Ranking and Unranking of the Planar Embeddings of a Planar Graph
Fuente:
arXiv
Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866916482692153344 |
|---|---|
| author | Di Battista, Giuseppe Grosso, Fabrizio Maragno, Giulia Patrignani, Maurizio |
| author_facet | Di Battista, Giuseppe Grosso, Fabrizio Maragno, Giulia Patrignani, Maurizio |
| contents | Let $\mathcal{G}$ be the set of all the planar embeddings of a (not necessarily connected) $n$-vertex graph $G$. We present a bijection $Φ$ from $\mathcal{G}$ to the natural numbers in the interval $[0 \dots |\mathcal{G}| - 1]$. Given a planar embedding $\mathcal{E}$ of $G$, we show that $Φ(\mathcal{E})$ can be decomposed into a sequence of $O(n)$ natural numbers each describing a specific feature of $\mathcal{E}$. The function $Φ$, which is a ranking function for $\mathcal{G}$, can be computed in $O(n)$ time, while its inverse unranking function $Φ^{-1}$ can be computed in $O(n α(n))$ time. The results of this paper can be of practical use to uniformly at random generating the planar embeddings of a graph $G$ or to enumerating such embeddings with amortized constant delay. Also, they can be used to counting, enumerating or uniformly at random generating constrained planar embeddings of $G$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_10319 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Ranking and Unranking of the Planar Embeddings of a Planar Graph Di Battista, Giuseppe Grosso, Fabrizio Maragno, Giulia Patrignani, Maurizio Computational Geometry Data Structures and Algorithms Let $\mathcal{G}$ be the set of all the planar embeddings of a (not necessarily connected) $n$-vertex graph $G$. We present a bijection $Φ$ from $\mathcal{G}$ to the natural numbers in the interval $[0 \dots |\mathcal{G}| - 1]$. Given a planar embedding $\mathcal{E}$ of $G$, we show that $Φ(\mathcal{E})$ can be decomposed into a sequence of $O(n)$ natural numbers each describing a specific feature of $\mathcal{E}$. The function $Φ$, which is a ranking function for $\mathcal{G}$, can be computed in $O(n)$ time, while its inverse unranking function $Φ^{-1}$ can be computed in $O(n α(n))$ time. The results of this paper can be of practical use to uniformly at random generating the planar embeddings of a graph $G$ or to enumerating such embeddings with amortized constant delay. Also, they can be used to counting, enumerating or uniformly at random generating constrained planar embeddings of $G$. |
| title | Ranking and Unranking of the Planar Embeddings of a Planar Graph |
| topic | Computational Geometry Data Structures and Algorithms |
| url | https://arxiv.org/abs/2411.10319 |