Dynamic DFS in Undirected Graphs: breaking the O( m ) barrier
Surender Baswana, Shreejit Ray Chaudhury, Keerti Choudhary, Shahbaz Khan · 2015
Given an undirected graph G = (V, E) on n vertices and m edges, we address the problem of maintaining a DFS tree when the graph is undergoing updates (insertion and deletion of vertices or edges). We present the following results for this problem. 1. Fault tolerant DFS tree: There exists a data structure of size Õ(m)1 such that given any set ℱ of failed vertices or edges, a DFS tree of the graph G\ℱ can be reported in Õ(n|ℱ|) time. 2. Fully dynamic DFS tree: There exists a fully dynamic algorithm for maintaining a DFS tree that takes worst case time per update for any arbitrary online sequence of updates. 3. Incremental DFS tree: Given any arbitrary online sequence of edge insertions, we can maintain a DFS tree in Õ(n) worst case time per edge insertion. These are the first o(m) worst case time results for maintaining a DFS tree in a dynamic environment. Moreover, our fully dynamic algorithm provides, in a seamless manner, the first deterministic algorithm with O(1) query time and o(m) worst case update time for the dynamic subgraph connectivity, biconnectivity, and 2-edge connectivity.