Semi-Dynamic Shortest Paths and Breadth-First Search in Digraphs

In this paper we study the problem of maintaining a single source shortest path tree or a breadth-rst 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 (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 rst 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 also show how maintain a breadth-rst search tree of a graph G during a sequence of edge insertions or during a sequence of edge deletions in total time O(nm), where m is the nal number of edges of G in the case of insertions or the initial number of edges of G in the case of deletions. These are the rst results concerning the dynamic maintenance of a breadth-rst search tree.