Euclidean versus Hyperbolic Congestion in Idealized versus Experimental Networks

Jonckheere, Edmond; Lou, Mingji; Bonahon, Francis; Baryshnikov, Yuliy

doi:10.1080/15427951.2010.554320

Cited by 62 publications

(65 citation statements)

References 19 publications

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“…When the network is uniformly distributed in a disk of radius R, the maximum traffic load by shortest path routing occurs in the center [18] and is in the order of Θ(n √ n) [11]. Notice that in fact the maximum traffic load is already asymptotically optimal.…”

Section: Scaling Law In Disks Spheres Half Spheresmentioning

confidence: 99%

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Load balanced short path routing in large-scale wireless networks using area-preserving maps

Goswami

Ban

et al. 2014

Proceedings of the 15th ACM International Symposium on Mobile Ad Hoc Networking and Computing

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Load balanced routing in a network, i.e., minimizing the maximum traffic load any node carries for unsplittable flows, is a well known NP-hard problem. Finding practical algorithms remains a long standing challenge. In this paper we propose greedy routing using virtual coordinates that achieves both small path stretch ratio (compared to shortest path) and small load balancing ratio (compared to optimal load balanced routing), in a large scale wireless sensor network deployed densely inside a geometric domain with complex shape. We first provide a greedy routing scheme on a disk with a stretch ratio of at most 2, and under which the maximum load is a factor 4 √ 2 smaller than the maximum load under shortest path routing. This is the first simple routing scheme with a small stretch that has been proven to outperform shortest path routing in terms of load balancing. Then we transform a network of arbitrary shape to a disk by an area preserving map φ. We show that both the path length and the maximum traffic load in the original network only increases by an additional factor of d 2 , where d is the maximum length stretch of φ. Combined with the result on a disk we again achieve both bounded stretch and bounded load balancing ratio. Our simulation results evaluated the practical performance on both quality measures.

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Section: Scaling Law In Disks Spheres Half Spheresmentioning

confidence: 99%

“…In polar coordinates (R, Θ), R for node zij lying on contour γi is justfi, i.e R(zij) =fi, ∀j. Now we will find the angular coordinate for each node of γi by computing a (discrete) integral along γi, which approximates the integral in Equation 11.…”

Section: The Path L Is Mapped To Any Given Radius Of Dmentioning

confidence: 99%

Load balanced short path routing in large-scale wireless networks using area-preserving maps

Goswami

Ban

et al. 2014

Proceedings of the 15th ACM International Symposium on Mobile Ad Hoc Networking and Computing

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“…Ideas related to hyperbolicity have been applied in numerous other networks applications, e.g., to problems such as distance estimation, sensor networks, and traffic flow and congestion minimization [30,14,15,27,3], as well as large-scale data visualization [22,26,31]. The latter applications typically take important advantage of the idea that data are often hierarchical or tree-like and that there is "more room" in hyperbolic spaces of a given dimension than corresponding Euclidean spaces.…”

Section: Introductionmentioning

confidence: 99%

On the Hyperbolicity of Small-World and Treelike Random Graphs

Chen¹,

Fang²,

Hu³

et al. 2013

Internet Mathematics

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Abstract. Hyperbolicity is a property of a graph that may be viewed as being a "soft" version of a tree, and recent empirical and theoretical work has suggested that many graphs arising in Internet and related data applications have hyperbolic properties. Here, we consider Gromov's notion of δ-hyperbolicity, and we establish several positive and negative results for small-world and tree-like random graph models. In particular, we show that small-world random graphs built from underlying grid structures do not have strong improvement in hyperbolicity, even when the rewiring greatly improves decentralized navigation. On the other hand, for a class of tree-like graphs called ringed trees that have constant hyperbolicity, adding random links among the leaves in a manner similar to the small-world graph constructions may easily destroy the hyperbolicity of the graphs, except for a class of random edges added using an exponentially decaying probability function based on the ring distance among the leaves. Our study provides the first significant analytical results on the hyperbolicity of a rich class of random graphs, which shed light on the relationship between hyperbolicity and navigability of random graphs, as well as on the sensitivity of hyperbolic δ to noises in random graphs.

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“…Jonckheere et al [9] examined the negative curvature and load congestion in both artificial data and real networks. They proposed to use Yamabe flow with free boundary condition to alleviate network congestion.…”

Section: Introductionmentioning

confidence: 99%

Spherical representation and polyhedron routing for load balancing in wireless sensor networks

Ban

Zeng

et al. 2011

2011 Proceedings IEEE INFOCOM

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Abstract-In this paper we address the problem of scalable and load balanced routing for wireless sensor networks. Motivated by the analog of the continuous setting that geodesic routing on a sphere gives perfect load balancing, we embed sensor nodes on a convex polyhedron in 3D and use greedy routing to deliver messages between any pair of nodes with guaranteed success. This embedding is known to exist by the Koebe-Andreev-Thurston Theorem for any 3-connected planar graphs. In our paper we use discrete Ricci flow to develop a distributed algorithm to compute this embedding. Further, such an embedding is not unique and differs from one another by a Möbius transformation. We employ an optimization routine to look for the Möbius transformation such that the nodes are spread on the polyhedron as uniformly as possible. We evaluated the load balancing property of this greedy routing scheme and showed favorable comparison with previous schemes.

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Euclidean versus Hyperbolic Congestion in Idealized versus Experimental Networks

Cited by 62 publications

References 19 publications

Load balanced short path routing in large-scale wireless networks using area-preserving maps

Load balanced short path routing in large-scale wireless networks using area-preserving maps

On the Hyperbolicity of Small-World and Treelike Random Graphs

Spherical representation and polyhedron routing for load balancing in wireless sensor networks

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