Incremental Sheaf Cohomology on Cellular Complexes: O(1)-in-n Lazy Edit Processing under Bounded Local Geometry
2026-06-02 • Data Structures and Algorithms
Data Structures and AlgorithmsArtificial Intelligence
AI summaryⓘ
The authors developed a new method to quickly update a mathematical object called first sheaf cohomology when a network-like structure changes bit by bit. Normally, recalculating this object from scratch after each change is very slow, but their approach focuses only on the small part affected by each edit, making updates very fast. They show their algorithm handles millions of edits efficiently with guaranteed accuracy after synchronization steps. They also explain why this local update trick doesn't work for more complicated sheaves. Their experiments demonstrate practical speed on large graphs.
sheaf cohomologycellular complexcoboundary matrixincremental maintenancelocal geometryBarabasi-Albert graphlazy updatingMayer-Vietoris assemblyrestriction mapamortized complexity
Authors
Jason L. Volk
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.