) Abstract We consider the problem of maintaining the distances and the shortest paths from a single source in either a directed or an undirected graph with positive real edge weights, handling insertions, deletions and cost updates of edges. We propose fully dynamic algorithms with optimal space requirements and query time. The cost of update operations depends on the class of the considered graph and on the number of vertices that, due to an edge modification, either change their distance from the source or change their parent in the shortest path tree. In the case of graphs with bounded genus (including planar graphs), bounded degree graphs, bounded treewidth graphs and fi-near-planar graphs with bounded fi, the update procedures require O(log n) amortized time per vertex update, while for general graphs with n vertices and m edges they require O( p m log n) amortized time per vertex update. The solution is based on a dynamization of Dijkstra's algorithm [6] and requires simple ...

In this paper we study the problem of maintaining a single source shortest path tree or a breadth-first search tree for a directed graph, in either an incremental or decremental setting. We maintain a single source shortest path tree of a directed graph G with unit edge weights during a sequence of edge deletions in total time O(mn), thus obtaining a O(n) amortized time for each deletion if the sequence has length\Omega\Gamma m), where n is the number of vertices of G and m is the initial number of edges of G. To the best of our knowledge, this is the first known decremental algorithm for directed graphs with unit edge weights that is asymptotically faster than recomputing the single source shortest path tree from scratch after each deletion, which can be accomplished in O(m 2 ) total time. This result is extended to handle the case of integer edge weights in [1; C], allowing to maintain a single source shortest path tree during a sequence of edge deletions in total time O(Cmn). We...