Incremental Sheaf Cohomology on Cellular Complexes: O(1)-in-n Lazy Edit Processing under Bounded Local Geometry

Researchers present an algorithmic framework for efficiently maintaining sheaf cohomology computations on dynamically evolving cellular complexes, reducing edit processing time from O(mn³) to O(1) per operation under bounded local geometry assumptions. The method demonstrates practical viability through experiments on large-scale graphs with millions of vertices and streaming edits, achieving microsecond-level latency while maintaining zero computational drift.

This arXiv paper addresses a specialized computational topology problem with potential applications to distributed systems, network analysis, and data structures that rely on persistent cohomological invariants. The research tackles the computational bottleneck of maintaining first sheaf cohomology during dynamic graph evolution—a problem relevant to systems where topology changes incrementally rather than requiring full recomputation.

The algorithmic contribution centers on exploiting bounded local geometry constraints to achieve constant-time lazy processing of individual edits. By deferring global assembly operations to synchronization points (Flush) and maintaining only locally affected coboundary blocks, the framework transforms an inherently global computation into one with local scope. This architectural approach mirrors lazy evaluation strategies used in functional programming and incremental computation frameworks.

Practical validation through experiments on Barabasi-Albert graphs with up to 5×10⁶ vertices demonstrates sub-100 microsecond median latency per edit, suggesting real-world applicability. The reported zero drift through verified batches validates correctness while the O(|E|) streaming construction for cellular decomposition extends the framework's utility. The adversarial algebraic-RAM barrier argument provides theoretical grounding by identifying fundamental limits for non-partitioned sheaves, establishing that this result is not trivially generalizable.

For developers building systems requiring topological invariants—such as persistent homology engines, network analysis tools, or algebraic data structure libraries—this work provides both practical algorithms and theoretical understanding of feasibility boundaries. The distinction between partitioned and unpartitioned sheaves clarifies where such optimizations remain possible.

• →Incremental sheaf cohomology computation achieves O(1) per-edit time by leveraging bounded local geometry properties and deferring global assembly.
• →Experimental validation on million-scale graphs confirms sub-100 microsecond latency with zero measured computational drift across verified batches.
• →The framework partitions coboundary computations locally, processing streaming edits lazily while maintaining correctness at synchronization checkpoints.
• →Theoretical analysis identifies fundamental barriers preventing similar optimizations for unpartitioned sheaves with non-trivial restriction maps.
• →Practical applicability extends to network analysis, persistent homology computation, and dynamic topological data structure maintenance systems.