A Space-Efficient Algorithm for the Dynamic DFS Problem in Undirected Graphs

Kengo Nakamura, Kunihiko Sadakane

Abstract: Depth-first search (DFS) is a well-known graph traversal algorithm and can be performed in $$O(n+m)$$ time for a graph with n vertices and m edges. We consider dynamic DFS problem, that is, to maintain a DFS tree of an undirected graph G when edges and vertices are gradually inserted into or deleted from G. We present an algorithm for this problem which takes worst case $$O(\sqrt{mn}\cdot \mathrm {polylog}(n))$$ time per update and requires only $$(3m+o(m))\log n+O(m)$$ bits of space. This is the first sublinear worst case update time algorithm for this problem which requires only $$O(m\log n)$$ bits. Moreover, the time complexity of this algorithm is close to, or under particular condition better than, the state-of-the-art algorithm of Chen et al. [5], which requires $$O(m\log ^2 n)$$ bits of space.