Parallel Minimum Cuts in Near-linear Work and Low Depth

Geissmann, Barbara; Gianinazzi, Lukas

doi:10.1145/3210377.3210393

Cited by 15 publications

(25 citation statements)

References 31 publications

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“…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.…”

Section: Introductionsupporting

confidence: 66%

“…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.…”

Section: Introductionmentioning

confidence: 99%

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

Anderson

Blelloch

2021

Proceedings of the 33rd ACM Symposium on Parallelism in Algorithms and Architectures

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We present a randomized 𝑂 (𝑚 log 2 𝑛) work, 𝑂 (polylog 𝑛) depth parallel algorithm for minimum cut. This algorithm matches the work bounds of a recent sequential algorithm by Gawrychowski, Mozes, and Weimann [ICALP'20], and improves on the previously best parallel algorithm by Geissmann and Gianinazzi [SPAA'18], which performs 𝑂 (𝑚 log 4 𝑛) work in 𝑂 (polylog 𝑛) depth.Our algorithm makes use of three components that might be of independent interest. Firstly, we design a parallel data structure that efficiently supports batched mixed queries and updates on trees. It generalizes and improves the work bounds of a previous data structure of Geissmann and Gianinazzi and is work efficient with respect to the best sequential algorithm. Secondly, we design a parallel algorithm for approximate minimum cut that improves on previous results by Karger and Motwani. We use this algorithm to give a work-efficient procedure to produce a tree packing, as in Karger's sequential algorithm for minimum cuts. Lastly, we design an efficient parallel algorithm for solving the minimum 2-respecting cut problem. CCS CONCEPTS• Theory of computation → Graph algorithms analysis; Parallel algorithms.

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“…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.…”

Section: Introductionsupporting

confidence: 66%

Section: Introductionmentioning

confidence: 99%

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

Anderson

Blelloch

2021

Proceedings of the 33rd ACM Symposium on Parallelism in Algorithms and Architectures

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“…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.…”

Section: Improvements In Pram Parallel Algorithmsmentioning

confidence: 89%

“…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.…”

Section: Improvements In Pram Parallel Algorithmsmentioning

confidence: 96%

“…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].…”

Section: Pram Modelmentioning

confidence: 99%

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Faster Algorithms for Edge Connectivity via Random 2-Out Contractions

Ghaffari

Nowicki

Thorup

2020

Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms

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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 probability 1 :• Two sequential algorithms with complexities O(m log n) and O(m+n log 3 n). These improve on a long line of developments including a celebrated O(m log 3 n) algorithm of Karger [STOC'96] and the state of the art O(m log 2 n(log log n) 2 ) algorithm of Henzinger et al. [SODA'17]. Moreover, our O(m + n log 3 n) algorithm is optimal when m = Ω(n log 3 n).• AnÕ(n 0.8 D 0.2 + n 0.9 ) 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 ofÕ(n 1−1/353 D 1/353 + n 1−1/706 ), 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.

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Minimum Cut in $$O(m\log ^2 n)$$ Time

Gawrychowski,

Mozes,

Weimann

2024

Theory Comput Syst

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We give a randomized algorithm that finds a minimum cut in an undirected weighted m-edge n-vertex graph G with high probability in $$O(m \log ^2 n)$$ O ( m log 2 n ) time. This is the first improvement to Karger’s celebrated $$O(m \log ^3 n)$$ O ( m log 3 n ) time algorithm from 1996. Our main technical contribution is a deterministic $$O(m \log n)$$ O ( m log n ) time algorithm that, given a spanning tree T of G, finds a minimum cut of G that 2-respects (cuts two edges of) T.

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Parallel Minimum Cuts in Near-linear Work and Low Depth

Cited by 15 publications

References 31 publications

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

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

Faster Algorithms for Edge Connectivity via Random 2-Out Contractions

Minimum Cut in $$O(m\log ^2 n)$$ Time

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Parallel Minimum Cuts in Near-linear Work and Low Depth

Cited by 15 publications

References 31 publications

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

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

Faster Algorithms for Edge Connectivity via Random 2-Out Contractions

Minimum Cut in $$O(m\log ^2 n)$$ Time

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Product

Resources

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

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