Expediting Parallel Graph Connectivity Algorithms

Finding whether a graph is k-connected, and the identification of its k-connected components is a fundamental problem in graph theory. For this reason, there have been several algorithms for this problem in both the sequential and parallel settings. Several recent sequential and parallel algorithms for k-connectivity rely on one or more breadth-first traversals of the input graph. While BFS can be made very efficient in a sequential setting, the same cannot be said in the case of parallel environments. A major factor in this difficulty is due to the inherent requirement to use a shared queue, balance work among multiple threads in every round, synchronization, and the like. Optimizing the execution of BFS on many current parallel architectures is therefore quite challenging. For this reason, it can be noticed that the time spent by the current parallel graph connectivity algorithms on BFS operations is usually a significant portion of their overall runtime. In this paper, we study how one can, in the context of algorithms for graph connectivity, mitigate the practical inefficiency of relying on BFS operations in parallel. Our technique suggests that such algorithms may not require a BFS of the input graph but actually can work with a sparse spanning subgraph of the input graph. The incorrectness introduced by not using a BFS spanning tree can then be offset by further post-processing steps on suitably defined small auxiliary graphs. Our experiments on finding the 2, and 3-connectivity of graphs on Nvidia K40c GPUs improve the state-of-the-art on the corresponding problems by a factor 2.2x, and 2.1x respectively.