On oriented diameter of (n,k)-star graphs

Assignment of one of two possible directions to every edge of an undirected graph G=(V,E) is called an orientation of G. The resulting directed graph is denoted by G⃗. A strong orientation is one in which every vertex is reachable from every other vertex via a directed path in G⃗. The diameter of G⃗, i.e., the maximum distance from any vertex to any other vertex, depends on the particular orientation. The minimum diameter among all possible orientations of G is called the oriented diameter diam⃗(G) of G. Let n,k be two integers such that 1≤k<n. In the realm of interconnection networks of processing elements, an (n,k)-star graph S_n,k offers a topology that permits to circumvent the lack of scalability of n-star graphs S_n. The oriented diameter quantifies an upper limit on the delay in communication over interconnection networks. In this paper, we present a strong orientation scheme for S_n,k that combines approaches suggested by Cheng and Lipman (2002) for S_n,k with the one proposed by Fujita (2013) for S_n, reaping benefits from both worlds. Next, we propose a distributed routing algorithm for S_n,k⃗ inspired by an algorithm proposed in Kumar et al. (2021) for S_n⃗. With the aid of both the orientation scheme and the routing algorithm, we show that diam⃗(S_n,k)≤⌊[Formula presented.]⌋+2k+6−δ(n,k) where δ(n,k) is a non-negative function. The function δ(n,k) takes on values 2k−n, 0, and [Formula presented.] respectively for three disjoint intervals k>[Formula presented.], [Formula presented.]<k≤[Formula presented.] and k≤[Formula presented.]. For every value of n, k, our upper bound performs better than all known bounds in literature.