Optimal Electrical Oblivious Routing on Expanders
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , , , |
|---|---|
| Format: | Preprint |
| Publié: |
2024
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866913386376200192 |
|---|---|
| author | Florescu, Cella Kyng, Rasmus Gutenberg, Maximilian Probst Sachdeva, Sushant |
| author_facet | Florescu, Cella Kyng, Rasmus Gutenberg, Maximilian Probst Sachdeva, Sushant |
| contents | In this paper, we investigate the question of whether the electrical flow routing is a good oblivious routing scheme on an $m$-edge graph $G = (V, E)$ that is a $Φ$-expander, i.e. where $\lvert \partial S \rvert \geq Φ\cdot \mathrm{vol}(S)$ for every $S \subseteq V, \mathrm{vol}(S) \leq \mathrm{vol}(V)/2$. Beyond its simplicity and structural importance, this question is well-motivated by the current state-of-the-art of fast algorithms for $\ell_{\infty}$ oblivious routings that reduce to the expander-case which is in turn solved by electrical flow routing.
Our main result proves that the electrical routing is an $O(Φ^{-1} \log m)$-competitive oblivious routing in the $\ell_1$- and $\ell_\infty$-norms. We further observe that the oblivious routing is $O(\log^2 m)$-competitive in the $\ell_2$-norm and, in fact, $O(\log m)$-competitive if $\ell_2$-localization is $O(\log m)$ which is widely believed.
Using these three upper bounds, we can smoothly interpolate to obtain upper bounds for every $p \in [2, \infty]$ and $q$ given by $1/p + 1/q = 1$. Assuming $\ell_2$-localization in $O(\log m)$, we obtain that in $\ell_p$ and $\ell_q$, the electrical oblivious routing is $O(Φ^{-(1-2/p)}\log m)$ competitive. Using the currently known result for $\ell_2$-localization, this ratio deteriorates by at most a sublogarithmic factor for every $p, q \neq 2$.
We complement our upper bounds with lower bounds that show that the electrical routing for any such $p$ and $q$ is $Ω(Φ^{-(1-2/p)}\log m)$-competitive. This renders our results in $\ell_1$ and $\ell_{\infty}$ unconditionally tight up to constants, and the result in any $\ell_p$- and $\ell_q$-norm to be tight in case of $\ell_2$-localization in $O(\log m)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_07252 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Optimal Electrical Oblivious Routing on Expanders Florescu, Cella Kyng, Rasmus Gutenberg, Maximilian Probst Sachdeva, Sushant Data Structures and Algorithms In this paper, we investigate the question of whether the electrical flow routing is a good oblivious routing scheme on an $m$-edge graph $G = (V, E)$ that is a $Φ$-expander, i.e. where $\lvert \partial S \rvert \geq Φ\cdot \mathrm{vol}(S)$ for every $S \subseteq V, \mathrm{vol}(S) \leq \mathrm{vol}(V)/2$. Beyond its simplicity and structural importance, this question is well-motivated by the current state-of-the-art of fast algorithms for $\ell_{\infty}$ oblivious routings that reduce to the expander-case which is in turn solved by electrical flow routing. Our main result proves that the electrical routing is an $O(Φ^{-1} \log m)$-competitive oblivious routing in the $\ell_1$- and $\ell_\infty$-norms. We further observe that the oblivious routing is $O(\log^2 m)$-competitive in the $\ell_2$-norm and, in fact, $O(\log m)$-competitive if $\ell_2$-localization is $O(\log m)$ which is widely believed. Using these three upper bounds, we can smoothly interpolate to obtain upper bounds for every $p \in [2, \infty]$ and $q$ given by $1/p + 1/q = 1$. Assuming $\ell_2$-localization in $O(\log m)$, we obtain that in $\ell_p$ and $\ell_q$, the electrical oblivious routing is $O(Φ^{-(1-2/p)}\log m)$ competitive. Using the currently known result for $\ell_2$-localization, this ratio deteriorates by at most a sublogarithmic factor for every $p, q \neq 2$. We complement our upper bounds with lower bounds that show that the electrical routing for any such $p$ and $q$ is $Ω(Φ^{-(1-2/p)}\log m)$-competitive. This renders our results in $\ell_1$ and $\ell_{\infty}$ unconditionally tight up to constants, and the result in any $\ell_p$- and $\ell_q$-norm to be tight in case of $\ell_2$-localization in $O(\log m)$. |
| title | Optimal Electrical Oblivious Routing on Expanders |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2406.07252 |