A new pair routing algorithm for transmitting messages in multiprocessor systems and networks-on-chip based on circulant networks of arbitrary dimension is proposed. It allows using all reserve shortest paths in the presence of destructive factors (deadlocks, livelocks, starvation, failures) at the nodes and channels of the communication network. A distinctive feature of the proposed algorithm is the absence of using the routing tables with fixed shortest paths when message is transmitted. It becomes possible to determine the set of the shortest paths for routing due to the relative addresses of destination nodes based on a parametric description of the network. Estimates of the number of reserve shortest paths are obtained, and an effective algorithm for using these paths to prevent dynamic topology changes and network congestions is proposed. To reduce the required memory in networks-on-chip with a circulant topology, we proposed a version of the routing algorithm for two-dimensional optimal circulants. We experimentally found the minimum number of reference nodes (nodes containing mapping tables) for them and estimates of memory for mapping tables, as well as the average path length for the routing algorithm using the reference nodes.