The Bandwidths of a Matrix. A Survey of Algorithms

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The bandwidth, average bandwidth, envelope, profile and antibandwidth of the matrices have been the subjects of study for at least 45 years. These problems have generated considerable interest over the years because of them practical relevance in areas like: solving the system of equations, finite element methods, circuit design, hypertext layout, chemical kinetics, numerical geophysics etc. In this paper a brief description of these problems are made in terms of their definitions, followed by a comparative study of them, using both approaches: matrix geometry and graph theory. Time evolution of the corresponding algorithms as well as a short description of them are made. The work also contains concrete real applications for which a large part of presented algorithms were developed.

[1] Ababei C., Yan Feng, Brent Goplen, Hushrav Mogal, Tianpei Zhang, Kia Bazargan, and Sachin S. Sapatnekar, Placement and Routing in 3D Integrated Circuits, Design and Test of Computers, Vol. 22, Issue: 6, IEEE ISSN: 0740-7475, 2005, (2005), 520-531

[2] Alway G.G. and Martin D.W., An algorithm for reducing the bandwidth of matrix of symetrical configuration, The Computer Journal, Oxford Journals, 8/3, (1965), 264-272

[3] Antoine G., A. Kahoum, L. Grigori, and M. Sosonkina, A partitioning algo- rithm for block-diagonal matrices with overlap, Parallel Computing 34/6-8 Elsevier, 2008, (2008), 332-344

[4] Arany I, L. Szoda, and W.F. Smyth, Minimizing the bandwidth of sparse matrices, Annales Univ. Sci. Budapest., Sect. Comp., vol. 9, (1973)

[5] Arany I, Another method for finding pseudo-peripheral nodes, Annales Universitatis Scientiarum Budapestinensis de RolandoEtvos nominatae, tom4, (1983), 39-49

[6] Arany I, The method of Gibbs-Poole-Stockmeyer is non-heuristic, Annales Univ. Sci. Budapest., Sect. Comp., vol. 4, (1983)

[7] Arbenz P., Cleary A., Dongarra J., and Hegland M., Parallel numerical linear algebra. chapter: A comparison of parallel solvers for diagonally dominant and general narrow banded linear systemss, Parallel numerical linear algebra, Nova Science Publishers, Inc., Commack, (2001), 35-56

[8] Bansal .R and Srivastava K., Memetic algorithm for the antibandwidth maxi- mization problem, Journal of Heuristics - HEURISTICS , vol. 17, no. 1, Springer, 2011, (2011), 39-60

[9] Barnand S.T., Pothen A., and Simon H.D., A spectral algorithm for envelope reduction of sparse matrices, Journal Num. Lin. Alg. With Appl. 2, (1995), 311-334

[10] Baumann N., P. Fleishmann, and O. Mutzbauer, Double ordering and fill-in for the LU Factorization, SIAM J. Matrix Analysis and Applications, 25, (2003), 630-641

[11] Berry M., B. Hendrickson, and P. Raghavan, Sparse matrix reordering schemes for browsing hypertext, S. Smale, J. Renegar, M. Shub (Eds.), Lectures in Applied Mathematics, 32: The Mathematics of Numerical Analysis, AMS, Providence, RI, 1996, (1996), 99-123

[12] Benitez A. and Branas F., The Go-Away algorithm for Block Factorization of a Sparse Matrix, Course on algorithms for Sparse Large Scale Linear Algebraic Systems, NATO ASI SERIES, Vol. 508, Kluwer, Londres, 1998, (1998), 107-117

[13] Bhatt S.N. and Leighton F.T., A framework for solving VLSI graph layout problems, Computer and System Sciences, Vol. 28, 1984, (1984), 300-343

[14] Blum C., M. J. Blesa Aguilera, A. Roli, and M. Sampels, Hybrid Meta- heuristics, An Emerging Approach to Optimization, volume 114 of Studies in Computational Intelligence. Springer, 2008, (2012)

[15] Bolanos M.E., S. Aviyente, and H. Radha, Graph entropy rate minimization and the compressibility of undirected binary graphs, IEEE Statistical Signal Processing Workshop (SSP), 2012, (2012)

[16] Botafogo R.A., Cluster analysis for hypertext systems, Proceedings of the 16th Annul International ACM-SIGIR Conference on Research and Development in Information Retrieval, ISBN:0-89791-605-0, 1993, (1993), 116-125

[17] Boman E.G. and Hendrickson B., A multilevel algorithm for reducing the enve- lope of sparse matrices, Tech. Rep. SCCM-96-14, Stanford University, 1996, (1996)

[18] Boutora Y., R. Ibtiouen, S. Mezani, N. Takorabet, and A. Rezzoug, A new fast method of profile and wavefront reduction for cylindrical structures in finite elements method analysis, Progress In Electromagnetics Research B, Vol. 27, 2011, (2011), 349-363

[19] Campos V., Pinana E., and Marti R., Adaptive Memory Programming for Matrix Bandwidth Minimization, Annals of Operations Research, March 2011, Vol. 183, Issue 1, Springer 2011, (2011), 7-23

[20] Caprara A. and Salazar-Gonzales J.J., Laying Out Sparse Graphs with Prov- ably Minimum Bandwidth, INFORMS Journal on Computing Vol. 17, No. 3, Summer 2005, ISSN:1091-9856, (2005), 356-373

[21] Caproni A., F. Cervelli, M. Mongiardo, L. Tarricone, and F. Malucelli, Bandwidth Reduced Full-Wave Simulation of Lossless and Thin Planar Microstrip Circuits, ACES JOURNAL, vol. 13, no. 2, (1998), 197-204

[22] Chan W.M. and George A., A linear time implementation of the Reverse Cuthill- McKee algorithm, BIT Numerical Mathematics, vol. 20, no. 1, (1980), 8-14

[23] Chan G.K. and M. Head-Gordon, Highly correlated calculations with a polyno- mial cost algorithm: A study of the density matrix renormalization group, The Journal of Chemical Physics, vol. 116, issue 11, (2002); American Institute of Physics Publishing, doi: 10.1063/1.1449459, 2002, (2002)

[24] Clift S.S., Simon H.D., and Tang Wei-Pai, Spectral Ordering Techniques for Incomplete LU Preconditioners for CG Methods, RIACS Technical Report 95.20 September 1995, Queens Univ., (1995)

[25] Corso G.D. and Manzini G., Finding exact solutions to the bandwidth mini- mization problem, Computing 62, 3, (1999), 189-203

[26] Corso G.D. and Romani F., Heuristic spectral techniques for the reduction of bandwidth and work-bound of sparse matrices, Numerical Algorithms 28, (2001), 127-136

[27] Crisan G.C. and Pintea C.M., A hybrid technique for matrix bandwidth prob- lem, University of Bacu Faculty of Sciences, Scientific Studies and Research, Series Mathematics and Informatics, Vol. 21 (2011), No. 1, (2011), 113-120

[28] Cuthill E. and J. McKee, Reducing the bandwidth of sparse symmetric matrices, Proc. of ACM, (1969), 157-172.

[29] Czibula G., Crisan G.C., Pintea C.M., and Czibula I.G., Soft Computing approaches on the Bandwidth Problem, INFORMATICA 24/2 2013, (2013), 169-180

[30] E. F. D'Azevedo, P. A. Forsyth, and Wei-Pai Tang, Ordering methods for preconditioned conjugate gradient methods applied to unstructured grid problems, SIAM Journal on Matrix Analysis and Applications, Volume 13 Issue 3, July 1992, (1992), 944-961

[31] Diaz J., The fi operator, Fundamentals of Computation Theory, Ed. Akademie- Verlag, (1979), 105-111

[32] Duarte A., R. Marti, M.G.C. Resende, and R.M.A. Silva, GRASP with path relinking heuristics for the antibandwidth problem, Networks, (2011). doi: 10.1002/net.20418., (2011)

[33] Dueck G.H. and Jefis J., A heuristic bandwidth minimization algorithm, Journal of Combinatorial Mathematics and Combinatorial Computing 18, (1995), 97-108

[34] DufiIain S. and Gerard A. Meurant, The efiect of ordering on preconditioned conjugate gradients, BIT Numerical Mathematics 1989, Volume 29, Issue 4, (1989), 635-657

[35] Dutot A., D. Olivier, and G. Savin, The fioperator, Proceedings of EPNACS 2011 within ECCS'11 Emergent Properties in Natural and Artificial Complex Systems, Vienna, Austria - September 15, 2011, (2011)

[36] Ellen M.B. Cavalheiro, Daniele C. Silva, and Sheila M. de Almeida, Aplica- cao de Algoritmos Geneticos no Reordenamento deMatrizes Esparsas, Anais do Congresso de Matematica Aplicada e Computacional CMAC de Nordeste, ISSN:2317-3297, 2013, (2013)

[37] Esposito A. and Tarricone L., Parallel heuristics for bandwidth reduction of sparse matrices with IBM SP2 and Cray T3D, Springer, Applied Parallel Computing Industrial Computation and Optimization LNCS Vol. 1184, (1996), 239-246

[38] Esposito A., M.S. Catalano, F. Malucelli, and L. Tarricone, Sparse matrix bandwidth reduction: Algorithms, applications and real industrial cases in electro- magnetics, high performance algorithms for structured matrix problems, Advances in the theory of Computation and Computational Mathematics 2, (1998), 27-45

[39] Esposito A., F. Malucelli, and Tarricone L., Bandwidth and Profile reduction of Sparse Matrices: An Experimental Comparison of New Heuristics, Proc. of Algorithms and Experiment (ALEX98) febr. 1998, R.Battiti and A.A. Bertossi (Eds), (1998), 19-26

[40] Everstine G.C., The BANDIT program for the reduction of matrix bandwidth for NASTRAN, Naval Ship Research development center, computation and mathematics, Department research and development report, Report 3827/1972, (1972)

[41] Everstine G.C., Finite Element formulation of structural acoustics problems, Computers and Srrucmres Vol. 65, No. 3, Elsevier, 1997, (1997), 307-321

[42] Fernando L. Alvaro and Zian Wang, Direct Sparse Interval Hull Computations for Thin Non-M-Matrices, Interval Computations No 2, 1993, (1993)

[43] Garey M., Graham R., and Knuth D., Complexity results for bandwidth min- imization, SIAM J. Appl. Math. 34, (1978), 477-495

[44] George J.A., Computer implementation of the finite element method, Standford Computer Science Dept., Tech. Report STAn-CS-71-208, Standford, California, (1971)

[45] George A. and Pothen A., An Analysis of Spectral Envelope Reduction via Quadratic Assignment Problems, SIAM. J. Matrix Anal. and Appl., 18/3, (2006), 706-732

[46] Ghidetti K., L. Catabriga, M.C. Boeres, and M.C. Rangel, A study of the inuence of sparse matrices reordering algorithms on Krylov-type preconditioned iterative methods, Mecanica Computacional Vol XXIX, 2010, (2010), 2323-2343

[47] Gibbs N.E., Poole W.G., and Stockmeyer P.K., An algorithm for reducing the bandwidth and proe of a sparse matrix, SIAM Journal on Numerical Analysis, 13, (1976), 236-250

[48] Glover F. and Laguna M., "Tabu Search" in Modern Heuristic Techniques for Combinatorial Problems, Ed. C. Reeves, Blackwell Sc. Publ., Oxford, (1993), 70-150

[49] Glover F., Scatter search and path relinking, New Ideas in Optimization, McGraw- Hill Ltd., 1999., (1999), 297-316

[50] Golub G.H. and Plemonts R.J., Large-Scale Geodetic Least-Squares Adjustment by Dissection and Orthogonal Decomposition, Linear Algebra and its Application, 34, Elsevier, (1980), 3-27

[51] Grama A., Naumov M., and Sameh A., Evaluating sparse linear system solvers on scalable parallel architectures, AFRL-RI-RS-TR-2008-273 Final Technical Report October 2008, Purdue University, (2008)

[52] Greiner D. and Winter G., Sparse Matrices Reordering using Evolutionary Algo- rithms:A Seeded Approach, Proc. ERCOFTAC 2006 Design Optimization: Methods and Applications, Las Palmas de Gran Canaria, 2006, (2006)

[53] Guevara R.L., Reducing the Bandwidth of a Sparse Symmetric Matrix with Ge- netic Algorithmic, GEM, CSREA Press, (2010), (2010), 209-214

[54] Gupta P.K., For eficient parallel solution of load ow problem, Project report, Dept. Of. El. Eng., Indian Institute of Science, Bangalore, (1995)

[55] Gurari E. and Sudborough I., Improved dynamic programming algorithms for bandwidth minimization and the min-cut linear arrangement problem, Journal of Algorithms, Vol. 5, No. 4, Elsevier, (1984), 531-546

[56] Hager W.W., Minimizing the profile of a symmetric matrix, Siam J. Sci. Comput. vol. 23, no. 5, (2002), 1799-1816

[57] Harary F., Graph Theory, Addison-Wesley, (1969) [58] Harper L.H., Optimal assignments of numbers to vertices, J. Soc. Indust. Appl. Math., Vol. 12, No. 1, (1964), 131-135

[59] Harper L.H., Optimal numberings and isoperimetric problems on graphs, J. Combin. Theory, Vol. 1, (1966), 385-393

[60] Huang H., J.M. Dennis, L. Wang, and P. Chen, A scalable parallel LSQR algorithm for solving large-scale linear system for tomographic problems: a case study in seismic tomography, ICCS 2013, Procedia Computer Science 18, Elsevier, 2013, (2013), 581-590

[61] Isazadeh A., Izadkhah H., and Mokarram A.H., A Learning based Evolution- ary Approach for Minimization of Matrix Bandwidth Problem, Appl. Math. Inf. Sci. 6, No. 1, (2012), 51-57

[62] Jennings A., A compact storage scheme for the solution of symmetric linear simul- taneous equations, Computer Journal 9, (1966), 281-285

[63] Karp R., Mapping the Genome: Some Combinatorial Problems Arising in Molecular Biology, STOC'93 Proceedings of the twenty-fifth annual ACM symposium on Theory of computing, 1993, (1993)

[64] Kaveh A. and Sharafi P., A simple ANT algorithm for profile optimization of sparse matrix, Asian Journal of Civil Engineering vol. 9, no. 1, (2007), 35-46

[65] Kaveh A. and Sharafi P., Ordering for bandwidth and profile minimization prob- lems via charged system search algorithm, IJST, Transactions of Civil Engineering, Vol. 36, No. C1, (2012), 39-52

[66] Kennedy J. and Eberhart R., Particle Swarm Optimization, IEEE International Conference on Neural Networks (Perth, Australia), IEEE Service Center, Piscataway, NJ, IV, (1995), 1942-1948

[67] Kendall R., Incidence Matrices, Interval Graphs and Seriation in archaeologys, Pacific Journal of Mathematics, vol. 28, no. 3 1969, (1969)

[68] King P., An automatic reordering scheme for simultaneous equations derived from network systems, Int. J. Numer. meth. Eng. 2, (1970), 523-533

[69] Konig G., M. Moldaschl, and W.N. Gansterer, Computing eigenvectors of block tridiagonal matrices based on twisted block factorizations, Journal of Computational and Applied Mathematics 236, Elsevier, 2012, (2012)

[70] Koohestani B. and Corne D.W., An Improved Fitness Function and Mutation Operator for Metaheuristic Approaches to the Bandwidth Minimization Problem, AIP Conference Proceedings 1117, 21, doi: 10.1063/1.3130627, 2009, (2009)

[71] Koohestani B. and Poli R., A Genetic Programming Approach to the Matrix Bandwidth-Minimization Problem, R. Schaefer et al. (Eds.): PPSN XI, Part II, LNCS 6239, Springer-Verlag Berlin Heidelberg 2010, (2010), 482-491

[72] Koohestani B. and Poli R., A Hyper-Heuristic Approach to Evolving Algorithms for Bandwidth Reduction Based on Genetic Programming, SGAI Conf., Springer 2011, (2011), 93-100

[73] Koohestani B. and Poli R., A Genetic Programming Approach for Evolving Highly-Competitive General Algorithms for Envelope Reduction in Sparse Matrices, C.A. Coello Coello et al. (Eds.): PPSN 2012, Part II, LNCS 7492, Springer, 2012., (2012), 287-296

[74] Kratsch D., Finding the minimum bandwidth of an interval graph, Inform. Comput. 74, (1987), 140-187

[75] Kumfert G. and Pothen A., Two improved algorithms for envelope and wavefront reduction, BIT Numerical Mathematics, 37/3, (1997), 559-590

[76] Leung J.Y.-T, Vornberger O., and Witthofi J.D, On some variants of the bandwidth minimization problem, SIAM J. on Computing, 13, (1984), 650-667

[77] Levin M.P., Compound algorithm for decreasing of matrix profile size, Trends in Mathematics, Inform. Center for Math. Sc., vol 9, no. 1, June 2006, (2006), 141-148

[78] Lim A., Lin J., and Xiao F., Particle Swarm Optimization and Hill Climbing to Solve the Bandwidth Minimization Problem, MIC2003: The Fifth Metaheuristics International Conference, Kyoto, Japan, August 2528, 2003, (2003)

[79] Lim A., Rodrigues B., and Xiao F., Integrated Genetic Algorithm with Hill Climbing for Bandwidth Minimization Problem, E. Cantu-Paz et al. (Eds.): GECCO 2003, LNCS 2724, Springer-Verlag Berlin Heidelberg, 2003, (2003), 1594-1595

[80] Lim A., Rodrigues B., and Xiao F., A Centroid-based approach to solve the Bandwidth Minimization Problem, Proceedings of the 37th Hawaii International Conference on System Sciences, CPS 0-7695-2056-1/04 IEEE, 2004, (2004)

[81] Lim A., Lin J., Rodrigues B., and Xiao F., Ant colony optimization with hill climbing for the bandwidth minimization problem, Elsevier, Applied Soft Computing 6/2, (2006), 180-188

[82] Lim A., Rodrigues B., and Xiao F., Heuristics for matrix bandwidth reduction, Elsevier, European Journal of Operational Research 174, (2006), 69-91

[83] Lozano M., Duarte A., Gortazar F., and Marti R, Variable neighborhood search with ejection chains for the antibandwidth problem, Journal of Heuristics, 18/6, Springer, (2012), 919-938

[84] Lozano M., Duarte A., Gortazar F., and Marti R, A hybrid metaheuristic for the cyclic antibandwidth problem, Knowledge-Based Systems 54, Elsevier, (2013), 103-113

[85] Luo J.C., Algorithms for reducing the bandwidth and profile of a sparse matrix, Computers and Structures 44 (1992), (1992), 535-548

[86] Mafteiu-Scai L.O., Bandwidth reduction on sparse matrix, West University of Timisoara Annals, XLVIII/3, (2010)

[87] Mafteiu-Scai L.O., Negru V., Zaharie D., and Aritoni O., Average bandwidth reduction in sparse matrices using hybrid heuristics, Studia Universitatis Babes- Bolyai University, Cluj Napoca, 3/2011, (2011)

[88] Mafteiu-Scai L.O., Negru V., Zaharie D., and Aritoni O., Average bandwidth reduction in sparse matrices using hybrid heuristics-extended version, Proc. KEPT 2011, selected papers, ed. M. Frentiu et. all, Cluj-Napoca, July 4-6, 2011, Presa Universitara Clujeana, ISSN 2067-1180, (2011), 379-389

[89] Mafteiu-Scai L.O., Interchange opportunity in average bandwidth reduction in sparse matrix, West Univ. of Timisoara Annals, Timisoara, Romania, ISSN:1841-3293, (2012)

[90] Mafteiu-Scai L.O., Average Bandwidth Relevance n Parallel Solving Systems of Linear Equations, IJERA Vol. 3, Issue 1, January-February 2013, ISSN 2248-9622, (2013), 1898-1907

[91] Mafteiu-Scai L.O., Experiments and Recommendations for Partitioning Systems of Equations, West Univ. of Timisoara Annals, Timisoara, Romania, LI 1/2014, ISSN:1841-3293, (2014), 141-156

[92] Maheswaran M., K.J. Webb, and H.J. Siegel, A Modified Conjugate Gradient Squared Algorithm for Nonsymmetric Linear Systems, The Journal of Supercomputing, 14, Kluwer Academic Publishers , 1999, (1999), 257-280

[93] Marti R., M. Laguna, F. Glover, and V. Campos, Reducing the bandwidth of a sparse matrix with tabu search, European Journal of Operational Research 135 (2), (2001), 211-220

[94] Marti R., Campos V., and Pinana E., A Branch and Bound Algorithm for the Matrix Bandwidth Minimization, European Journal of Operational Research, 186/2, (2008), 513-528

[95] Maruster S., Negru V., and Mafteiu-Scai L.O., Experimental study on parallel methods for solving systems of equations, SYNACS Timisoara, 2012, IEEE Xplore CPS ISBN: 978-1-4673-5026-6, DOI: 10.1109/SYNASC.2012.7, (2013)

[96] Mecke S. and Wagner D., Solving Geometric Covering Problems by Data Re- duction, Algorithms ESA 2004, Lecture Notes in Computer Science Volume 3221, 2004, (2004), 760-771

[97] N. Mladenovic, D. Urosevic, D. Perez-Brito, and C.G. Garcia-Gonzalez, Variable neighbourhood search for bandwidth reduction, European Journal of Operational Research 200, Elsevier, (2010), 14-27

[98] Mueller C., Sparse Matrix Reordering Algorithms for Cluster Identification, For I532, Machine Learning in Bioinformatics, December 17, 2004, (2004)

[99] Osipov P., Simple heuristic algorithm for profile reduction of arbitrary sparse ma- trix, Applied Mathematics and Computation, Vol 168/2, Elsevier, (2005), 848-857

[100] Pan V., Optimum parallel computation with band matrices, TR-93-061, NSF Grant CRR9020690 1993, (1993)

[101] Papadimitriou C.H., The NP-completeness of the bandwidth minimization prob- lem, Computing 16, 3, (1976), 263-270

[102] Pinana E., Plana I., Campos V., and Marti R., GRASP and path relinking for the matrix bandwidth minimization, European Journal of Operational Research 153 (1,16), (2004), 200-210

[103] Pinar A. and Heath M.T., Improving performance of sparse matrix-vector mul- tiplication, Proceeding SC '99 Proceedings of the 1999 ACM/IEEE conference on Supercomputing Article No. 30, ISBN:1-58113-091-0, (1999)

[104] Pintea C.M., Crisan G.C., and Chira C., A Hybrid ACO Approach to the Matrix Bandwidth Minimization Problem, M. Graa Romay et al. (Eds.): HAIS 2010, Part I, LNAI 6076, Springer-Verlag Berlin Heidelberg 2010, (2010), 407-414

[105] Pintea C.M. and Vescan A., Bio-Inspired Components for Bandwidth Problem, "Vasile Alecsandri" University of Bacu, Faculty of Sciences, Scientific Studies and Research, Series Mathematics and Informatics, Vol. 21 (2011), No. 1, (2011), 185-192

[106] Pintea C.M., Advances in Bio-inspired Computing for Combinatorial Optimization Problems, Intelligent Systems Reference Library, Vol. 57, Springer, ISBN 978-3-642-40178-7, (2014)

[107] Poli R., Covariant tarpeian method for bloat control in genetic programming, Riolo, R., McConaghy, T., Vladislavleva, E. (eds.) Genetic Programming Theory and Practice VIII, Genetic and Evolutionary Computation, vol. 8, chap. 5, Springer, (2014), 71-90

[108] Pop P. and Matei O., An Improved Heuristic for the Bandwidth Minimization Based on genetic programming, Proceedings part II Hybrid Artificial Intelligent Systems: 6th International Conference, HAIS Poland 2011, (2011), 67-75

[109] Pop P. and Matei O., Reducing the bandwidth of a sparse matrix with a genetic algorithm, Optimization: A Journal of Mathematical Programming and Operations Research, Vol. 63, Issue 12, 2014, (2014), 1851-1876

[110] Pop P. and Matei O., An Eficient Metaheuristic Approach for Solving a Class of Matrix Optimization Problems, Proceedings of the 15th EU/ME 2014 Workshop, ISBN 978-605-85313-0-7, (2014), 17-25

[111] Quoct V. and O'Learys J.R., AUTOMATIC NODE RESEQUENCING WITH CONSTRAINTS, Computers and Structures Vol. 18, No. 1, (1984), 55-69

[112] Rainer G., Bandwidth reduction on sparse matrices by introducing new variables, Ingeniare. Revista chilena de ingeniera, vol. 18 No. 3, (2010), 395-400

[113] Ramon P. and Benitez A., On Reducing Bandwidth of Matrices in the Go-Away Algorithm for Regular Grids, Divulgaciones Matematicas Vol. 7 No. 1, 1999, (1999), 1-12

[114] Raspaud A., Schroder H, Sykora O., Torok L., and Vrto I., Antibandwidth and cyclic antibandwidth of meshes and hypercubes, Discrete Mathematics, 309, (2009), 3541-3552

[115] Ravi R., Agrawal A., and Klein P., Ordering problems approximated: single- processor scheduling and interval graph Completition, Proc. Automata, Languages and Programming, Springer, ISBN: 0-387-54233-7, (1991), 751-762

[116] Reid J.K. and Scott J.A., Reducing the total bandwidth of a sparse unsymmetric matrix, RAL-TR-2005-001 CCLRC ISSN 1358-6254, (2005)

[117] Rosen R., Matrix bandwidth minimization, In Proc. 23rd Nat. Conf. ACM, (1968), 585-595

[118] Saxe J.B., GRASP with Path Relinking for the SumCut Problem, International Journal of Combinatorial Optimization Problems and Informatics, Vol. 3, No. 1, Jan-Aprilie 2012, ISSN: 2007-1558, (2012), 3-11

[119] Saxe J.B., Dynamic programming algorithms for recognizing small bandwidth graphs in polynomial time, SIAM Journal of Algebraic and Discrete Methods 1, (1980), 363-369

[120] Sloan S.M., An algorithm for profile and wavefront reduction of sparse matri- ces, International Journal for Numerical Methods in Engineering. Vol. 23, Issue 2, February, 1985, (1985), 239-251

[121] Smyth W.F. and Arany I., Another algorithm for reducing bandwidth and profile of a sparse matrix, AFIPS '76 Proceedings, ACM 1976, (1976)

[122] Smyth W.F., Algorithms for the reduction of matrix bandwidth and profile, Journal of Computational and Applied Mathematics 12-13, (1985), 551-561

[123] Snay R.A., Reducing the profile of sparse symmetric matrices, NOAA Technical Memorandum NOS NGS-4, 1976, (1976)

[124] Taniguchi T., Bandwidth Minimization Algorithm for Finite Element Mesh, Memoirs of the School of Engineering, Okayama University, Vol. 16, No.1, (1981)

[125] Taniguchi T., Reordering Algorithm for Skyline Method (Numerical Algorithms of Large Linear Problems), KURENAI : Kyoto University Research Information Repository, 1985-02, (1985), 30-49

[126] Rodriguez-Tello E. and Betancourt L.C., An Improved Memetic Algorithm for the Antibandwidth Problem, Artificial Evolution, LNCS Springer, Vol. 7401, 2012, (2012), 121-132

[127] Tewarson R.P., Sparse Matrices, Elsevier, Academic press, 99, (1973)

[128] Torok L. and Vrto I., Antibandwidth of 3-dimensional meshes, Electronic Notes in Discrete Mathematics, 28, (2007), 161-167

[129] Torres-Jimenez J. and Rodriguez-Tello E., A New Measure for the Bandwidth Minimization Problem, Advances in Artificial Intelligence, LNCS, vol. 1952, 2000, Springer, ISBN 978-3-540-41276-2, (2000), 477-486

[130] Rodriguez-Tello E., Hao K-K., and Torres-Jimenez J., An Improved Simu- lated Annealing Algorithm for Bandwidth Minimization, European Journal of Operational Research 185, (2008), 1319-1335

[131] Tsao Y.P. and Chang G.D., Profile minimization on compositions of graphs, NCTS/TPE-Math Technical Report 2006-007, (2005)

[132] Ullman J.D., Computational aspects of VLSI, Computer Science Press, Rockville, Md., 1983, (1983)

[133] Wai-Hung L. and Sherman A.H., Comparative analysis of the CutHill-McKee and the Reverse CutHill-Mckee ordering algorithms for sparse matrices, SIAM J. Numer. Anal., 13, (1976), 198-213

[134] Wang Q., Y. C. Guo, and X. W. Shi, An improved algorithm for matrix band- width and profile reduction in finite element analysis, Progress In Electromagnetics Research Letters, 9, (2009), 29-38

[135] Weise T., Global Optimization Algorithms " Theory and Application ", e-book, 3rd edition,, (2011)

[136] Xu S., H.X. Lin, and W. Xue, Sparse Matrix-Vector Multiplication Optimiza- tions based on Matrix Bandwidth Reduction using NVIDIA CUDA, Int. Sym. on Distrib. Comp. and App. to Business, Eng. and Science, IEEE CPS, ISBN:978-1-4244-7539-1, 2010, (2010), 609-614

[137] Yixun L. and Jinjiang Y., The dual bandwdith problem for graphs, Journal of Zhengzhou University, 35 (2003), (2003), 1-5

[138] Zabih R., Some Applications of Graph Bandwidth to Constraint Satisfaction Prob- lems, AAAI-90 Proceedings, AAAI (, 1990, (1990)

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