Absorbing Lexicographic Products in Metarouting Eric Parsonage, Hung X. Nguyen, Matthew Roughan University of Adelaide, Australia {eric.parsonage,hung.nguyen,matthew.roughan}@adelaide.edu.au Abstract—Modern treatments of routing protocols use alge- braic techniques to derive the protocol’s properties, permitting a semantic richness more flexible than simple numerical “shortest paths”. Many such routing protocols make preference decisions based on multiple criteria. This fits well with an algebraic formulation with each strata in the decision process modeled as an algebraic structure, that are combined to create the full routing protocol. Routing protocols constructed in this manner are the focus of this paper. To implement such a routing protocol we must understand the properties needed on each of the algebraic formulations representing a strata. In this paper we examine a stratified routing algebra based on a recently suggested absorbing product and provide the necessary and sufficient conditions required by each of the operands to guarantee that such a routing language ensures globally optimal paths will be found. I. I NTRODUCTION Convergence of a routing protocol to an optimum is crucial for stability and efficiency. Proofs of convergence have their roots in the work of Carr´ e [1] who in 1971 showed that several different kinds of pathfinding problems could be solved by variants of classical methods in linear algebra using matrices over semirings. Since then it has been possible to view a routing protocol as comprising two distinct components: routing protocol = routing language + algorithm, where a protocol’s routing language is used to configure a network and the protocol’s algorithm computes routing solutions using the network’s configuration. This abstraction is very useful as we know that if a routing language has certain properties then the associated protocol will converge. However, more recently, it was discovered that the Border Gateway Protocol (BGP), that is used universally for inter- domain routing in the Internet, does not always converge [2], [3], despite the fact that the properties needed for such convergence were known. It might reasonably be concluded that although the theory was available, it was too hard to apply to the complex routing protocols needed to implement the flexible policies of modern inter-domain routing. Carr´ e’s formulation has been extended in various ways in the intervening years [4]–[6]. Griffin and Sobrinho [6], in particular, extended the standard formulation to allow for combinations of simple building blocks into larger, more complex routing protocols. The process, that is referred to as metarouting, allows one to derive the properties of the combined algebra from its components without the tedious or difficult process of deriving these properties ab initio. The most prominent example of such a construction is the lexicographic product, which has been used to model BGP-like protocols where the choice of best routes descends through a series of metrics, using those after the first only as a tie-break if the preceding metrics are equal. In this paper we examine a structure based on an absorbing lexicographic product proposed by Gurney [7] and further elaborated upon by Griffin [8], specifically designed to over- come practical difficulties that arise when implementing a routing language using a simple lexicographic product. Several examples of the use of the absorbing lexicographic product to model subsets of BGP policy can be found in [8]. One of these considers a simplified model of BGP with only two components. The first component is used to model the business relationships between Autonomous Systems (ASs) (call it local preference to match the conventions of BGP) and the second models AS path length. The path selection decision process is based on policy first (via the local prefer- ence metric) with tiebreaks then decided by the shortest AS path. The classification of business relationships between ASs into customer-provider, peer-peer, and downstream leads to import/export rules such as: (1) an AS does not export to a provider or peer routes that it has learned from other providers and other peers; and (2) an AS can export to its customers any route it knows of. The model implements these policies by assigning functions to links in the network that represent the combined import and export policies of the routers at each end of a link. Thus as paths are propagated these functions combine to cause invalid paths (paths that are filtered out by the import/export rules) to be marked with an absorbing zero thus excluding them from further consideration. A less conventional example is the calculation of shared- risk groups across paths using Martelli’s algebra [9]. Here the size of the state information that needs be conveyed increases as path lengths increase. Thus, eliminating routes of no interest from consideration would reduce the computational and communications overheads of the protocol. This could be achieved by using an absorbing lexicographic product where the first component is used to absorb paths of no interest and the second component is an instance of Martelli’s algebra. Unfortunately the increased flexibility of the absorbing product comes at the cost of disallowing the use of several powerful theorems for the proof of its properties. Sufficient conditions for a routing algebra formed from the absorbing product always to converge to a globally optimal solution were previously known [8]. Our main result is the provision of necessary and sufficient conditions. Discovery of necessary conditions is crucial if we want to create these routing algebras with all the possible flexi- bility that is allowed given the requirement for convergence.