Near Optimal Parallel Algorithms for Dynamic DFS in Undirected Graphs

Abstract

Depth first search (DFS) tree is a fundamental data structure for solving various graph problems. The classical algorithm [54] 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. [5] presented a simple algorithm for updating the DFS tree of an undirected graph after an edge/vertex update in Õ( n ) 1 time. However, their algorithm is strictly sequential. We present an algorithm achieving similar bounds that can be easily adopted to the parallel environment. In the parallel environment, a DFS tree can be computed from scratch in expected Õ(1) time [2] on an EREW PRAM, whereas the best deterministic algorithm takes Õ(√ n ) time [2, 27] on a CRCW PRAM. Our algorithm can be used to develop optimal time (to poly log n factors) deterministic parallel algorithms for maintaining fully dynamic DFS and fault tolerant DFS of an undirected graph. (1) Parallel Fully Dynamic DFS : Given an arbitrary online sequence of vertex or edge updates, we can maintain a DFS tree of an undirected graph in Õ(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 any set of k updates (where k is constant) in the graph, a DFS tree of the updated graph can be computed in Õ(1) time using n processors on an EREW PRAM. For constant k , this is also work optimal (to poly log n factors). Moreover, our fully dynamic DFS algorithm provides, in a seamless manner, nearly optimal (to poly log 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.