Let G = (V, E) be an n-nodes non-negatively real-weighted undirected graph. In this paper we show how to enrich a single-source shortest-path tree (SPT) of G with a sparse set of auxiliary edges selected from E, in order to create a structure which tolerates effectively a path failure in the SPT. This consists of a simultaneous fault of a set F of at most f adjacent edges along a shortest path emanating from the source, and it is recognized as one of the most frequent disruption in an SPT. We show that, for any integer parameter k ≥ 1, it is possible to provide a very sparse (i.e., of size O(kn · f 1+1/k )) auxiliary structure that carefully approximates (i.e., within a stretch factor of (2k − 1)(2|F | + 1)) the true shortest paths from the source during the lifetime of the failure. Moreover, we show that our construction can be further refined to get a stretch factor of 3 and a size of O(n log n) for the special case f = 2, and that it can be converted into a very efficient approximate-distance sensitivity oracle, that allows to quickly (even in optimal time, if k = 1) reconstruct the shortest paths (w.r.t. our structure) from the source after a path failure, thus permitting to perform promptly the needed rerouting operations. Our structure compares favorably with previous known solutions, as we discuss in the paper, and moreover it is also very effective in practice, as we assess through a large set of experiments.The natural solution is that of modeling the network as a graph (nodes as vertices and links as edges) and building a (fast and compact) structure to be used to transmit the data. In particular, the most common approach of this kind is that of computing a shortest-path tree (SPT), rooted at the desired source node, of such graph.However, the SPT, as any tree-based topology, is prone to unpredictable events that might occur in practice, such as failures of nodes and/or links. Therefore, the use of SPTs might result in a high sensitivity to malfunctioning, which unavoidably causes the undesired effect of disconnecting sets of nodes from the source and thus the interruption of the broadcasting service.Therefore, a general approach to cope with this scenario is to make the SPT fault-tolerant against a given number of simultaneous component failures, by adding to it a set of suitably selected edges from the underlying graph, so that the resulting structure will remain connected w.r.t. the source. In other words, the selected edges can be used to build up alternative paths from the root, each one of them in replacement of a corresponding original shortest path which was affected by the failure. However, if these paths are constrained to be shortest, then it can be easily seen that for a non-negatively real weighted and undirected graph of n nodes and m edges, this may require as much as Θ(m) additional edges, also in the case in which m = Θ(n 2 ). In other words, the set-up costs of the strengthened network may become unaffordable.Thus, a reasonable compromise is that of building sparse and fault-tolerant s...

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A dynamic graph algorithm is called batch if it is able to update efficiently the solution of a given graph problem after multiple updates at a time (i.e., a batch) take place on the input graph. In this article, we study batch algorithms for maintaining a single-source shortest-path tree in graphs with positive real edge weights. In particular, we focus our attention on homogeneous batches, that is, either incremental (containing only edge insertion and weight decrease operations) or decremental (containing only edge deletion and weight increase operations) batches, which model realistic dynamic scenarios like transient vertex failures in communication networks and traffic congestion/decongestion phenomena in road networks.We propose two new algorithms to process either incremental or decremental batches, respectively, and a combination of these two algorithms that is able to process arbitrary sequences of incremental and decremental batches. All these algorithms are update sensitive; namely, they are efficient with respect to the number of vertices in the shortest-path tree that change their parents and/or their distances from the source as a consequence of a batch. This makes unfeasible an effective comparison on a theoretical basis of our new algorithms with the solutions known in the literature, which in turn are analyzed with respect to others and different parameters. For this reason, in order to evaluate the quality of our approach, we provide also an extensive experimental study including our new algorithms and the most efficient previous batch algorithms. Our experimental results complement previous studies and show that the various solutions can be consistently ranked on the basis of the type of homogeneous batch and of the underlying network. As a result, our work can be helpful in selecting a proper solution depending on the specific application scenario. . 2015. Dynamic maintenance of a shortest-path tree on homogeneous batches of updates: New algorithms and experiments.