Is it common to try to improve an algorithm by decomposing its action on a topological piece of data (e.g. graphs, geometric data) into a series of steps, each of which only makes a local change/perturbation in a confined neighborhood of the data while fixing the rest?
One benefit of this "breakdown" is improved conceptual understanding, and it also facilitates induction proofs (possibly based on invariants that are preserved by each step). Does this "decomposition" have a name in algorithm design? The idea might be similar to Divide and Conquer.
One might say that every individual step in an algorithm, like deleting an edge or labeling a vertex of a graph, is small enough to be considered "local", but I'm talking more about the "decomposition" in a higher-level description, like how truncating a polyhedron is local but sorting a list is not (even though at the lowest level, comparisons, insertions and deletions are local).
Examples of decomposing into local transformations
Augustin Cauchy's proof of his Rigidity Theorem relied on his "Arm Lemma" that opened up a convex planar chain by sequentially opening at each vertex. Although his proof was found to be wrong, he decomposed the opening algorithm into a series of "local opening moves", and he tried to prove that each local move preserved convexity of the chain. By induction, convexity would still hold after iterating over all vertices.
The pivot algorithm for self-avoiding walks (SAW) transforms a SAW into another by rotating or reflecting part of the original SAW. Markov Chain Monte Carlo can employ this transformation as a transition between SAW states to sample the space of SAWs.
As an application of the above, one technique of protein folding prediction is to force an amino acid chain into several known protein structures, choose the one with the lowest energy (due to intermolecular forces), then let the chosen structure "relax" into a configuation with lower energy. Relaxation involves the prescription of a few "local moves", where small parts of the chain can "wriggle around" (e.g. subchains pivoting around their ends), and random wriggles are allowed for a period of time to let the chain "stabilize". Modelling the chain as an SAW allows pivots to be used as a "wriggle".