Optimal routing algorithm for double loop networks with restricted steps

Double loop networks (DLN, for simplicity) G (n; h), where n is the number of its nodes and 1 and h are its steps, are important interconnection networks. Some optimal routing algorithms for G (n; h) are proposed, their time complexity is at least O (√n). DLN with restricted steps are considered. It is proved that if the step h satisfies a certain inequality, then the formula for its diameter is given, and the constant time optimal routing is obtained, precisely, at most 6 arithmetic operations or comparisons are needed to determine a shortest path from the origin 0 to any destination node. The results can be applied to 66 infinite families of tight optimal DLN and 30 infinite families of nearly tight optimal DLN, such that for every n, 5&le n&le300, except n=99 and 187, there is G (n; h) contained in above some family.