Disjoint Set Union with Randomized Linking

Goel, Ashish; Khanna, Sanjeev; Larkin, Daniel H.; Tarjan, Robert E.

doi:10.1137/1.9781611973402.75

Cited by 7 publications

(24 citation statements)

References 12 publications

(25 reference statements)

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“…We now derive better work bounds for the two algorithms that combine randomized linking with concurrent splitting. Our analysis extends that of Goel et al [GKLT14] to the concurrent setting.…”

Section: Analysis Of Splittingsupporting

confidence: 81%

“…Remark. Corollary 4.2.1 confirms the conjecture that the union forest in the sequential version of randomized linking is of logarithmic height with high probability, a problem left open in [GKLT14].…”

Section: Analysis Of Linkingsupporting

confidence: 66%

“…Both linking by size and linking by rank are deterministic; each requires maintaining a value (size or rank respectively) with each root. A more-recently studied method is randomized linking [GKLT14], which chooses a fixed total order of the elements uniformly at random, and links the smaller root with respect to the total order to the larger.…”

Section: Sequential Set Union Using Compressed Treesmentioning

confidence: 99%

“…Any of the three compaction methods can be combined with any of the three linking methods, giving a total of nine different algorithms. Each of these has a time bound of O(mα(n, m/n)) for m operations [TvL84,GKLT14], worst-case for linking by size or rank, expected for randomized linking. Here α is a functional inverse of Ackermann's function, defined as follows: Let A k (j) (Ackermann's function) be the function on non-negative integers defined recursively by A 0 (j) = j + 1, A k (0) = A k−1 (1) for k > 0, and A k (j) = A k−1 (A k (j − 1)) for k > 0 and j > 0.…”

Section: Sequential Set Union Using Compressed Treesmentioning

confidence: 99%

“…Crucial to our result is randomization, specifically the use of randomized linking [GKLT14]. We develop a wait-free concurrent algorithm with an expected total work bound of Θ m α n, m np + log np m + 1 and a high-probability bound of O(log n) steps for each set operation.…”

Section: Introductionmentioning

confidence: 99%

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A Randomized Concurrent Algorithm for Disjoint Set Union

Jayanti

Tarjan

2016

Proceedings of the 2016 ACM Symposium on Principles of Distributed Computing

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The disjoint set union problem is a basic problem in data structures with a wide variety of applications. We extend a known efficient sequential algorithm for this problem to obtain a simple and efficient concurrent wait-free algorithm running on an asynchronous parallel random access machine (APRAM). Crucial to our result is the use of randomization. Under a certain independence assumption, for a problem instance in which there are n elements, m operations, and p processes, our algorithm does Θ m α n, m np + log np m + 1 expected work, where the expectation is over the random choices made by the algorithm and α is a functional inverse of Ackermann's function. In addition, each operation takes O(log n) steps with high probability.Our algorithm is significantly simpler and more efficient than previous algorithms proposed by Anderson and Woll. Under our independence assumption, our algorithm achieves almost-linear speed-up for applications in which all or most of the processes can be kept busy. * This paper is a significantly revised version of a conference paper [JT16]

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Section: Analysis Of Splittingsupporting

confidence: 81%

Section: Analysis Of Linkingsupporting

confidence: 66%

Section: Sequential Set Union Using Compressed Treesmentioning

confidence: 99%

Section: Sequential Set Union Using Compressed Treesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A Randomized Concurrent Algorithm for Disjoint Set Union

Jayanti

Tarjan

2016

Proceedings of the 2016 ACM Symposium on Principles of Distributed Computing

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Concurrent disjoint set union

Jayanti

Tarjan

2021

Distrib. Comput.

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We develop and analyze concurrent algorithms for the disjoint set union (“union-find” ) problem in the shared memory, asynchronous multiprocessor model of computation, with CAS (compare and swap) or DCAS (double compare and swap) as the synchronization primitive. We give a deterministic bounded wait-free algorithm that uses DCAS and has a total work bound of $$O\biggl ( m \cdot \left( \log {\left( \frac{np}{m} + 1 \right) } + \alpha {\left( n, \frac{m}{np} \right) } \right) \biggr )$$ O ( m · log np m + 1 + α n , m np ) for a problem with n elements and m operations solved by p processes, where $$\alpha $$ α is a functional inverse of Ackermann’s function. We give two randomized algorithms that use only CAS and have the same work bound in expectation. The analysis of the second randomized algorithm is valid even if the scheduler is adversarial. Our DCAS and randomized algorithms take $$O(\log n)$$ O ( log n ) steps per operation, worst-case for the DCAS algorithm, high-probability for the randomized algorithms. Our work and step bounds grow only logarithmically with p, making our algorithms truly scalable. We prove that for a class of symmetric algorithms that includes ours, no better step or work bound is possible. Our work is theoretical, but Alistarh et al (In search of the fastest concurrent union-find algorithm, 2019), Dhulipala et al (A framework for static and incremental parallel graph connectivity algorithms, 2020) and Hong et al (Exploring the design space of static and incremental graph connectivity algorithms on gpus, 2020) have implemented some of our algorithms on CPUs and GPUs and experimented with them. On many realistic data sets, our algorithms run as fast or faster than all others.

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