🤖 AI Summary
This work addresses the challenge of efficiently maintaining first-layer sheaf cohomology under dynamic edits to one-dimensional cellular complexes, where traditional methods incur $O(n^3)$ recomputation time per edit and thus struggle with large-scale streaming updates. Under a local geometric boundedness assumption, the authors propose a lazy-update mechanism that combines cellular sheaf theory, local block decomposition, and the Mayer–Vietoris sequence to defer local feature resolution and global assembly until necessary. This approach achieves, for the first time, $O(1)$ edit latency—constant with respect to the total complex size $n$—for sheaf cohomology in dynamic settings. Theoretical analysis shows that such locality is unattainable for nontrivial sheaves without partitioning. Experiments on a Barabási–Albert graph with $5 \times 10^6$ vertices and $1.7 \times 10^7$ edits confirm the method’s efficacy: median edit latency is only 35 microseconds, with results exactly matching those of batch recomputation.
📝 Abstract
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.