Combinatorial constructions of fault-tolerant routings with levelled minimum optical indices

The design of fault-tolerant routings with levelled minimum optical indices plays an important role in the context of optical networks. However, not much is known about the existence of optimal routings with levelled minimum optical indices besides the results established by Dinitz, Ling and Stinson via the partitionable Steiner quadruple systems approach. In this paper, we introduce a new concept of a large set of even levelled P"3@?-design of order v and index 2, denoted by (v,P"3@?,2)-LELD, which is equivalent to an optimal, levelled (v-2)-fault-tolerant routing with levelled minimum optical indices of the complete network with v nodes. On the basis of the theory of three-wise balanced designs and partitionable candelabra systems, several infinite classes of (v,P"3@?,2)-LELDs are constructed. As a consequence, the existence problem for optimal routings with levelled minimum optical indices is solved for nearly a third of the cases.

[1]  L. Ji.  On the 3BD-closed set B3( 4, 5 ) , 2004, Discret. Math..

[2]  L. Ji,et al.  A new existence proof for large sets of disjoint Steiner triple systems , 2005, J. Comb. Theory, Ser. A.

[3]  Haim Hanani,et al.  On Quadruple Systems , 1960, Canadian Journal of Mathematics.

[4]  C. Colbourn,et al.  The CRC handbook of combinatorial designs , edited by Charles J. Colbourn and Jeffrey H. Dinitz. Pp. 784. $89.95. 1996. ISBN 0-8493-8948-8 (CRC). , 1997, The Mathematical Gazette.

[5]  P. Hell,et al.  Graph Problems Arising from Wavelength-Routing in All-Optical Networks , 2004 .

[6]  Haim Hanani,et al.  On Some Tactical Configurations , 1963, Canadian Journal of Mathematics.

[7]  Stéphane Pérennes,et al.  Efficient collective communication in optical networks , 1996, Theor. Comput. Sci..

[8]  Douglas R. Stinson,et al.  Fault-tolerant routings with minimum optical index , 2006 .

[9]  Bruno Beauquier,et al.  All-to-all communication for some wavelength-routed all-optical networks , 1999, Networks.

[10]  Lijun Ji.  An improvement on H design , 2009 .

[11]  Ján Manuch,et al.  Fault Tolerant Forwarding and Optical Indexes: A Design Theory Approach , 2004, SIROCCO.

[12]  Large set of P3‐decompositions , 2002 .

[13]  Ján Manuch,et al.  On f-wise Arc Forwarding Index and Wavelength Allocations in Faulty All-optical Hypercubes , 2003, RAIRO Theor. Informatics Appl..

[14]  Alok Aggarwal,et al.  Efficient routing in optical networks , 1996, JACM.

[15]  Lijun Ji.  On the 3BD-closed set B3({4,5,6}) , 2004 .

[16]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[17]  Bruno Beauquier.  All-to-All Communication for some Wavelength-Routed All-Optical Networks , 1999 .