On Non-Commutative Routing

The complexity of routing requirements leads to increasingly intricate routing metrics. Existing routing algebra theories have demonstrated that convergent and optimal routing algorithms can be designed only when path metrics satisfy certain properties such as monotonicity and isotonicity. Furthermore, some non-isotonic metrics can be converted into isotonic forms on partial orders through reduction. However, practical scenarios often involve non-commutative algebraic properties, which are overlooked by existing theories. For these problems, there lacks a unified framework to study their solvability, a systematic method for their reduction, and an efficient algorithm to compute optimal routes. In this work, we extend routing algebra to accommodate non-commutative routing problems, propose general reduction methods for them, and explore their solvability. In addition, we design a link state algorithm that converge fast on a reduced partial order. All these discussions are supported by concrete examples, theoretical proofs, and simulations on various network topologies.