Title: Lazy Processing of Incremental Sheaf Cohomology on Cellular Complexes with Bounded Local Geometry: Achieving O(1) Edit Latency

Abstract: This paper introduces an algorithmic framework designed for the incremental maintenance of the first sheaf cohomology group, $H^1(X; \mathcal{F})$, on 1-dimensional cellular complexes that are subject to dynamic changes and equipped with finite-dimensional cellular sheaves. Traditional methods for computing $H^1$, which rely on the factorization of the coboundary matrix, demand $O(n^3)$ computational time. Consequently, if a complex undergoes a sequence of $m$ edits, performing a full recomputation after every single edit would result in a cumulative cost of $O(mn^3)$.

However, by assuming bounded local geometry—specifically limits on cell size ($v_{\max}$), stalk dimension ($d$), and nerve degree ($D$)—we demonstrate that each individual edit (such as vertex or edge insertion, or updates to restriction maps) influences only a restricted set of local coboundary blocks. Leveraging this locality, our algorithm handles streaming edits lazily in $O(1)$ time relative to the total size of the complex, $n$. The computational cost depends polynomially on the local geometry parameters $v_{\max}$, $d$, and $D$, which are treated as constants independent of $n$. This approach postpones local eigensolves and the global assembly via Mayer-Vietoris to specific synchronization points, referred to as "Flush" operations.

At these synchronization points, the maintained state is guaranteed to align with the result of batch assembling the partitioned sheaf model. Empirical testing confirms zero measurable drift in all runs verified against batch processing, even for complexes with up to $V = 10^6$ vertices. Additionally, we provide an amortized $O(|E|)$ method for constructing the cellular decomposition from a stream. We also address an adversarial algebraic-RAM barrier, arguing that non-trivial sheaves (defined by $d \geq 2$ and non-identity restriction maps) lack the same locality properties when unpartitioned.

Experimental results on Barabasi-Albert graphs, containing up to $5 \times 10^6$ vertices and $1.7 \times 10^7$ streaming edits, show a median lazy update latency of 35 $\mu$s per edit, excluding flush operations. The query time, corresponding to global assembly during synchronization, operates in $O(n)$ per flush for the implemented full-traversal path. Detailed reports on exact synchronization costs are provided separately.

Source: arXiv Generated at: 2026-06-04 00:00:00 UTC