Near Optimal Parallel Algorithms for Dynamic DFS in Undirected Graphs

Depth first search (DFS) tree is a fundamental data structure for solving graph problems. The classical algorithm [SiComp74] for building a DFS tree requires $O(m+n)$ time for a given graph $G$ having $n$ vertices and $m$ edges. Recently, Baswana et al. [SODA16] presented a simple algorithm for updating DFS tree of an undirected graph after an edge/vertex update in $\tilde{O}(n)$ 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 model, a DFS tree can be computed from scratch using $m$ processors in expected $\tilde{O}(1)$ time [SiComp90] on an EREW PRAM, whereas the best deterministic algorithm takes $\tilde{O}(\sqrt{n})$ time [SiComp90,JAlg93] 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 an arbitrary online sequence of vertex/edge updates, we can maintain a DFS tree of an undirected graph in $\tilde{O}(1)$ 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, the updated DFS tree can be computed in $\tilde{O}(1)$ time using $n$ processors on an EREW PRAM. Moreover, our fully dynamic DFS algorithm provides, in a seamless manner, nearly optimal (upto polylog n factors) algorithms for maintaining a DFS tree in semi-streaming model and a restricted distributed model. These are the first parallel, semi-streaming and distributed algorithms for maintaining a DFS tree in the dynamic setting.Comment: Accepted to appear in SPAA'17, 32 Pages, 5 Figure

Parallel Real-Time Computation: Sometimes Quantity Means Quality

The primary purpose of parallel computation is the fast execution of computational tasks that require an inordinate amount of time to perform sequentially. As a consequence, interest in parallel computation to date has naturally focused on the speedup provided by parallel algorithms over their sequential counterparts. The thesis of this paper is that a second equally important motivation for using parallel computers exists. Specifically, the following question is posed: Can parallel computers, thanks to their multiple processors, do more than simply speed up the solution to a problem? We show that within the paradigm of real-time computation, some classes of problems have the property that a solution to a problem in the class, when computed in parallel, is far superior in quality than the best one obtained on a sequential computer. What constitutes a better solution depends on the problem under consideration. Thus, `better' means `closer to optimal' for optimization problems, `more secure' for cryptographic problems, and `more accurate' for numerical problems. Examples from these classes are presented. In each case, the solution obtained in parallel is significantly, provably, and consistently better than a sequential one. It is important to note that the purpose of this paper is not to demonstrate merely that a parallel computer can obtain a better solution to a computational problem than one derived sequentially. The latter is an interesting (and often surprising) observation in its own right, but we wish to go further. It is shown here that the improvement in quality can be arbitrarily high (and certainly superlinear in the number of processors used by the parallel computer). This result is akin to superlinear speedup --- a phenomenon itself originally thought to be impossible