Given a graph $G(V,E)$ having $n$ vertices and $m$ edges, we maintain its Breadth-First Search (BFS) tree from source $s$ under an online sequence of edge updates in the prediction model. Our approach leverages a predicted update sequence aiding online processing. We present algorithms for incremental (insertions-only), decremental (deletions-only), and fully dynamic (insertions and deletions) settings that maintain a BFS tree (parent and level information). Classically, the incremental and decremental BFS tree requires total $O(mn)$ time [JACM81], with amortized $O(n)$ and worst-case $O(m)$ update time. The combinatorial BMM conjecture restricts any polynomial improvement [FOCS14] even when the updates are known in advance [STOC15]. For fully dynamic BFS trees, only the trivial $O(m)$ time recomputation is known. Our complexity bounds are expressed in prediction error measures, where error vertices are those having incorrectly predicted distances, with the corresponding difference as their error. The vertex prediction error $η_{v}$ is the sum of degrees of error vertices, weighted vertex prediction error $η^*_{v}$ is error-weighted sum of degrees of error vertices, and $η_e$ counts the incorrectly predicted updates. For incremental and decremental BFS, our algorithm requires respectively $O(η_v + η_e)$ and $O(\min\{m,η^*_v + η_e\})$ worst case update time using $O(mn)$ preprocessing time and space, and total update time of $O(η^*_v + η_e)$. For fully-dynamic updates, our algorithm requires $O(\min\{m,η^*_v+η_e\})$ worst case update time. At its core, we extend the classical ES Trees [JACM81] for batch updates and fully dynamic updates. This simple extension is sufficient to give a competitive prediction algorithm, which may be generalized to other graph problems. We also consider space optimizations and error correction to improve our results.

翻译：给定一个包含 $n$ 个顶点和 $m$ 条边的图 $G(V,E)$，我们在预测模型下维护从源点 $s$ 出发的广度优先搜索（BFS）树，以应对在线边更新序列。我们的方法利用预测的更新序列来辅助在线处理。针对增量（仅插入）、减量（仅删除）和全动态（插入与删除）场景，我们提出了维护BFS树（父节点与层级信息）的算法。经典情况下，增量与减量BFS树需要总时间 $O(mn)$ [JACM81]，分摊更新时间为 $O(n)$，最坏情况更新时间为 $O(m)$。组合BMM猜想限制了任何多项式级别的改进 [FOCS14]，即使更新序列事先已知 [STOC15]。对于全动态BFS树，目前仅有平凡的 $O(m)$ 时间重计算方法。我们的复杂度界限以预测误差度量表示，其中误差顶点是指距离预测不正确的顶点，其对应差值即为误差。顶点预测误差 $\eta_{v}$ 为误差顶点的度数之和，加权顶点预测误差 $\eta^*_{v}$ 为误差顶点度数的误差加权和，而 $\eta_e$ 统计预测不正确的更新次数。对于增量和减量BFS，我们的算法分别需要 $O(\eta_v + \eta_e)$ 和 $O(\min\{m,\eta^*_v + \eta_e\})$ 的最坏情况更新时间，预处理时间和空间为 $O(mn)$，总更新时间为 $O(\eta^*_v + \eta_e)$。对于全动态更新，我们的算法需要 $O(\min\{m,\eta^*_v+\eta_e\})$ 的最坏情况更新时间。其核心在于，我们将经典ES树 [JACM81] 扩展至批量更新和全动态更新。这一简单扩展足以提供有竞争力的预测算法，并可推广至其他图问题。我们还考虑了空间优化与误差校正，以改进结果。