Routing and wavelength assignment for exchanged crossed cubes on ring-topology optical networks

The exchanged crossed cube, denoted by ECQ(s,t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textit{ECQ}(s, t)$$\end{document}, is a novel graph with fewer edges and smaller diameter compared to other variations of the corresponding hypercube. The ring topology, denoted by Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_n$$\end{document}, is one of the most popular topologies in Wavelength division multiplexing optical networks. This paper addresses the routing and wavelength assignment problem for realizing ECQ(s,t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textit{ECQ}(s, t)$$\end{document} communication pattern on Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_n$$\end{document}, where n=s+t+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n=s+t+1$$\end{document}. We propose an embedding scheme. Base on the embedding scheme, a wavelength assignment algorithm using 2s+t-2+⌊2t/3⌋\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2^{s+t-2}+\lfloor 2^t/3\rfloor $$\end{document} wavelengths is devised. We show that the wavelength assignment algorithm uses no more than 1.25 times of wavelengths compared to the optimal wavelength number, i.e., it is a factor 1.25 approximation algorithm. Moreover, the number of additional required wavelengths is no more than ⌊2t-1/3⌋\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lfloor 2^{t-1}/3\rfloor $$\end{document}.