f$-DSO) with stretch $σ\ge 1$ is a data structure that preprocesses a given undirected, unweighted graph $G$ with $n$ vertices and $m$ edges, and a positive integer $f$. When queried with a pair of vertices $s, t$ and a set $F$ of at most $f$ edges, it returns a $σ$-approximation of the $s$-$t$-distance in $G-F$. We study $f$-DSOs that take subquadratic space. Thorup and Zwick [JACM 2005] showed that this is only possible for $σ\ge 3$. We present, for any constant $f \ge 1$ and $α\in (0, \frac{1}{2})$, and any $\varepsilon > 0$, a randomized $f$-DSO with stretch $ 3 + \varepsilon$ that w.h.p. takes $\widetilde{O}(n^{2-\fracα{f+1}}) \cdot O(\log n/\varepsilon)^{f+2}$ space and has an $O(n^α/\varepsilon^2)$ query time. The time to build the oracle is $\widetilde{O}(mn^{2-\fracα{f+1}}) \cdot O(\log n/\varepsilon)^{f+1}$. We also give an improved construction for graphs with diameter at most $D$. For any positive integer $k$, we devise an $f$-DSO with stretch $2k-1$ that w.h.p. takes $O(D^{f+o(1)} n^{1+1/k})$ space and has $\widetilde{O}(D^{o(1)})$ query time, with a preprocessing time of $O(D^{f+o(1)} mn^{1/k})$. Chechik, Cohen, Fiat, and Kaplan [SODA 2017] devised an $f$-DSO with stretch $1{+}\varepsilon$ and preprocessing time $O(n^{5+o(1)}/\varepsilon^f)$, albeit with a super-quadratic space requirement. We show how to reduce their preprocessing time to $O(mn^{2+o(1)}/\varepsilon^f)$. |
| title | Approximate Distance Sensitivity Oracles in Subquadratic Space |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2305.11580 |