区间图族的简洁数据结构
Succinct Data Structures for Families of Interval Graphs
摘要：
我们考虑的问题，设计简洁的数据结构的区间图与$n$顶点，同时支持度，邻接，邻域和最短路径查询在最佳时间的$Θ(\log n)$位字RAM模型。度查询报告在恒定时间内到给定顶点的关联边的数量，如果在恒定时间内两个顶点之间存在边，则邻接查询返回true，邻域查询报告与所查询顶点的度成比例的所有相邻顶点的时间集合，并且最短路径查询返回与其长度成比例的时间最短路径，因此这些查询的运行时间是最佳的。为了展示简洁性，我们首先证明了至少$n\log{n} - 2n\log\log n - O(n)$位是表示任何具有$n$顶点的未标记区间图$G$所必需的，回答了Yang和Pippenger的一个公开问题[Proc. Amer. Math. Soc. 2017]。这通过大小为$n\log{n} +O(n)$位的数据结构来增强，同时不仅最佳地支持上述查询，而且还能够执行各种组合算法（如适当着色、最大独立集等）。在输入区间图上有效地。最后，我们将我们的想法扩展到其他变体的区间图，例如，适当/单位区间图，k-适当和k-不适当的区间图，圆弧图，并设计简洁/紧凑的数据结构，这些图形类，以及随着支持他们的查询效率。
英文摘要：
We consider the problem of designing succinct data structures for interval graphs with $n$ vertices while supporting degree, adjacency, neighborhood and shortest path queries in optimal time in the $Θ(\log n)$-bit word RAM model. The degree query reports the number of incident edges to a given vertex in constant time, the adjacency query returns true if there is an edge between two vertices in constant time, the neighborhood query reports the set of all adjacent vertices in time proportional to the degree of the queried vertex, and the shortest path query returns a shortest path in time proportional to its length, thus the running times of these queries are optimal. Towards showing succinctness, we first show that at least $n\log{n} - 2n\log\log n - O(n)$ bits are necessary to represent any unlabeled interval graph $G$ with $n$ vertices, answering an open problem of Yang and Pippenger [Proc. Amer. Math. Soc. 2017]. This is augmented by a data structure of size $n\log{n} +O(n)$ bits while supporting not only the aforementioned queries optimally but also capable of executing various combinatorial algorithms (like proper coloring, maximum independent set etc.) on the input interval graph efficiently. Finally, we extend our ideas to other variants of interval graphs, for example, proper/unit interval graphs, k-proper and k-improper interval graphs, and circular-arc graphs, and design succinct/compact data structures for these graph classes as well along with supporting queries on them efficiently.
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。
