Algorithmic complexity trade-offs between digit-map computations and Smith normal form

Ascertain the comparative computational complexity of obtaining arithmetic barcodes via successive mod-^k digit connecting maps versus computing Smith normal form of the coboundary over a discrete valuation ring, with particular attention to sparse or structured matrices, and determine structural conditions under which the digit-map approach outperforms Smith normal form.

Background

The paper’s Digit-SNF Dictionary provides two algorithmic routes to the same invariants: computing digit map image dimensions across precision levels or computing Smith normal form directly. Practical performance may depend on sparsity, structure, and precision regimes.

A formal complexity analysis would guide implementation choices, indicating scenarios where hierarchical mod-^k computations are more efficient than global Smith normal form, and vice versa.