Oriented Diameter of Star Graphs
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An orientation of an undirected graph G is an assignment of exactly one direction to each edge of G. Converting two-way traffic networks to one-way traffic networks and bidirectional communication networks to unidirectional communication networks are practical instances of graph orientations. In these contexts minimising the diameter of the resulting oriented graph is of prime interest. The n-star network topology was proposed as an alternative to the hypercube network topology for multiprocessor systems by Akers and Krishnamurthy [IEEE Trans. on Computers (1989)]. The n-star graph S_n consists of n! vertices, each labelled with a distinct permutation of [n]. Two vertices are adjacent if their labels differ exactly in the first and one other position. S_n is an (n-1)-regular, vertex-transitive graph with diameter 3(n-1)/2. Orientations of S_n, called unidirectional star graphs and distributed routing protocols over them were studied by Day and Tripathi [Information Processing Letters (1993)] and Fujita [The First International Symposium on Computing and Networking (CANDAR 2013)]. Fujita showed that the (directed) diameter of this unidirectional star graph S_n is at most 5n/2 + 2. In this paper, we propose a new distributed routing algorithm for the same S_n analysed by Fujita, which routes a packet from any node s to any node t at an undirected distance d from s using at most min{4d+4, 2n+4} hops. This shows that the (directed) diameter of S_n is at most 2n+4. We also show that the diameter of S_n is at least 2n when n ≥ 7, thereby showing that our upper bound is tight up to an additive factor.
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