Link state routing protocols such as OSPF or IS-IS currently use only best paths to forward IP packets throughout a domain. The optimality of sub-paths ensures consistency of hop by hop forwarding although paths, calculated using Dijkstra algorithm, are recursively composed. According to the link metric, the diversity of existing paths can be underestimated using only best paths. Hence, it reduces potential benefits of multipath applications such as load balancing and fast rerouting. In this paper, we propose a low time complexity multipath computation algorithm able to calculate at least two paths with a different first hop between all pairs of nodes in the network if such next hops exist. Using real and generated topologies, we evaluate and compare the complexity of our proposition with several techniques. Simulation results suggest that the path diversity achieved with our proposition is approximatively the same that the one obtained using consecutive Dijsktra computations, but with a lower time complexity.
Routing is one of the key components of the Internet. Despite the potential benefits of multipath routing (e.g. [5] or [6]), most backbone networks still use unipath routing such as OSPF or IS-IS or their ECMP feature (Equal Cost MultiPath). With these routing protocols, the forwarding only changes upon topology variations and not upon traffic variations. Dynamic multipath routing (e.g. [16], [15], [8] or [3]) is able to provide several services such as load balancing, to reduce delays and improve throughput, and fast rerouting schemes in case of failures. The reliability of an IP network against failures and congestions depends on the reaction time necessary for the convergence of the underlying routing protocol. Proactive multiple paths calculation allows to accelerate this reaction time: pre-computed alternate paths can be directly used as backup paths without waiting for the routing protocol convergence. This proactive mechanism can improve the network response in case of troubles where such backup paths exist. To provide these functionalities, the set of forwarding alternatives has to be large enough to achieve a good path diversity. However, current routers only support ECMP. This feature corresponds to a simple variant of Dijkstra where equal cost paths are inherited along the shortest path tree (SPT). The optimality condition of sub-paths computed with ECMP restricts the number of loopfree paths and so reduces potential advantages of multipath routing. In order to use multiple unequal cost paths between a pair of ingress and egress routers, there are two forwarding possibilities. On the one hand, source multipath forwarding schemes can use MPLS with a path signaling protocol (such as RSVP-TE [4]) to establish any desired paths. With this kind of approach, either the deployment is generalized in the whole network and does not scale very well (proportional to the square of the number of routers), either the reaction time can be as long as the notification delay on the return path. On the other hand, multipath routing protocols with hop by hop forwarding needs to validate a set of next hops such that the recursive composition between neighbor routers does not create forwarding loops (see [14], [15] and [17]). The first limitation is the complexity in time, space and the number of messages exchanged to compute and validate loopfree paths. In this paper, we propose a simple hop by hop scheme that does not require a signaling protocol to validate loopfree paths. If the validation procedure, whose goal is to verify the absence of loops, is local (without exchanging any message) and does not involve all routers, then the deployment can be incremental. Our approach is equivalent to ECMP in terms of time, space and message exchange complexity but allows to compute a greater diversity of forwarding alternatives. In this paper, we propose the following contributions:
-a new graph decomposition analysis.
-two variants of the Dijkstra algorithm: Dijkstra-Transverse (DT) and multi-Dijkstra-Transverse (mDT). -a proof that they compute at least two distinct next hops from the calculating node towards each node of the graph if such next hops exist. -an evaluation of the efficiency and the complexity of our proposition compared to existing techniques. This paper is organized as follows. Section II summarizes basic multipath routing notions and related work. Section III introduces our algorithms and their properties. Section IV presents our simulation results to underline the relevance and the low time complexity of our proposition.
Table I lists the graph definitions used in the paper. Notations are related to the multipath hop by hop forwarding context: computed paths are loopfree and first hop distinct. We order paths according to an additive metric C, and we focus on the best paths having distinct first hops. To distinguish equal cost paths, we consider the lexicographical order of first hops. For simplicity reasons we do not consider the multigraph issue: a first hop is equivalent to a successor node, the next hop. The valuation w denotes the weight of each directed link With hop by hop link state multipath routing using multiple unequal cost paths, two phases may be necessary to ensure loopfree routing: a path computation algorithm and a validation process. We do not consider validation processes using a signaling protocol (such as it can be done with distance vector routing messages, see [15] for example).
With unipath or ECMP routing, the sub-path optimality condition guarantees the correctness of next hop composition. To increase the number of valid alternatives, the simplest rule to select a next hop v on a router s (such that v ∈ succ(s)) is the downstream criteria which can be expressed as follows:
This rule is referenced in the IS-IS standard ISO 8473, is used in OSPF-OMP [14] and is denoted LFI in [15] (with the particularity of avoiding routing loops even in transient periods of topology changes). T
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