Graph bandwidth B(G): place vertices on a line {1,...,n}, minimize the maximum stretch of any edge. B(P_n)=1, B(C_n)=2, B(K_n)=n-1, B(K_{p,q})=p+⌊q/2⌋-1. NP-hard to compute. Applications: sparse matrix reordering, VLSI.
How to Use
Select graph:
- B: Bandwidth
- Labeling: Optimal order
- Bounds: δ, density
Applications
Sparse matrix bandwidth: reorder rows/cols to move nonzeros near diagonal. Cuthill-McKee and reverse Cuthill-McKee algorithms. VLSI layout: minimize wire length. Profile minimization in finite elements.
Bounds
B(G) ≥ ⌈(n-1)/diam(G)⌉. B(G) ≥ δ(G)/2. B(G) ≥ density lower bound. For trees: B can be computed in polynomial time (special case). For caterpillars: exact formula known.
Step-by-Step Instructions
- 1Select graph.
- 2Compute bandwidth.
- 3Find labeling.
- 4Check bounds.
- 5Apply Cuthill-McKee.