AI 中文总结
研究动态边连通性问题，提出随机算法，其最坏情况更新和查询时间为$\tilde{O}(n^{12/13})$。还设计了两种确定性算法，分别用于简单图和无加权多重图，在更新和查询时间上有多项式改进或特定时间复杂度提升。
AI 中文摘要
在（完全）动态边连通性问题中，目标是维护一个$n$顶点图$G$在边插入和删除操作下的边连通性$\lambda_G$。主要成果是一种随机算法，用于在动态简单图中维护边连通性，其最坏情况更新和查询时间为$\tilde{O}(n^{12/13})$，适用于所有$\lambda_G$值。这是首个更新和查询时间为$o(n)$的算法。还利用为此开发的工具设计了另外两种算法。一种是确定性算法，用于相同任务，最坏情况更新和查询时间为$n^{1+o(1)}$或$\tilde{O}(n)$摊销时间；另一种是用于动态无加权多重图相同任务的确定性算法，表示为$\tilde{O}(n^{3/2})$最坏情况更新和查询时间。
英文摘要
In the (fully) dynamic edge connectivity problem, the goal is to maintain the edge connectivity $λ_G$ of an $n$-vertex graph $G$ that undergoes edge insertions and deletions. Our main result is a randomized algorithm for maintaining edge connectivity in dynamic simple graphs using worst-case update and query time $\tilde{O}(n^{12/13})$, for all values of $λ_G$. This is the first algorithm that has $o(n)$ update and query time, as all existing algorithms achieve this only when $λ_G$ is below $n^{1/11}$ or above $n^{1/2}$ (up to polylogarithmic factors). We then use the tools developed for this purpose to design two additional algorithms. The first one is a deterministic algorithm for the exact same task, that uses $n^{1+o(1)}$ worst-case update and query time or $\tilde{O}(n)$ amortized update and query time; this gives a polynomial improvement over existing deterministic algorithms. The second one is a deterministic algorithm for the same task but in dynamic unweighted multigraphs, that uses $\tilde{O}(n^{3/2})$ worst-case update and query time.
