Parallel Minimum Cuts in Near-linear Work and Low Depth

Abstract

We present the first near-linear work and poly-logritharithmic depth algorithm for computing a minimum cut in a graph, while previous parallel algorithms with poly-logarithmic depth required at least quadratic work in the number of vertices.In a graph with n vertices and m edges, our algorithm computes the correct result with high probability in O(m log 4 n) work and O(log 3 n) depth. This result is obtained by parallelizing a data structure that aggregates weights along paths in a tree and by exploiting the connection between minimum cuts and approximate maximum packings of spanning trees.In addition, our algorithm improves upon bounds on the number of cache misses incurred to compute a minimum cut.

Parallel Minimum Cuts in O ( m log 2 n ) Work and Low Depth

“…This result generalizes and improves on Geissmann and Gianinazzi [14] who give an algorithm for evaluating a batch of 𝑘 pathweight updates and queries in Ω(𝑘 log 2 𝑛) work.…”

“…A new wave of interest in the problem has recently pushed these frontiers. Geissmann and Gianinazzi [14] design a parallel algorithm for minimum 2-respecting cuts that performs 𝑂 (𝑚 log 3 𝑛) work in 𝑂 (log 2 𝑛) depth. Their algorithm is based on parallelizing Karger's algorithm by replacing a sequential data structure for the so-called minimum path problem, based on dynamic trees, with a data structure that can evaluate a batch of updates and queries in parallel.…”

Faster Algorithms for Edge Connectivity via Random 2-Out Contractions

“…We get an algorithm with O(log 3 n) depth and O(m log n + n log 4 n) work. This improves on the work complexity of the state of the art algorithm of Geissman and Gianinazzi [GG18], which has O(log 3 n) depth and O(m log 4 n) work.…”

“…For the standard PRAM model of parallel algorithms (Concurrent Write Exclusive Read), our algorithm improves the total work while achieving the same depth complexity as the state of the art [GG18]. We get an algorithm with O(log 3 n) depth and O(m log n + n log 4 n) work.…”

“…The implementation is rather straightforward and boils down to solving several Connected Components problems in parallel, which we do with O(m log n) work and O(log n) depth [PR99]. Then, we use an approach proposed in Section 2.4 to compute an O(n) edge graph that preserves all non singleton (2 − ε)-minimum cuts, and find a minimum cut of computed graph with state of the art algorithm for multigraphs in O(n log 4 n) work and O(log 3 n) depth [GG18].…”