
Dynamic Approximate Shortest Paths and Beyond: Subquadratic and WorstCase Update Time
Consider the following distance query for an nnode graph G undergoing e...
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Computing Lengths of Shortest NonCrossing Paths in Planar Graphs
Given a plane undirected graph G with nonnegative edge weights and a se...
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Expected Complexity of Routing in Θ 6 and HalfΘ 6 Graphs
We study online routing algorithms on the Θ6graph and the halfΘ6graph...
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Routing in Strongly Hyperbolic Unit Disk Graphs
Greedy routing has been studied successfully on Euclidean unit disk grap...
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Labeling Algorithm and Compact Routing Scheme for a Small World Network Model
This paper presents a small world networks generative model and a labeli...
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Small World Model based on a Sphere Homeomorphic Geometry
We define a small world model over the octahedron surface and relate its...
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A FastConvergence Routing of the HotPotato
Interactions between the intra and interdomain routing protocols recei...
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The idemetric property: when most distances are (almost) the same
We introduce the idemetric property, which formalises the idea that most nodes in a graph have similar distances between them, and which turns out to be quite standard amongst smallworld network models. Modulo reasonable sparsity assumptions, we are then able to show that a strong form of idemetricity is actually equivalent to a very weak expander condition (PUMP). This provides a direct way of providing short proofs that smallworld network models such as the WattsStrogatz model are strongly idemetric (for a wide range of parameters), and also provides further evidence that being idemetric is a common property. We then consider how satisfaction of the idemetric property is relevant to algorithm design. For idemetric graphs we observe, for example, that a single breadthfirst search provides a solution to the allpairs shortest paths problem, so long as one is prepared to accept paths which are of stretch close to 2 with high probability. Since we are able to show that Kleinberg's model is idemetric, these results contrast nicely with the well known negative results of Kleinberg concerning efficient decentralised algorithms for finding short paths: for precisely the same model as Kleinberg's negative results hold, we are able to show that very efficient (and decentralised) algorithms exist if one allows for reasonable preprocessing. For deterministic distributed routing algorithms we are also able to obtain results proving that less routing information is required for idemetric graphs than the worst case in order to achieve stretch less than 3 with high probability: while Ω(n^2) routing information is required in the worst case for stretch strictly less than 3 on almost all pairs, for idemetric graphs the total routing information required is O(nlog(n)).
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