Greedy and Local Ratio Algorithms in the MapReduce Model

Harvey, Nicholas J. A.; Liaw, Christopher; Liu, Paul

doi:10.1145/3210377.3210386

Cited by 20 publications

(17 citation statements)

References 40 publications

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“…Furthermore, Parter [42] designed an MPC algorithm that uses O(n) memory per machine and finds a (∆ + 1) coloring in O(log log ∆ · log * (n)) rounds 2 . Our Result 4 improves these results significantly: both the number of used colors as well as per machine memory compared to [26], and round-complexity (with at most polylog(n)-factor more per-machine memory) compared to [42].…”

Section: Our Contributionsmentioning

confidence: 90%

“…Starting with Luby's celebrated distributed/parallel algorithm for the maximal independent set problem [36], there have been numerous attempts in adapting these greedy algorithms to different models of computation including the models considered in this paper (see, e.g. [5,26,28,34,35,40]). Typically these adaptations require multiple passes/rounds of computation, and this is for the fundamental reason that most greedy algorithms are inherently sequential: they require accessing the input graph in an adaptive manner based on decisions made thus far, which, although limited, still results in requiring multiple passes/rounds over the input.…”

Section: Our Contributionsmentioning

confidence: 99%

See 1 more Smart Citation

Sublinear Algorithms for (Δ + 1) Vertex Coloring

Assadi¹,

Chen²,

Khanna³

2019

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

195

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Any graph with maximum degree ∆ admits a proper vertex coloring with ∆ + 1 colors that can be found via a simple sequential greedy algorithm in linear time and space. But can one find such a coloring via a sublinear algorithm?We answer this fundamental question in the affirmative for several canonical classes of sublinear algorithms including graph streaming, sublinear time, and massively parallel computation (MPC) algorithms. In particular, we design:

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Section: Our Contributionsmentioning

confidence: 90%

Section: Our Contributionsmentioning

confidence: 99%

Sublinear Algorithms for (Δ + 1) Vertex Coloring

Assadi¹,

Chen²,

Khanna³

2019

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

195

View full text Add to dashboard Cite

show abstract

“…The model appeared some years after the programming paradigm was popularized by Google [31]. It has been successfully employed in practice for massive-scale algorithms [15,19,20,41,44,48,68,69]. Algorithms in MapReduce use map and reduce functions, executed in sequence.…”

Section: Mapreduce Algorithmmentioning

confidence: 99%

Triangle Centrality

Burkhardt¹

2021

Preprint

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Triangle centrality is introduced for finding important vertices in a graph based on the concentration of triangles surrounding each vertex. An important vertex in triangle centrality is at the center of many triangles, and therefore it may be in many triangles or none at all.Given a simple, undirected graph G = (V, E), with n = |V | vertices and m = |E| edges, where N (v) is the neighborhood set of v, N △ (v) is the set of neighbors that are in triangles with v, and N + △ (v) is the closed set that includes v, then the triangle centrality for v, where △(v) and △(G) denote the respective triangle counts of v and G, is given byIt follows that the vector of triangle centralities for all vertices iswhere A is adjacency matrix of G, I is the identity matrix, 1 is the vectors of all ones, and the elementwise product A 2 • A gives T , and Ť is the binarized analog of T .We give optimal algorithms that compute triangle centrality in O(m 3/2 ) time and O(m + n) space. Using fast matrix multiplication it takes n ω+o(1) time where ω is the matrix product exponent.On a Concurrent Read Exclusive Write (CREW) Parallel Random Access Memory (PRAM) machine, we give a near work-optimal algorithm that takes O(log n) time using O(m √ m) CREW PRAM processors. In MapReduce, we show it takes four rounds using O(m √ m) communication bits, and is therefore optimal. We also give a deterministic algorithm to find the triangle neighborhood and triangle count of each vertex in O(m √ m) time and O(m + n) space. It can be also easily be computed in O(m δ(G)) expected time, where δ(G) is the average graph degeneracy and is related to the arboricity. We leave it as an open problem to deterministically compute triangle neighbors in O(m δ(G)) time and O(m + n) space.Our empirical results demonstrate that triangle centrality uniquely identified central vertices thirtypercent of the time in comparison to five other well-known centrality measures, while being asymptotically faster to compute on sparse graphs than all but the most trivial of these other measures.

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“…In [32], MapReduce algorithms for well-known problems are proposed, where they show the theoretical approximation ratio is two for the minimum vertex cover problem. On the other hand, our first algorithm may lead to solutions of worse quality for special cases than the MapReduce algorithm in the literature.…”

Section: Introductionmentioning

confidence: 99%

A MapReduce Algorithm for Minimum Vertex Cover Problems and Its Randomization

Nakamura

Kinjo

Yoshida

2020

cai

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MapReduce is a programming paradigm for large-scale distributed information processing. This paper proposes a MapReduce algorithm for the minimum vertex cover problem, which is known to be NP-hard. The MapReduce algorithm can efficiently obtain a minimal vertex cover in a small number of rounds. We show the effectiveness of the algorithm through experimental evaluation and comparison with exact and approximate algorithms which demonstrates a high quality in a small number of MapReduce rounds. We also confirm from experimentation that the algorithm has good scalability allowing high-quality solutions under restricted computation times due to increased graph size. Moreover, we extend our algorithm to randomized one to obtain a good expected approximate ratio.

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Greedy and Local Ratio Algorithms in the MapReduce Model

Cited by 20 publications

References 40 publications

Sublinear Algorithms for (Δ + 1) Vertex Coloring

Sublinear Algorithms for (Δ + 1) Vertex Coloring

Triangle Centrality

A MapReduce Algorithm for Minimum Vertex Cover Problems and Its Randomization

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