基于随机游走的动态边定向：从树到外平面图及其他
Dynamic Edge Orientation via Random Walks: From Trees to Outerplanar Graphs and Beyond
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中文总结 AI 辅助
该研究针对完全动态边定向问题，基于随机游走范式，将森林的动态边定向算法扩展到外平面图及无K_{2,t}子式图，实现了常数出度与O(log n)最坏情况更新时间的维护。
中文摘要 AI 辅助
我们研究完全动态边定向问题，重点关注最坏情况时间复杂度。无向图会经历边的插入和删除操作，目标是维护一个具有较小最大出度（下文简称出度）和较小最坏情况更新时间的定向。任意定向的出度至少为α-1，其中α是图的荫度，即其边集可划分成的最少森林数量。当α=O(1)时，长期以来已知出度和最坏情况更新时间都可被O(log n)界定。尽管有大量后续研究，但即使对于非常基础的图族，也没有已知能在最坏情况更新时间达到o(log³n)的算法来维护常数出度，唯一值得注意的例外是森林。对于森林，一个简单的民间算法通过随机游走维护出度为2：当插入操作产生一个出度为3的顶点时，算法会重复选择一条均匀随机的出边，直到到达一个出度至多为1的顶点，然后翻转得到的有向路径。由于基础图无环，路径长度的期望很容易被证明为O(log n)，对于多项式长度的更新序列也以高概率成立。我们证明这种简单的随机游走范式可扩展到外平面图。我们的算法维护常数出度，最坏情况更新时间为O(log n)，该时间界在期望意义下成立，对于多项式长度的更新序列也以高概率成立。我们给出了严格分析：出度为4时可实现O(log n)长度的路径，而出度为3时会产生poly(n)长度的路径。我们还将该论证扩展到任意t≥2的无K_{2,t}子式图，出度界仅取决于t，且具有相同的更新时间保证。[...]
英文摘要
We study the \emph{fully dynamic edge orientation problem}, focusing on \emph{worst-case} time bounds. An undirected graph undergoes edge insertions and deletions, and the goal is to maintain an orientation with small {\em maximum outdegree} (hereafter, outdegree) and small worst-case update time. The outdegree of any orientation is at least $α-1$, where $α$ is the graph's \emph{arboricity}, i.e., the minimum number of forests into which its edge set can be partitioned. When $α= O(1)$, it is long known that both the outdegree and the worst-case update time can be bounded by $O(\log n)$. Despite numerous follow-ups, no $o(\log^3 n)$ worst-case update time is known for maintaining constant outdegree, even for very basic graph families---with a notable exception, \emph{forests}. For forests, a \emph{simple folklore} algorithm maintains outdegree 2 via \emph{random walks}: When an insertion creates a vertex of outdegree 3, the algorithm repeatedly chooses a uniformly random outgoing edge until reaching a vertex of outdegree at most 1, and then flips the resulting directed path. As the underlying graph is cycle-free, the path length is easily shown to be $O(\log n)$ in expectation, and also with high probability for polynomially long update sequences. We prove that this simple random walk paradigm extends to \emph{outerplanar graphs}. Our algorithm maintains constant outdegree with $O(\log n)$ worst-case update time, where the time bound holds in expectation, and also with high probability for polynomially long update sequences. We give a \emph{tight analysis}: outdegree 4 is achievable with $O(\log n)$-length paths, while outdegree 3 incurs $\mathtt{poly}(n)$-length paths. We also extend the argument to $K_{2,t}$-minor-free graphs, for any $t \ge 2$, with the outdegree bound depending only on $t$ and with the same update time guarantees. The locality of [...]
