Abstract
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 diameterdiam⃗(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 Sn,k offers a topology that permits to circumvent the lack of scalability of n-star graphs Sn. 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 Sn,k that combines approaches suggested by Cheng and Lipman (2002) for Sn,k with the one proposed by Fujita (2013) for Sn, reaping benefits from both worlds. Next, we propose a distributed routing algorithm for Sn,k⃗ inspired by an algorithm proposed in Kumar et al. (2021) for Sn⃗. With the aid of both the orientation scheme and the routing algorithm, we show that diam⃗(Sn,k)≤⌊n+k2⌋+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 n−3k2 respectively for three disjoint intervals k>n2, n3<k≤n2 and k≤n3. For every value of n, k, our upper bound performs better than all known bounds in literature.
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