Work-Efficient Parallel Union-Find with Applications to Incremental Graph Connectivity

Simsiri, Natcha; Tangwongsan, Kanat; Tirthapura, Srikanta

doi:10.1007/978-3-319-43659-3_41

Cited by 15 publications

(10 citation statements)

References 20 publications

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“…The worst-case work and depth of their algorithm is the same as that of a breadth-first search on the graph. Simsiri et al give a work-efficient, logarithmic-depth algorithm for batch incremental (no deletions) dynamic connectivity [45]. Kopelowitz et al have recently shown that the sparsified version of Frederickson's algorithm [17,15] can be parallelized nearly work-efficiently for a single update [27].…”

Section: Parallel Dynamic Connectivitymentioning

confidence: 99%

Batch-Parallel Euler Tour Trees

Tseng¹,

Dhulipala²,

Blelloch³

2019

2019 Proceedings of the Twenty-First Workshop on Algorithm Engineering and Experiments (ALENEX)

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The dynamic trees problem is to maintain a forest undergoing edge insertions and deletions while supporting queries for information such as connectivity. There are many existing data structures for this problem, but few of them are capable of exploiting parallelism in the batch setting, in which large batches of edges are inserted or deleted from the forest at once. In this paper, we demonstrate that the Euler tour tree, an existing sequential dynamic trees data structure, can be parallelized in the batch setting. For a batch of k updates over a forest of n vertices, our parallel Euler tour trees achieve O(k log(1 + n/k)) expected work and O(log n) depth with high probability. Our work bound is asymptotically optimal, and we improve on the depth bound achieved by Acar et al. for the batch-parallel dynamic trees problem [1].Our main building block for parallelizing Euler tour trees is a batch-parallel skip list data structure, which we believe may be of independent interest. Euler tour trees require a sequence data structure capable of joins and splits. Traditionally, balanced binary trees are used, but they are difficult to join or split in parallel when processing batches of updates. We show that skip lists, on the other hand, support batches of joins or splits of size k over n elements with O(k log(1 + n/k)) work in expectation and O(log n) depth with high probability. We also achieve the same efficiency bounds for augmented skip lists, which allows us to augment our Euler tour trees to support subtree queries.Our data structures achieve between 67-96× self-relative speedup on 72 cores with hyperthreading on large batch sizes.

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Section: Parallel Dynamic Connectivitymentioning

confidence: 99%

Batch-Parallel Euler Tour Trees

Tseng¹,

Dhulipala²,

Blelloch³

2019

2019 Proceedings of the Twenty-First Workshop on Algorithm Engineering and Experiments (ALENEX)

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“…An algorithm for a dynamic graph is called incremental if it can efficiently handle insertion of edges, decremental if it can handle deletion of edges, and fully dynamic if it can handle both insertions and deletions. For example, a parallel algorithm due to Simsiri et al [36] is an incremental algorithm for graph connectivity, an algorithm due to Thorup [40] is a decremental algorithm, and one due to Wulff-Nilsen [43] is a fully dynamic algorithm. Our algorithms can be viewed as a change-sensitive incremental algorithm for maximal cliques, and a change-sensitive decremental algorithm for maximal cliques.…”

Section: Preliminariesmentioning

confidence: 99%

Incremental maintenance of maximal cliques in a dynamic graph

2019

Self Cite

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We consider the maintenance of the set of all maximal cliques in a dynamic graph that is changing through the addition or deletion of edges. We present nearly tight bounds on the magnitude of change in the set of maximal cliques, as well as the first change-sensitive algorithms for clique maintenance, whose runtime is proportional to the magnitude of the change in the set of maximal cliques. We present experimental results showing these algorithms are efficient in practice, and are faster than prior work by two to three orders of magnitude.

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“…Motivated by such dynamic applications, there has been recent interest in developing parallel batch-dynamic algorithms [2,57,1,62]. In the batch-dynamic framework, instead of applying one update or query at a time a whole batch is applied.…”

Section: Introductionmentioning

confidence: 99%

“…Understanding the connectivity structure of graphs is of significant practical interest, for example, due to its use as a primitive for clustering the vertices of a graph [52]. Existing batch-dynamic connectivity algorithms are designed for restricted settings, e.g., in the incremental setting when all updates are edge insertions [57], or when the underlying graph is a forest [50,1,62]. In practice, since most real-world graphs change over time, many implementations of parallel batch-dynamic connectivity algorithms have been implemented [40,32,65,33,53,64].…”

Section: Introductionmentioning

confidence: 99%

Parallel Batch-Dynamic Graph Connectivity

Acar,

Anderson,

Blelloch

et al. 2019

Preprint

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With the rapid growth of graph datasets over the past decade, a new kind of dynamic algorithm, supporting the ability to ingest batches of updates and exploit parallelism is needed in order to efficiently process large streams of updates. In this paper, we study batch and parallel algorithms for the dynamic connectivity problem, a fundamental problem that has received considerable attention in sequential setting. Perhaps the best known sequential algorithm is the elegant level-set algorithm of Holm, de Lichtenberg and Thorup (HDT), which achieves O(lg 2 n) amortized time per edge insertion or deletion, and O(lg n) time per query.In this paper, we design a parallel batch-dynamic connectivity algorithm that is work-efficient with respect to the HDT algorithm for small batch sizes, and is asymptotically faster when the average batch size is sufficiently large. Given a sequence of batched updates, where ∆ is the average batch size of all deletions, our algorithm achieves O(lg n lg(1 + n/∆)) expected amortized work per edge insertion and deletion and O(lg 3 n) depth w.h.p. Our algorithm answers a batch of k connectivity queries in O(k lg(1 + n/k)) expected work and O(lg n) depth w.h.p. To the best of our knowledge, our algorithm is the first parallel batch-dynamic algorithm for connectivity.

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Work-Efficient Parallel Union-Find with Applications to Incremental Graph Connectivity

Cited by 15 publications

References 20 publications

Batch-Parallel Euler Tour Trees

Batch-Parallel Euler Tour Trees

Incremental maintenance of maximal cliques in a dynamic graph

Parallel Batch-Dynamic Graph Connectivity

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