TY - GEN

T1 - Faster algorithms for edge connectivity via random 2-out contractions

AU - Ghaffari, Mohsen

AU - Nowicki, Krzysztof

AU - Thorup, Mikkel

PY - 2020

Y1 - 2020

N2 - We provide a simple new randomized contraction approach to the global minimum cut problem for simple undirected graphs. The contractions exploit 2-out edge sampling from each vertex rather than the standard uniform edge sampling. We demonstrate the power of our new approach by obtaining better algorithms for sequential, distributed, and parallel models of computation. Our end results include the following randomized algorithms for computing edge connectivity, with high probability1: • Two sequential algorithms with complexities O(m log n) and O(m + n log3 n). These improve on a long line of developments including a celebrated O(m log3 n) algorithm of Karger [STOC'96] and the state of the art O(m log2 n(log log n)2) algorithm of Henzinger et al. [SODA'17]. Moreover, our O(m + n log3 n) algorithm is optimal when m = Ω(n log3 n). • An Õ(n0.8D0.2 + n0.9) round distributed algorithm, where D denotes the graph diameter. This improves substantially on a recent breakthrough of Daga et al.[STOC'19], which achieved a round complexity of Õ(n1−1/353D1/353 + n1−1/706), hence providing the first sublinear distributed algorithm for exactly computing the edge connectivity. • The first O(1) round algorithm for the massively parallel computation setting with linear memory per machine.

AB - We provide a simple new randomized contraction approach to the global minimum cut problem for simple undirected graphs. The contractions exploit 2-out edge sampling from each vertex rather than the standard uniform edge sampling. We demonstrate the power of our new approach by obtaining better algorithms for sequential, distributed, and parallel models of computation. Our end results include the following randomized algorithms for computing edge connectivity, with high probability1: • Two sequential algorithms with complexities O(m log n) and O(m + n log3 n). These improve on a long line of developments including a celebrated O(m log3 n) algorithm of Karger [STOC'96] and the state of the art O(m log2 n(log log n)2) algorithm of Henzinger et al. [SODA'17]. Moreover, our O(m + n log3 n) algorithm is optimal when m = Ω(n log3 n). • An Õ(n0.8D0.2 + n0.9) round distributed algorithm, where D denotes the graph diameter. This improves substantially on a recent breakthrough of Daga et al.[STOC'19], which achieved a round complexity of Õ(n1−1/353D1/353 + n1−1/706), hence providing the first sublinear distributed algorithm for exactly computing the edge connectivity. • The first O(1) round algorithm for the massively parallel computation setting with linear memory per machine.

UR - http://www.scopus.com/inward/record.url?scp=85084036820&partnerID=8YFLogxK

U2 - 10.1137/1.9781611975994.77

DO - 10.1137/1.9781611975994.77

M3 - Article in proceedings

AN - SCOPUS:85084036820

SP - 1260

EP - 1279

BT - 31st Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2020

A2 - Chawla, Shuchi

PB - Association for Computing Machinery

T2 - 31st Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2020

Y2 - 5 January 2020 through 8 January 2020

ER -