One of the simplest problems on directed graphs is that of identifying the set of vertices reachable from a designated source vertex. This problem can be solved easily sequentially by performing a graph search, but efficient parallel algorithms have eluded researchers for decades. For sparse high-diameter graphs in particular, there is no previously known parallel algorithm with both nearly linear work and nontrivial parallelism. This talk presents the first such algorithm. Specifically, this talk presents a randomized parallel algorithm for digraph reachability with work O(m*polylog(n)) and span O(n^{2/3}*polylog(n)), with high probability, where n is the number of vertices and m is the number of arcs.

The main technical contribution is an efficient Monte Carlo algorithm that, through the addition of O(n*polylog(n)) shortcuts, reduces the diameter of the graph to O(n^{2/3}*polylog(n)) with high probability. While there are existing algorithms that achieve similar guarantees, they are not efficient; the sequential algorithms have running times much worse than linear. This talk presents a surprisingly simple sequential algorithm with running time O(m log^2(n)) that achieves the stated diameter reduction. Parallelizing that algorithm yields the main result, but doing so involves overcoming several additional challenges.

Jeremy Fineman is an Associate Professor of Computer Science at Georgetown University. His main research interest is in algorithm design and analysis, with a focus on data structures, parallel algorithms, memory-efficient algorithms, and scheduling. He received his Ph.D. from MIT in 2009 and did a postdoc at CMU before starting at Georgetown in 2011.