The optimal routing of augmented cubes.
Résumé
A routing in a graph G is a set of paths connecting each ordered pair of vertices. Load of an edge e is the number of times it appears on these paths. The edge-forwarding index of G is the smallest of maximum loads over all routings. Augmented cube of dimension n, AQ n , is the Cayley graph (Z n 2 , {e 1 , e 2 ,. .. , e n , J 2 ,. .. , J n }) where e i 's are the vectors of the standard basis and J i = n j=n−i+1 e j. S.A. Choudum and V. Sunitha showed that the greedy algorithm provides a shortest path between each pair of vertices of AQ n. Min Xu and Jun-Ming Xu claimed that this routing also proves that the edge-forwarding index of AQ n is 2 n−1. Here we disprove this claim, by showing that in this specific routing some edges are repeated nearly 4 3 2 n−1 times (to be precise, 2 n+1 3 for even values of n and 2 n+1 3 for odd values of n). However, by providing other routings, we prove that 2 n−1 is indeed the edge-forwarding index of AQ n .
Fichier principal
OptimalRoutingAugmentedCubesRevised18Feb2018.pdf (266.86 Ko)
Télécharger le fichier
Origine | Fichiers produits par l'(les) auteur(s) |
---|