Graph profile: sum of 'local bandwidths' over all vertices. profile(G) = min_σ Σ_v (σ(v) - min_{u∈N[v]} σ(u)). Minimizes total envelope width. Closely related to sparse matrix storage. Cuthill-McKee: classic heuristic for profile reduction.
How to Use
Select graph:
- Profile: Sum value
- vs bw: Compare
- CM: Cuthill-McKee
Sparse Matrix Connection
Symmetric sparse matrix A: profile = sum of distances from diagonal to first nonzero in each row. Minimizing profile reduces storage and fill-in during Cholesky factorization. Cuthill-McKee: BFS-based reordering heuristic.
Algorithms
Profile minimization is NP-hard (like bandwidth). Heuristics: Cuthill-McKee (BFS from peripheral vertex), reverse Cuthill-McKee (often better), Sloan's algorithm (combines BFS with wavefront). Widely used in FEM software.
Step-by-Step Instructions
- 1Select graph.
- 2Compute profile.
- 3Apply Cuthill-McKee.
- 4Compare orderings.
- 5Minimize total stretch.