Algorithms for Fault‐Tolerant Routing in Circuit‐Switched Networks

Abstract

In this paper we consider the k edge‐disjoint paths problem (k‐EDP), a generalization of the well‐known edge‐disjoint paths problem. Given a graph $G=(V,E)$ and a set of terminal pairs (or requests) T, the problem is to find a maximum subset of the pairs in T for which it is possible to select paths such that each pair is connected by k edge‐disjoint paths and the paths for different pairs are mutually disjoint. To the best of our knowledge, no nontrivial result is known for this problem for $k>1$. To measure the performance of our algorithms we use the recently introduced flow number F of a graph. This parameter is known to fulfill $F=O(\Delta \alpha^{-1} \log n)$, where $\Delta$ is the maximum degree, $\alpha$ is the edge expansion of G, and n is the number of vertices in G. We show that a simple greedy online algorithm achieves a competitive ratio of $O(k^3 F)$ which naturally extends the best known bound of $O(F)$ for $k=1$ to higher k. To achieve this competitive ratio, we introduce a new method of converting a system of k disjoint paths into a system of k length‐bounded disjoint paths. We also show that any deterministic online algorithm has a competitive ratio of $\Omega(k F)$. In addition, we study the k disjoint flows problem (k‐DFP), which is a generalization of the previously studied unsplittable flow problem. The difference between the k‐DFP and the k‐EDP is that now we consider a graph with edge capacities and our requests are allowed to have arbitrary demands $d_i$. The aim is to find a subset of requests of maximum total demand for which it is possible to select flow paths such that all the capacity constraints are maintained and each selected request with demand $d_i$ is connected by k disjoint paths, each of flow value $d_i/k$. The k‐EDP and k‐DFP problems have important applications in fault‐tolerant (virtual) circuit switching, which plays a key role in optical networks.