Near Optimal Parallel Algorithms for Dynamic DFS in Undirected Graphs

Depth first search (DFS) tree is a fundamental data structure for solving various graph problems. The classical algorithm [SIAMCOMP74] for building a DFS tree requires O(m+n) time for a given undirected graph G having n vertices and m edges. Recently, Baswana et al. [SODA16] presented a simple algorithm for updating the DFS tree of an undirected graph after an edge/vertex update in O (n) time. However, their algorithm is strictly sequential. We present an algorithm achieving similar bounds, that can be adopted easily to the parallel environment. In the parallel environment, a DFS tree can be computed from scratch using O(m) processors in expected O (1) time [SICOMP90] on an EREW PRAM, whereas the best deterministic algorithm takes O (√n) time [SIAMCOMP90,JAL93] on a CRCW PRAM. Our algorithm can be used to develop optimal (upto polylog n factors) deterministic algorithms for maintaining fully dynamic DFS and fault tolerant DFS, of an undirected graph. 1- Parallel Fully Dynamic DFS - Given any arbitrary online sequence of vertex or edge updates, we can maintain a DFS tree of an undirected graph in O (1) time per update using m processors on an EREW PRAM. 2- Parallel Fault tolerant DFS - An undirected graph can be preprocessed to build a data structure of size O(m) such that for a set of k updates (where k is constant) in the graph, a DFS tree of the updated graph can be computed in O (1) time using n processors on an EREW PRAM. For constant k, this is also work optimal (upto polylog n factors) Moreover, our fully dynamic DFS algorithm provides, in a seamless manner, nearly optimal (upto polylog n factors) algorithms for maintaining a DFS tree in the semi-streaming environment and a restricted distributed model. These are the first parallel, semi-streaming and distributed algorithms for maintaining a DFS tree in the dynamic setting.