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arXiv 2311.02427cs.DS
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具有$d$维$t$-表示的图的简洁数据结构

Succinct Data Structure for Graphs with $d$-Dimensional $t$-Representation

Girish Balakrishnan, Sankardeep Chakraborty, Seungbum Jo, N S Narayanaswamy, Kunihiko Sadakane

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21 pages, 5 figures
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AI总结：

本文针对具有$d$维$t$-表示的未标记图类$\mathcal{G}_{t,d}$，在$td^2$为$o(n/\log n)$时，证明了编码下界并设计了简洁数据结构，支持高效导航查询，同时推广到有界boxicity和区间数的图。

摘要：

Erdős和West（Discrete Mathematics'85）考虑了$n$个顶点的交图类，这些图具有$d$维$t$-表示，即该类中每个图的每个顶点都有一个由至多$t$个$d$维轴平行盒子组成的关联集合。特别地，对于图$G$和每个$d \geq 1$，他们定义$i_d(G)$为使$G$具有这种表示的最小$t$。对于固定的$t$和$d$，他们考虑了$i_d(G) \leq t$的$n$个顶点的标记图类，并证明了该类大小对数的上界为$(2nt+\frac{1}{2})d \log n - (n - \frac{1}{2})d \log(4\pi t)$。\n\n在本工作中，对于固定的$t$和$d$，我们考虑具有$d$维$t$-表示的$n$个顶点的未标记图类，记为$\mathcal{G}_{t,d}$。我们研究为$\mathcal{G}_{t,d}$类设计简洁数据结构的问题，旨在推广近期关于区间图简洁数据结构的结果（Algorithmica'21）。为此，对于每个满足$td^2$在$o(n / \log n)$中的$n$，我们首先证明编码属于$\mathcal{G}_{t,d}$的任意图所需的任何数据结构大小的下界为$(2dt-1)n \log n - O(ndt \log \log n)$比特。\n\n然后，我们为$\mathcal{G}_{t,d}$提出一个$((2dt-1)n \log n + dt\log t + o(ndt \log n))$-比特的数据结构，该结构能高效支持导航查询。将此数据结构与我们的下界论证进行对比，我们证明对于每个固定的$t$和$d$，以及所有满足$td^2$在$o(n/\log n)$中的$n \geq 0$，我们为$\mathcal{G}_{t,d}$设计的数据结构是简洁的。\n\n作为副产品，当$td^2$在$o(n/\log n)$中时，我们还获得了有界boxicity（记为$d$且$t=1$）的图和有界区间数（记为$t$且$d=1$）的图的简洁数据结构。

英文摘要：

Erdős and West (Discrete Mathematics'85) considered the class of $n$ vertex intersection graphs which have a {\em $d$-dimensional} {\em $t$-representation}, that is, each vertex of a graph in the class has an associated set consisting of at most $t$ $d$-dimensional axis-parallel boxes. In particular, for a graph $G$ and for each $d \geq 1$, they consider $i_d(G)$ to be the minimum $t$ for which $G$ has such a representation. For fixed $t$ and $d$, they consider the class of $n$ vertex labeled graphs for which $i_d(G) \leq t$, and prove an upper bound of $(2nt+\frac{1}{2})d \log n - (n - \frac{1}{2})d \log(4πt)$ on the logarithm of size of the class. In this work, for fixed $t$ and $d$ we consider the class of $n$ vertex unlabeled graphs which have a {\em $d$-dimensional $t$-representation}, denoted by $\mathcal{G}_{t,d}$. We address the problem of designing a succinct data structure for the class $\mathcal{G}_{t,d}$ in an attempt to generalize the relatively recent results on succinct data structures for interval graphs (Algorithmica'21). To this end, for each $n$ such that $td^2$ is in $o(n / \log n)$, we first prove a lower bound of $(2dt-1)n \log n - O(ndt \log \log n)$-bits on the size of any data structure for encoding an arbitrary graph that belongs to $\mathcal{G}_{t,d}$. We then present a $((2dt-1)n \log n + dt\log t + o(ndt \log n))$-bit data structure for $\mathcal{G}_{t,d}$ that supports navigational queries efficiently. Contrasting this data structure with our lower bound argument, we show that for each fixed $t$ and $d$, and for all $n \geq 0$ when $td^2$ is in $o(n/\log n)$ our data structure for $\mathcal{G}_{t,d}$ is succinct. As a byproduct, we also obtain succinct data structures for graphs of bounded boxicity (denoted by $d$ and $t = 1$) and graphs of bounded interval number (denoted by $t$ and $d=1$) when $td^2$ is in $o(n/\log n)$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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