Bandwidth and profile minimization
For a class of bandwidth approximation algorithms (called level algorithms), we show that the ratio of the approximation and the exact bandwidth is Ω(log bw(G)-1(n)). Then we give a general approximation algorithm which tries to improve a given layout. It is based on a reordering of a previously generated layout. To generate initial layouts two further level algorithms are introduced.
To compare such algorithms we define two norms for the quality of bandwidth approximation algorithms. The first is influenced by a bound which is inherently given for all level algorithms. The second is influenced by a lower bound for the bandwidth. We have tested the improvement algorithm on many graphs and it provides substantial better bandwidth approximation than all level algorithms.
The bandwidth improvement algorithm is also used for minimizing the profile of a graph. We give a list of results obtained by our algorithm for 30 examples given by G. C. Everstine. So our algorithms becomes comparable with other algorithms which have used these examples as a test set.
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- [APSZ81]S. F. Assmann, G. W. Peck, M. M. Syslo, and J. Zak. The bandwidth of catterpilars with hairs of length 1 and 2. SIAM J. Algebraic and Discret Methodes, 2:387–393, 1981.Google Scholar
- [CCDG82]P. Z. Chinn, J. Chvátalová, A. K. Dewedney, and N. E. Gibbs. The bandwidth problem for graphs and matrices — a survey. Journal of Graph Theory, 6:223–254, 1982.Google Scholar
- [CM69]E. Cuthill and J. Mckee. Reducing the bandwidth of matrices. In Proc. 24th Nat. Conf. ACM, pages 157–166, 1969.Google Scholar
- [Eve79a]S. Even. Graph Algorithms. Computer Science Press, Inc., 1979.Google Scholar
- [Eve79b]G. C. Everstine. A comparision of three resequencing algorithm for the reduction of matrix profile and wavefront. International journal for numerical methods in engineering, 14:837–853, 1979.Google Scholar
- [Eve88]G. C. Everstine. Bandwidth test results. Personal communication, 1988.Google Scholar
- [GPS76]N. E. Gibbs, W. G. Pool Jr., and P. K. Stockmeyer. An algorithm for reducing the bandwidth and profile of a sparse matrix. SIAM J. Numerical Analysis, 13:236–250, 1976.Google Scholar
- [GS84]E. T. Gurari and I. H. Sudborough. Improved dynamic programming algorithms for bandwidth minimization and the mincut linear arrangement problem. Journal of Algorithms, 5:531–546, 1984.Google Scholar
- [HHH85]E. O. Hare, W. R. Hare, and S. T. Hendetniemi. Another upper bound for the bandwidths of trees. Congressus Numerantium 50, 77–83, 1985.Google Scholar
- [Kra86]D. Kratsch. Finding the Minimum Bandwidth of an Interval Graph. Manuskript N/86/18, Friedrich-Schiller-Universität Jena, 1986.Google Scholar
- [Lev71]R. Levy. Resequencing of the structural stiffness matrix to improve computational efficiency. Jet Propulsion Laboratory Quart. Tech. Review, 1:61–70, 1971.Google Scholar
- [LS76]W.-H. Liu and A. H. Sherman. Comparative analysis of the Cuthill-McKee and the reverse Cuthill-McKee ordering algorithms for sparse matrices. SIAM J. Numerical Analysis, 13:198–213, 1976.Google Scholar
- [Meh84]K. Mehlhorn. Data Structures and Algorithms. Volume 1, Sorting and Searching, EATCS Monographs on Theoretical Computer Sience, 1984.Google Scholar
- [Mon86]B. Monien. The bandwidth — minimization problem for caterpillars with hair length 3 is NP-Complete. SIAM J. Algebraic and Discret Methodes, 7:505–512, 1986.Google Scholar
- [Pap76]C. H. Papadimitriou. The NP-Completeness of the bandwidth minimization problem. Computing, 16:263–270, 1976.Google Scholar
- [Sax80]J. B. Saxe. Dynamic programming algorithms for recognizing small bandwidth graphs in polynominal time. SIAM J. Algebraic and Discret Methodes, 363–369, 1980.Google Scholar
- [Tur86]J. S. Turner. On the probable performance of heuristics for bandwidth minimization. SIAM J. Computing, 15:561–580, 1986.Google Scholar
- [Wie88]M. Wiegers. Computing Lower Bounds of the Bandwidth. Manuscript, Universität GH Paderborn, 1988.Google Scholar