Abstract
Let T be a symmetric directed tree, i.e., an undirected tree with each edge viewed as two opposite arcs. We prove that the minimum number of colours needed to colour the set of all directed paths in T, so that no two paths of the same colour use the same arc of T, is equal to the maximum number of paths passing through an arc of T. This result is applied to solve the all-to-all communication problem in wavelength-division-multiplexing (WDM) routing in all-optical networks, that is, we give an efficient algorithm to optimally assign wavelengths to the all the paths of a tree network. It is known that the problem of colouring a general subset of all possible paths in a symmetric directed tree is an NP-hard problem. We study conditions for a given set 5 of paths be coloured efficiently with the minimum possible number of colours/wavelengths.
Work partially supported by the Italian Ministry of the University and of the Scientific Research in the framework of the project: “Efficienza di Algoritmi e Progetto di Strutture Informative” and by Galileo Project.
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Gargano, L., Hell, P., Perennes, S. (1997). Colouring paths in directed symmetric trees with applications to WDM routing. In: Degano, P., Gorrieri, R., Marchetti-Spaccamela, A. (eds) Automata, Languages and Programming. ICALP 1997. Lecture Notes in Computer Science, vol 1256. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-63165-8_206
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DOI: https://doi.org/10.1007/3-540-63165-8_206
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