动态图谱上实现1阶层化上同调的瞬时更新，每步操作仅需常数时间。
Incremental Sheaf Cohomology on Cellular Complexes: O(1)-in-n Lazy Edit Processing under Bounded Local Geometry
- 基于局部几何有界性，将上同调计算分解为可延迟的局部块处理
- 在百万级顶点与千万级编辑下，单次更新延迟中位数仅35微秒
- 适合需要高实时性的拓扑数据分析场景，如动态网络监控
我们提出一种算法框架，用于在具有有限维单元层化结构的动态一维细胞复形上增量维护第一阶层化上同调 $H^1(X; \mathcal{F})$。经典方法通过边界矩阵分解计算 $H^1$ 需要 $O(n^3)$ 时间；当复形随 $m$ 次编辑演化时，每次重算总耗时达 $O(mn^3)$。在有界局部几何假设下——最大单元大小 $v_{\max}$、分支维数 $d$、神经度 $D$ 均有界——每次编辑（顶点插入、边插入、限制映射更新）仅影响有限局部边界块。因此，算法以 $O(1)$ 时间处理懒加载编辑流（代价为局部几何参数的多项式，视为与 $n$ 无关的常数），将局部特征值求解与梅耶-维托里斯全局组装推迟至同步点（Flush）。同步时维持状态与批量组装一致，所有批量验证运行中未观测到漂移（最高 $V = 10^6$）。实验显示，在含最多 $5 \times 10^6$ 顶点和 $1.7 \times 10^7$ 流式编辑的巴拉巴西-阿尔伯特图上，单次更新延迟中位数为35 μs（不含刷新）；查询时间（同步时全局组装）为每次刷新 $O(n)$（按完整遍历路径实现）。精确同步成本单独报告。
原文摘要 · Abstract (English)
We present an algorithmic framework for incremental maintenance of first sheaf cohomology $H^1(X; \mathcal{F})$ on dynamically evolving 1-dimensional cellular complexes equipped with finite-dimensional cellular sheaves. The classical computation of $H^1$ via factorization of the coboundary matrix requires $O(n^3)$ time; when the complex evolves with a stream of $m$ edits, full recomputation after each edit costs $O(mn^3)$. Under a bounded local geometry assumption -- bounded cell size $v_{\max}$, bounded stalk dimension $d$, and bounded nerve degree $D$ -- each edit (vertex insertion, edge insertion, restriction map update) affects only a bounded set of local coboundary blocks. The algorithm therefore processes lazy streaming edits in $O(1)$ time with respect to the total complex size $n$ (with cost polynomial in the local geometry parameters $v_{\max}$, $d$, and $D$, which are treated as constants independent of $n$), deferring local eigensolves and Mayer-Vietoris global assembly to synchronization points (Flush). At synchronization, the maintained state agrees with the corresponding batch assembly of the partitioned sheaf model; we observe zero measured drift in all batch-verified runs (through $V = 10^6$). We also give an amortized $O(|E|)$ streaming construction for the cellular decomposition and discuss an adversarial algebraic-RAM barrier arguing that unpartitioned non-trivial sheaves ($d \geq 2$, non-identity restriction maps) do not admit the same locality. Experiments on Barabasi-Albert graphs with up to $5 \times 10^6$ vertices and $1.7 \times 10^7$ streaming edits show 35 $μ$s median lazy per-edit update latency (excluding flush); query time (global assembly at synchronization) is $O(n)$ per flush in the implemented full-traversal path. Exact synchronization costs are reported separately.
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