Routing in Networks with Low Doubling Dimension

Abraham, Ittai; Gavoille, Cyril; Goldberg, Andrew V.; Malkhi, Dahlia

doi:10.1109/icdcs.2006.72

Cited by 74 publications

(102 citation statements)

References 13 publications

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“…For planar graphs, a shortest path routing labeling scheme which uses 8n + o(n) bits per vertex is developed in [22], and a multiplicative (1 + )-stretched routing labeling scheme for every > 0 which uses O( −1 log 3 n) bits per vertex is developed in [39]. Routing in graphs with doubling dimension α has been considered in [1,10,37,38]. It was shown that any graph with doubling dimension α admits a multiplicative (1 + )-stretched routing labeling scheme with labels of size −O(α) log 2 n bits.…”

Section: Some Known Strategiesmentioning

confidence: 99%

Navigating in a Graph by Aid of Its Spanning Tree

Dragan

Matamala

2008

Algorithms and Computation

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Abstract. Let G = (V, E) be a graph and T be a spanning tree of G. We consider the following strategy in advancing in G from a vertex x towards a target vertex y: from a current vertex z (initially, z = x), unless z = y, go to a neighbor of z in G that is closest to y in T (breaking ties arbitrarily). In this strategy, each vertex has full knowledge of its neighborhood in G and can use the distances in T to navigate in G. Thus, additionally to standard local information (the neighborhood NG(v)), the only global information that is available to each vertex v is the topology of the spanning tree T (in fact, v can know only a very small piece of information about T and still be able to infer from it the necessary tree-distances). For each source vertex x and target vertex y, this way, a path, called a greedy routing path, is produced. Denote by gG,T (x, y) the length of a longest greedy routing path that can be produced for x and y using this strategy and T . We say that a spanning tree T of a graph G is an additive r-carcass for G if gG,T (x, y) ≤ dG(x, y) + r for each ordered pair x, y ∈ V . In this paper, we investigate the problem, given a graph family F, whether a small integer r exists such that any graph G ∈ F admits an additive r-carcass. We show that rectilinear p × q grids, hypercubes, distance-hereditary graphs, dually chordal graphs (and, therefore, strongly chordal graphs and interval graphs), all admit additive 0-carcasses. Furthermore, every chordal graph G admits an additive (ω(G) + 1)-carcass (where ω(G) is the size of a maximum clique of G), each 3-sun-free chordal graph admits an additive 2-carcass, each chordal bipartite graph admits an additive 4-carcass. In particular, any k-tree admits an additive (k +2)-carcass. All those carcasses are easy to construct.

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Section: Some Known Strategiesmentioning

confidence: 99%

Navigating in a Graph by Aid of Its Spanning Tree

Dragan

Matamala

2008

Algorithms and Computation

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“…There are two popular models in the literature, the labeled routing model (in which naming and routing schemes are jointly considered) [9,10,27] and name-independent routing (in which node IDs are independent of the routing schemes) [2,16]. Generally speaking, the theoretical results in compact routing in a graph whose shortest path metric has a constant doubling dimension are able to obtain, with polylogarithmic routing table size, 1 + ε stretch routing in the labeled routing scheme (see [8] and many others in the reference therein), and constant stretch factor routing in the name-independent routing scheme [16,1] (getting a stretch factor of 3 − ε will require linear routing table size [1]). The schemes here are all by centralized constructions and aim to get the best asymptotic bounds.…”

Section: Introductionmentioning

confidence: 99%

Resilient and Low Stretch Routing through Embedding into Tree Metrics

Gao

Zhou

2011

Lecture Notes in Computer Science

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Abstract. Given a network, the simplest routing scheme is probably routing on a spanning tree. This method however does not provide good stretch -the route between two nodes can be much longer than their shortest distance, nor does it give good resilience -one node failure may disconnect quadratically many pairs. In this paper we use two trees to achieve both constant stretch and good resilience. Given a metric (e.g., as the shortest path metric of a given communication network), we build two hierarchical well-separated trees using randomization such that for any two nodes u, v, the shorter path of the two paths in the two respective trees gives a constant stretch of the metric distance of u, v, and the removal of any node only disconnect the routes between O(1/n) fraction of all pairs. Both bounds are in expectation and hold true as long as the metric follows certain geometric growth rate (the number of nodes within distance r is a polynomial function of r), which holds for many realistic network settings such as wireless ad hoc networks and Internet backbone graphs. The algorithms have been implemented and tested on real data.

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“…It seems that for some applications replacing distances by ranks can be meaningful. In particular, it is interesting to consider linear arrangement problem [18], closest pairs [24], distance labelling [66], shortest paths [3], detecting communities [61], and dimensionality reduction [15].…”

Section: Directions For Further Workmentioning

confidence: 99%

Combinatorial Framework for Similarity Search

Lifshits

2009

2009 Second International Workshop on Similarity Search and Applications

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Abstract-We present an overview of the combinatorial framework for similarity search. An algorithm is combinatorial if only direct comparisons between two pairwise similarity values are allowed. Namely, the input dataset is represented by a comparison oracle that given any three points x, y, z answers whether y or z is closer to x. We assume that the similarity order of the dataset satisfies the four variations of the following disorder inequality: if x is the a'th most similar object to y and y is the b'th most similar object to z, then x is among the D(a + b) most similar objects to z, where D is a relatively small disorder constant. Combinatorial algorithms for nearest neighbor search have two important advantages: (1) they do not map similarity values to artificial distance values and do not use triangle inequality for the latter, and (2) they work for arbitrarily complicated data representations and similarity functions.Ranwalk, the first known combinatorial solution for nearest neighbors, is randomized, exact, zero-error algorithm with query time that is logarithmic in number of objects. But Ranwalk preprocessing time is quadratic. Later on, another solution, called combinatorial nets, was discovered. It is deterministic and exact algorithm with near-linear time and space complexity of preprocessing, and near-logarithmic time complexity of search. Combinatorial nets also have a number of side applications. For near-duplicate detection they lead to the first known deterministic algorithm that requires just nearlinear time + time proportional to the size of output. For any dataset with small disorder combinatorial nets can be used to construct a visibility graph: the one in which greedy routing deterministically converges to the nearest neighbor of a target in logarithmic number of steps. The later result is the first known work-around for Navarro's impossibility of generalizing Delaunay graphs.

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Routing in Networks with Low Doubling Dimension

Cited by 74 publications

References 13 publications

Navigating in a Graph by Aid of Its Spanning Tree

Navigating in a Graph by Aid of Its Spanning Tree

Resilient and Low Stretch Routing through Embedding into Tree Metrics

Combinatorial Framework for Similarity Search

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