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Optimal Groomings with Grooming Ratios Six and Seven
Author(s) -
Ge Gennian,
Kolotoğlu Emre,
Wei Hengjia
Publication year - 2015
Publication title -
journal of combinatorial designs
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.618
H-Index - 34
eISSN - 1520-6610
pISSN - 1063-8539
DOI - 10.1002/jcd.21428
Subject(s) - traffic grooming , combinatorics , mathematics , upper and lower bounds , drop (telecommunication) , synchronous optical networking , graph , discrete mathematics , computer science , wavelength division multiplexing , computer network , telecommunications , physics , optics , mathematical analysis , wavelength
Grooming uniform all‐to‐all traffic in optical (SONET) rings with grooming ratio C requires the determination of a decomposition of the complete graph into subgraphs each having at most C edges. The drop cost of such a grooming is the total number of vertices of nonzero degree in these subgraphs, and the grooming is optimal when the drop cost is minimum. The determination of optimal C ‐groomings has been considered for 3 ≤ C ≤ 9 , and completely solved for 3 ≤ C ≤ 5 . For C = 6 , it has been shown that the lower bound for the drop cost of an optimal C ‐grooming can be attained for almost all orders with 5 exceptions and 308 possible exceptions. For C = 7 , there are infinitely many unsettled orders; especially the case n ≡ 2 ( mod 3 ) is far from complete. In this paper, we show that the lower bound for the drop cost of a 6‐grooming can be attained for almost all orders by reducing the 308 possible exceptions to 3, and that the lower bound for the drop cost of a 7‐grooming can be attained for almost all orders with seven exceptions and 16 possible exceptions. Moreover, for the unsettled orders, we give upper bounds for the minimum drop costs.