AI 中文总结
研究稀疏图中动态支配与独立问题，给出动态数据结构，能维护特定查询答案，随机且有错误概率界定，给出摊销更新时间。\(r = 1\)时可改进结构，还能维护支配集最小大小近似，有相应更新时间。
AI 中文摘要
设\(\mathscr{C}\)为有界扩张的图类，\(r,k\in \mathbb{N}\)为固定值。对于给定的通过边插入和删除更新且始终满足\(G\in \mathscr{C}\)的动态图\(G\)，我们给出一种动态数据结构，用于维护两个查询的答案：（a）\(G\)是否包含大小为\(k\)的距离\(-r\)支配集？（b）\(G\)是否包含大小为\(k\)的距离\(-r\)独立集？该数据结构是随机的，错误概率由\(\varepsilon\)界定。摊销更新时间为\(\log^c n\cdot \log \frac{1}{\varepsilon}\)。对于第一个查询，若存在，数据结构还可输出大小为\(k\)的距离\(-r\)支配集。当\(r = 1\)时，即使仅假设维护的图\(G\)的退化度由常数\(d\)界定，支配集查询的数据结构也可实现，且摊销更新时间改进为\(2^{k^{{\cal O}(d)}}\cdot \log^3 n\cdot \log \frac{1}{\varepsilon}\)。最后证明，在退化度至多为\(d\)的图中，可维护距离\(-1\)支配集最小大小的\({\cal O}(d^2)\)近似，摊销期望更新时间为\(d^{{\cal O}(1)}\cdot \log n\)。
英文摘要
Let $\mathscr{C}$ be a class of graphs of bounded expansion and $r,k\in \mathbb{N}$ be fixed. We give a dynamic data structure that for a given dynamic graph $G$, updated by edge insertions and deletions subject to the promise that $G\in \mathscr{C}$ at all times, maintains the answer to the following two queries: (a) Does $G$ contain a distance-$r$ dominating set of size $k$? (b) Does $G$ contain a distance-$r$ independent set of size $k$? The data structure is randomized with error probability bounded by $\varepsilon$, for a parameter $\varepsilon>0$ fixed upon the initialization. The amortized update time is $\log^c n\cdot \log \frac{1}{\varepsilon}$, where $n$ is the vertex count of $G$ and $c$ is a constant that depends only on $r$, $k$, and $\mathscr{C}$. In the case of the first query, the data structure can also output a distance-$r$ dominating set of size $k$, if existent. We also prove that when $r=1$, our data structure for the dominating set query can be implemented even if we only assume that the maintained graph $G$ has degeneracy bounded by a constant $d$, yielding a simpler data structure with an improved amortized update time of $2^{k^{{\cal O}(d)}}\cdot \log^3 n\cdot \log \frac{1}{\varepsilon}$. Finally, we prove that in graphs of degeneracy at most $d$, one can maintain an ${\cal O}(d^2)$-approximation of the minimum size of a (distance-$1$) dominating set with amortized expected update time $d^{{\cal O}(1)}\cdot \log n$.
Comments58 pages
