Faster Algorithms for Edge Connectivity via Random 2-Out Contractions

Ghaffari, Mohsen; Nowicki, Krzysztof; Thorup, Mikkel

doi:10.1137/1.9781611975994.77

Cited by 40 publications

(46 citation statements)

References 23 publications

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“…Based on the algorithm of Karzanov and Timofeev [110] and its parallel variant given by Naor and Vazirani [149], they show how to give the cactus representation of the graph in the same asymptotic time. Likewise, the recent algorithm of Ghaffari et al [74] finds all non-trivial minimum cuts (i.e. minimum cuts where each side contains at least two vertices) of a simple unweighted graph in O m log 2 n time.…”

Section: Finding All Global Minimum Cutsmentioning

confidence: 99%

Practical Minimum Cut Algorithms

Henzinger

Noe

Schulz

et al. 2018

ACM J. Exp. Algorithmics

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The minimum cut problem for an undirected edge-weighted graph asks us to divide its set of nodes into two blocks while minimizing the weight sum of the cut edges. Here, we introduce a linear-time algorithm to compute near-minimum cuts. Our algorithm is based on cluster contraction using label propagation and Padberg and Rinaldi's contraction heuristics [SIAM Review, 1991]. We give both sequential and shared-memory parallel implementations of our algorithm. Extensive experiments on both real-world and generated instances show that our algorithm finds the optimal cut on nearly all instances significantly faster than other state-of-the-art algorithms while our error rate is lower than that of other heuristic algorithms. In addition, our parallel algorithm shows good scalability.

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Section: Finding All Global Minimum Cutsmentioning

confidence: 99%

Practical Minimum Cut Algorithms

Henzinger

Noe

Schulz

et al. 2018

ACM J. Exp. Algorithmics

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show abstract

“…This point of view might provide an explanation for the inconsistency and ambiguity concerning the explicit restrictions on local computation in the low-space MPC model. Many of the prior work explicitly allow for an unlimited local computation, e.g., [1,2,5,6,22]. Other works only recommend having a polynomial time computation [21,23], and some explicitly restrict the local computation to be polynomial [11,19].…”

Section: Our Resultsmentioning

confidence: 99%

Improved Deterministic (Δ+1) Coloring in Low-Space MPC

Czumaj

Davies

Parter

2021

Proceedings of the 2021 ACM Symposium on Principles of Distributed Computing

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We present a deterministic (log log log )-round low-space Massively Parallel Computation (MPC) algorithm for the classical problem of (Δ + 1)-coloring on -vertex graphs. In this model, every machine has sublinear local space of size for any arbitrary constant ∈ (0, 1). Our algorithm works under the relaxed setting where each machine is allowed to perform exponential local computations, while respecting the space and bandwidth limitations. Our key technical contribution is a novel derandomization of the ingenious (Δ + 1)-coloring local algorithm by Chang-Li-Pettie (STOC 2018, SIAM J. Comput. 2020). The Chang-Li-Pettie algorithm runs in = (log log ) rounds, which sets the state-ofthe-art randomized round complexity for the problem in the local model. Our derandomization employs a combination of tools, notably pseudorandom generators (PRG) and bounded-independence hash functions.The achieved round complexity of (log log log ) rounds matches the bound of log(), which currently serves an upper bound barrier for all known randomized algorithms for locally-checkable problems in this model. Furthermore, no deterministic sublogarithmic low-space MPC algorithms for the (Δ + 1)-coloring problem have been known before. CCS CONCEPTS• Computing methodologies → Distributed algorithms; • Mathematics of computing → Graph algorithms; • Theory of computation → Pseudorandomness and derandomization.

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“…In a very recent paper [10], Ghaffari et al used 2-out sampling to get faster randomized algorithms for edge connectivity. One of their lemmas is that, with high probability, the number of components in a random 2-out subgraph is O(n/δ) where δ is the smallest degree.…”

Section: Previous Results In the K-out Modelmentioning

confidence: 99%

Random k-out Subgraph Leaves only O(n/k) Inter-Component Edges

Holm

King

Thorup

et al. 2019

2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)

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Each vertex of an arbitrary simple graph on n vertices chooses k random incident edges. What is the expected number of edges in the original graph that connect different connected components of the sampled subgraph? We prove that the answer is O(n/k), when k ≥ c log n, for some large enough c. We conjecture that the same holds for smaller values of k, possibly for any k ≥ 2. Such a result is best possible for any k ≥ 2. As an application, we use this sampling result to obtain a one-way communication protocol with private randomness for finding a spanning forest of a graph in which each vertex sends only O( √ n log n) bits to a referee.

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Faster Algorithms for Edge Connectivity via Random 2-Out Contractions

Cited by 40 publications

References 23 publications

Practical Minimum Cut Algorithms

Practical Minimum Cut Algorithms

Improved Deterministic (Δ+1) Coloring in Low-Space MPC

Random k-out Subgraph Leaves only O(n/k) Inter-Component Edges

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