Semi-dynamic breadth-first search in digraphs

In this paper we propose dynamic algorithms for maintaining a breadth-first search tree from a given source vertex of a directed graph G in either an incremental or a decremental setting. During a sequence of q edge insertions or a sequence of q edge deletions the total time required is O(m·min{q,n}), where n is the number of vertices of G, and m is the final number of edges of G in the case of insertions or the initial number of edges of G in the case of deletions. This gives O(n) amortized time for each operation if the sequence has length Ω(m). Our algorithms require O(n+m) space. These are the first results in the literature concerning the dynamic maintenance of a breadth-first search tree for directed graphs. As a straightforward application of such algorithms we can maintain a shortest path tree for a directed graph in the case of unit edge weights within the same time bounds. In this case distance queries can be answered in constant time, while shortest path queries can be answered in time linear in the length of the retrieved path.