Finding a long directed cycle

Gabow, Harold N.; Nie, Shuxin

doi:10.1145/1328911.1328918

Cited by 26 publications

(19 citation statements)

References 20 publications

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“…Theorem 18 (Gabow and Nie [10,11] The complexity given in Proposition 20 is certainly not optimal. For example, it can be improved for spindles with paths of small lengths.…”

Section: Subdivision Of Directed Cyclesmentioning

confidence: 99%

Finding a subdivision of a digraph

Bang‐Jensen

Havet

Maia

2015

Theoretical Computer Science

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“…Theorem 18 (Gabow and Nie [10,11] The complexity given in Proposition 20 is certainly not optimal. For example, it can be improved for spindles with paths of small lengths.…”

Section: Subdivision Of Directed Cyclesmentioning

confidence: 99%

Finding a subdivision of a digraph

Bang‐Jensen

Havet

Maia

2015

Theoretical Computer Science

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“…P U lop is detected by an application of a simple randomized algorithm (Theorem 2.2) proposed by [25] to find simple cycles (in our case, C 6 ). For cycles longer than or equal to length k = 8, we apply Theorem 4.1 from [26] and run DFS to check for long backedges. If DFS fails to detect long backedges, then Theorem 4.1 says a long simple cycle, if it exists, will have length between k and 2k − 4.…”

Section: Resultsmentioning

confidence: 99%

On the incompatibility of connectivity and local pooling in Erdős-Rényi Graphs

Wildman

Weber

2013

2013 51st Annual Allerton Conference on Communication, Control, and Computing (Allerton)

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For a wireless communications network, Local Pooling (LoP) is a desirable property due to its sufficiency for the optimality of low-complexity greedy scheduling techniques. However, LoP in network graphs with a primary interference model enforces an edge sparsity that may be prohibitive to other desirable properties in wireless networks, such as connectivity.In this paper, we investigate the impact of the edge density on both LoP and the size of the largest component under the primary interference model, as the number of nodes in the network grows large. For Erdős-Rényi graphs, we employ threshold functions to establish critical values for the edge probability necessary for these properties to hold. These thresholds demonstrate that LoP and connectivity (or even the presence of a giant component) cannot both hold asymptotically for a large class of edge probability functions. A similar incompatibility for random geometric graphs is suggested by our simulation results.

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“…contains an additional factor of 1/2 due to symmetry, but nevertheless is upper bounded by the expressions in (30) and (31).…”

Section: A Ancillary Lemmasmentioning

confidence: 99%

On Characterizing the Local Pooling Factor of Greedy Maximal Scheduling in Random Graphs

Wildman

Weber

2016

IEEE/ACM Trans. Networking

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Abstract-The study of the optimality of low-complexity greedy scheduling techniques in wireless communications networks is a very complex problem. The Local Pooling (LoP) factor provides a single-parameter means of expressing the achievable capacity region (and optimality) of one such scheme, greedy maximal scheduling (GMS). The exact LoP factor for an arbitrary network graph is generally difficult to obtain, but may be evaluated or bounded based on the network graph's particular structure. In this paper, we provide rigorous characterizations of the LoP factor in large networks modeled as Erdős-Rényi (ER) and random geometric (RG) graphs under the primary interference model. We employ threshold functions to establish critical values for either the edge probability or communication radius to yield useful bounds on the range and expectation of the LoP factor as the network grows large. For sufficiently dense random graphs, we find that the LoP factor is between 1/2 and 2/3, while sufficiently sparse random graphs permit GMS optimality (the LoP factor is 1) with high probability. We then place LoP within a larger context of commonly studied random graph properties centered around connectedness. We observe that edge densities permitting connectivity generally admit cycle subgraphs which forms the basis for the LoP factor upper bound of 2/3. We conclude with simulations to explore the regime of small networks, which suggest the probability that an ER or RG graph satisfies LoP and is connected decays quickly in network size.

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Finding a long directed cycle

Cited by 26 publications

References 20 publications

Finding a subdivision of a digraph

Finding a subdivision of a digraph

On the incompatibility of connectivity and local pooling in Erdős-Rényi Graphs

On Characterizing the Local Pooling Factor of Greedy Maximal Scheduling in Random Graphs

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Finding a long directed cycle

Cited by 26 publications

References 20 publications

Finding a subdivision of a digraph

Finding a subdivision of a digraph

On the incompatibility of connectivity and local pooling in Erd&#x0151;s-R&#x00E9;nyi Graphs

On Characterizing the Local Pooling Factor of Greedy Maximal Scheduling in Random Graphs

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On the incompatibility of connectivity and local pooling in Erdős-Rényi Graphs