On the connectivity of randomm-orientable graphs and digraphs

Fenner, Trevor; Frieze, Alan

doi:10.1007/bf02579431

Cited by 88 publications

(76 citation statements)

References 5 publications

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“…Some futher results are given in [L / uczak, 1988]. An entirely different model of random digraphs is studied in [Fenner & Frieze, 1988]. The paper [Schnorr, 1985] presents an algorithm for constructing the transitive closure of a random digraph.…”

Section: Introductionmentioning

confidence: 99%

The transitive closure of a random digraph

Karp

1990

Random Struct Algorithms

199

172

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In a random n-vertex digraph, each arc is present with probability p, independently of the presence or absence of other arcs. We investigate the structure of the strong components of a random digraph and present an algorithm for the construction of the transitive closure of a random digraph. We show that, when n is large and np is equal to a constant c greater than 1, it is very likely that all but one of the strong components are very small, and that the unique large strong component contains about Θ 2 n vertices, where Θ is the unique root in [0,1] of the equation 1 − x − e −cx = 0. Nearly all the vertices outside the large strong component lie in strong components of size 1. Provided that the expected degree of a vertex is bounded away from 1, our transitive closure algorithm runs in expected time O (n). For all choices of n and p, the expected execution time of the algorithm is O (w (n) (n log n) 4/3 ), where w (n) is an arbitrary nondecreasing unbounded function. To circumvent the fact that the size of the transitive closure may be Ω(n 2 ) the algorithm presents the transitive closure in the compact form (A × B) ∪ C, where A and B are sets of vertices, and C is a set of arcs.

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Section: Introductionmentioning

confidence: 99%

The transitive closure of a random digraph

Karp

1990

Random Struct Algorithms

199

172

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“…A random m-orientable graph, also called a random k-out graph, is obtained by allowing each node to choose a set of m neighbors with equal probability of 1/ n−1 m , independently of each other [23]. A pair of nodes i and j have an undirected edge between them if either node chooses the other node as a neighbor.…”

Section: Random M-orientable Graphsmentioning

confidence: 99%

“…For n ≥ñ( ), let κ(n) := k(n)/ log(n) ∈ [κ − , κ + ]. Define 23) where ζ 1 (n) := κ(n)e − √ 2/κ(n) > 1 because κ(n) ≥ κ − > β , and…”

Section: Appendix E Proof Of Lemma A3mentioning

confidence: 99%

A New Random Graph Model with Self-Optimizing Nodes: Connectivity and Diameter

Kabkab

2015

Internet Mathematics

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We introduce a new random graph model. In our model, n, n ≥ 2, vertices choose a subset of potential edges by considering the (estimated) benefits or utilities of the edges. More precisely, each vertex selects k, k ≥ 1, incident edges it wishes to set up, and an undirected edge between two vertices is present in the graph if and only if both of the end vertices choose the edge. First, we examine the scaling law of the smallest k needed for graph connectivity with increasing n and prove that it is (log(n)). Second, we study the diameter of the random graph and demonstrate that, under certain conditions on k, the diameter is close to log(n)/ log(log(n)) with high probability. In addition, as a byproduct of our findings, we show that, for all sufficiently large n, if k > β log(n), where β ≈ 2.4626, there exists a connected Erdös-Rényi random graph that is embedded in our random graph, with high probability.

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“…The resulting communication graph corresponds to a random m-orientable graph [28]. In this model, a digraph DG In Robber, m = 5, making it even more likely that the communication graph is connected.…”

Section: Lemma 1 Within a Single Collective Robber Eventually Creatmentioning

confidence: 99%

Collective Receiver-Initiated Multicast for Grid Applications

Burger

Kielmann

2011

IEEE Trans. Parallel Distrib. Syst.

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Abstract-Grid applications often need to distribute large amounts of data efficiently from one cluster to multiple others (multicast). Existing sender-initiated methods arrange nodes in optimized tree structures, based on external network monitoring data. This dependence on monitoring data severely impacts both ease of deployment and adaptivity to dynamically changing network conditions. In this paper we present Robber, a collective, receiver-initiated, high-throughput multicast approach inspired by the BitTorrent protocol. Unlike BitTorrent, Robber is specifically designed to maximize the throughput between multiple cluster computers. Nodes in the same cluster work together as a collective that tries to steal data from peer clusters. Instead of using potentially outdated monitoring data, Robber automatically adapts to the currently achievable bandwidth ratios. Within a collective, nodes automatically tune the amount of data they steal remotely to their relative performance. Our experimental evaluation compares Robber to BitTorrent, to Balanced Multicasting, and to its predecessor MOB. Balanced Multicasting optimizes multicast trees based on external monitoring data, while MOB uses collective, receiver-initiated multicast with static load balancing. We show that both Robber and MOB outperform BitTorrent. They are competitive with Balanced Multicasting as long as the network bandwidth remains stable, and outperform it by wide margins when bandwidth changes dynamically. In large environments and heterogeneous clusters, Robber outperforms MOB.

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On the connectivity of randomm-orientable graphs and digraphs

Cited by 88 publications

References 5 publications

The transitive closure of a random digraph

The transitive closure of a random digraph

A New Random Graph Model with Self-Optimizing Nodes: Connectivity and Diameter

Collective Receiver-Initiated Multicast for Grid Applications

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