Dynamic Maintenance of a Shortest-Path Tree on Homogeneous Batches of Updates

Abstract

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.