Communication-avoiding parallel minimum cuts and connected components

Abstract

We present novel scalable parallel algorithms for finding global minimum cuts and connected components, which are important and fundamental problems in graph processing. To take advantage of future massively parallel architectures, our algorithms are communication-avoiding : they reduce the costs of communication across the network and the cache hierarchy. The fundamental technique underlying our work is the randomized sparsification of a graph: removing a fraction of graph edges, deriving a solution for such a sparsified graph, and using the result to obtain a solution for the original input. We design and implement sparsification with O (1) synchronization steps. Our global minimum cut algorithm decreases communication costs and computation compared to the state-of-the-art, while our connected components algorithm incurs few cache misses and synchronization steps. We validate our approach by evaluating MPI implementations of the algorithms on a petascale supercomputer. We also provide an approximate variant of the minimum cut algorithm and show that it approximates the exact solutions well while using a fraction of cores in a fraction of time.

Practical Minimum Cut Algorithms

“…Thẽ O() notation ignores logarithmic factors. Gianinazzi et al [75] give a parallel implementation of the algorithm of Karger and Stein. Other than that, there are no parallel implementation of either algorithm known to us. More recently, the randomized contraction-based algorithm of Ghaffari et al [74] solves the minimum cut problem on unweighted graphs in O(m log n) or O m + n log 3 n .…”

“…While there are implementations of the algorithm of Karger and Stein [37,75] for the minimum cut problem, to the best of our knowledge there are no published implementations of either of the algorithms to find the cactus graph representing all minimum cuts.…”

Shared-Memory Exact Minimum Cuts

“…Gianinazzi et al [9] give a MPI implementation of the algorithm of Karger and Stein [17]. We performed preliminary experiments on small graphs which can be solved by NOI-HNSS, NOI-CGKLS and HO-CGKLS in less than 3 seconds.…”

“…Gianinazzi et al report a running time of 5 seconds for RMAT graphs with n = 16000 and an average degree of 4000, using 1536 cores. As NOI can find the minimum cut on RMAT graphs [19] of equal size in less than 2 seconds using a single core, we do not include the implementation in [9] in our experiments.…”

“…This is however unfeasible for large instances. There has been a MPI implementation of this algorithm by Gianinazzi et al [9]. However, there have been no parallel implementations of the algorithms of Hao et al [12] and Nagamochi et al [24,25], which outperformed other exact algorithms by orders of magnitude [7,13,15], both in real-world and generated networks.…”