We consider the problem of designing a succinct data structure for path graphs (which are a proper subclass of chordal graphs and a proper superclass of interval graphs) with $n$ vertices while supporting degree, adjacency, and neighborhood queries efficiently. We provide two solutions for this problem. Our first data structure is succinct and occupies $n \log n+o(n \log n)$ bits while answering adjacency query in $O(\log n)$ time, and neighborhood and degree queries in $O(d \log^2 n)$ time where $d$ is the degree of the queried vertex. Our second data structure answers adjacency queries faster at the expense of slightly more space. More specifically, we provide an $O(n \log^2 n)$ bit data structure that supports adjacency query in $O(1)$ time, and the neighborhood query in $O(d \log n)$ time where $d$ is the degree of the queried vertex. Central to our data structures is the usage of the classical heavy path decomposition, followed by a careful bookkeeping using an orthogonal range search data structure among others, which maybe of independent interest for designing succinct data structures for other graphs. It is the use of the results of Acan et al. in the second data structure that permits a simple and efficient implementation at the expense of more space.
翻译：我们考虑的问题是为路径图设计一个简洁的数据结构(这是一个适当的圆形小分类和适当的超级间距图表),上面有1美元,同时支持程度、相邻和邻区查询。我们为这一问题提供了两种解决办法。我们的第一个数据结构是简洁的,占用了$\log n+o(n\log n) 位元,同时用$O(log n) 美元的时间回答对相邻查询,用$O(d) log2 n) 来回答相邻和学位查询,用$(d) = log2 n) 的时间,用$(d) 美元作为被查询的顶点。我们的第二个数据结构的核心是使用传统的重度路径结构,以略微多一点的空间为代价回答对相邻查询更快的查询。更具体地说,我们提供了一个$O(n)\log2 n) 位数据结构,用$(n) 美元的时间来回答对相邻查询,用$(d) = $(d) 美元是被查询的顶点。我们数据结构的核心是使用传统的重路段结构,用精确的数据结构来设计其他图表的搜索范围, 可能是使用数据。在图表中,可能使用其他数据库中,用一个独立的搜索范围中,用一个独立的数据结构中,用一个独立的搜索数据,用其他数据库中的数据,用一个可能使用。
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