Improved Algorithms for Maintaining DFS Tree in Undirected Graphs

Depth first search (DFS) tree is one of the most well-known data structures for designing efficient graph algorithms. Given an undirected graph G = (V,E) with n vertices and m edges, the textbook algorithm takes O(n + m) time to construct a DFS tree. In this paper, we study the problem of maintaining a DFS tree when the graph is undergoing updates. Formally, we show the following results. • Incremental DFS tree: Given any arbitrary online sequence of edge or vertex insertions, we can report a DFS tree in O(n) worst case time per operation, and O ( min{m logn + n log n, n} ) preprocessing time. • Fault tolerant DFS tree: There exists a data structure of size O(m logn), such that given any set F of failed vertices or edges, it can report a DFS tree of the graph G \ F in O(n|F| log n) time. • Fully dynamic DFS tree: Given any arbitrary online sequence of edge or vertex updates, we can report a DFS tree in O( √ nm log n) worst case time per operation. • Offline Incremental DFS tree: There exists a data structure with construction time O ( min{m logn + n log n, n} ) such that given any set I of inserted vertices or edges, a DFS tree of the graph G + I (denotes the graph obtained applying the insertions in I to G) can be reported in O(n + |I|) time. Our result for incremental DFS tree improves the previous O(n log n) worst case update time by Baswana et al. [1], and matches the trivial Ω(n) lower bound when it is required to explicitly output the DFS tree. Our other results also improve the corresponding state-of-the-art results by Baswana et al. [1] as well. Our work builds on the framework of the breakthrough work by Baswana et al. [1], together with a novel use of a tree-partition lemma by Duan and Zhang [8], and the celebrated fractional cascading technique by Chazelle and Guibas [5, 6].