Algorithms for node disjoint paths in incomplete star networks

We give efficient algorithms for node disjoint path problems in incomplete star graphs which are defined in this paper to reduce the large gaps in the size of systems based on star graph topologies. Four disjoint path paradigms in incomplete star graphs are discussed: (1) disjoint paths between a pair of nodes s and t, (2) disjoint paths from a node s to a set T of nodes, (3) disjoint paths from a set S of nodes to a set T of nodes, and (4) disjoint paths between node pairs (s_i ,t_i). We give algorithms which can find the maximum number of disjoint paths for these paradigms in optimal time. For an n-dimensional incomplete star graph G_n,m, the length of the disjoint paths constructed by our algorithms is at most d(G_n,m)+c, where d(G_n,m) is the diameter of G and c is a small constant. This paper also shows that the k-wide-diameter d _n-2^W(G_m,n), k-Rabin-diameter d_n-2 ^R(G_m,n), k-set-diameter d_n-2^S(G_m,n), and k-pair-diameter d_n-2^P(G_m,n) of G_n,m are at d(G_n,m)+c