Keywords and Synonyms
Graph bandwidth; Approximation algorithms; Metric embeddings
Problem Definition
The graph bandwidth problem concerns producing a linear ordering of the vertices of a graph \( { G = (V,E) } \) so as to minimize the maximum “stretch” of any edge in the ordering. Formally, let \( { n = |V| } \), and consider any one-to-one mapping \( { \pi : V \to \{1,2, \dots, n\} } \). The bandwidth of this ordering is \( { \mathsf{bw}_\pi(G) = \max_{\{u,v\} \in E} |\pi(u)-\pi(v)| } \). The bandwidth of G is given by the bandwidth of the best possible ordering: \( { \mathsf{bw}(G) = \min_{\pi} \mathsf{bw}_\pi(G) } \).
The original motivation for this problem lies in the preprocessing of sparse symmetric square matrices. Let A be such an \( { n \times n } \) matrix, and consider the problem of finding a permutation matrix P such that the non-zero entries of \( { P^{\text{T}} A P } \)all lie in as narrow a band as possible about the diagonal. This problem is equivalent to...
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\( { N(0,1) }\) denotes a standard normal random variable with mean 0 and variance 1.
Recommended Reading
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Lee, J. (2008). Graph Bandwidth. In: Kao, MY. (eds) Encyclopedia of Algorithms. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-30162-4_169
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DOI: https://doi.org/10.1007/978-0-387-30162-4_169
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