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Low Distortion Spanners
"... Abstract. A spanner of an undirected unweighted graph is a subgraph that approximates the distance metric of the original graph with some specified accuracy. Specifically, we say H ⊆ G is an fspanner of G if any two vertices u, v at distance d in G are at distance at most f(d) in H. There is clearl ..."
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Cited by 16 (2 self)
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Abstract. A spanner of an undirected unweighted graph is a subgraph that approximates the distance metric of the original graph with some specified accuracy. Specifically, we say H ⊆ G is an fspanner of G if any two vertices u, v at distance d in G are at distance at most f(d) in H. There is clearly some tradeoff between the sparsity of H and the distortion function f, though the nature of this tradeoff is still poorly understood. In this paper we present a simple, modular framework for constructing sparse spanners that is based on interchangable components called connection schemes. By assembling connection schemes in different ways we can recreate the additive 2 and 6spanners of Aingworth et al. and Baswana et al. and improve on the (1+ɛ, β)spanners of Elkin and Peleg, the sublinear additive spanners of Thorup and Zwick, and the (non constant) additive spanners of Baswana et al. Our constructions rival the simplicity of all comparable algorithms and provide substantially better spanners, in some cases reducing the density doubly exponentially. 1
Distributed Algorithms for Ultrasparse Spanners and Linear Size Skeletons
"... We present efficient algorithms for computing very sparse low distortion spanners in distributed networks and prove some nontrivial lower bounds on the tradeoff between time, sparseness, and distortion. All of our algorithms assume a synchronized distributed network, where relatively short messages ..."
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Cited by 7 (0 self)
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We present efficient algorithms for computing very sparse low distortion spanners in distributed networks and prove some nontrivial lower bounds on the tradeoff between time, sparseness, and distortion. All of our algorithms assume a synchronized distributed network, where relatively short messages may be communicated in each time step. Our first result is a fast distributed algorithm for finding an O(2 log ∗ n log n)spanner with size O(n). Besides being nearly optimal in time and distortion, this algorithm appears to be the first that constructs an O(n)size skeleton without requiring unbounded length messages or time proportional to the diameter of the network. Our second result is a new class of efficiently constructible (α, β)spanners called Fibonacci spanners whose distortion improves with the distance being approximated. At their sparsest Fibonacci spanners can have nearly linear size O(n(log log n) φ) where φ = 1+ √ 5 2 is the golden ratio. As the distance increases the Fibonacci spanner’s multiplicative distortion passes through four discrete stages, moving from logarithmic to loglogarithmic, then into a period where it is constant, tending to 3, followed by another period tending to 1. On the lower bound side we prove that many recent sequential spanner constructions have no efficient counterparts in distributed networks, even if the desired distortion only needs to be achieved on the average or for a tiny fraction of the vertices. In particular, any distance preservers, purely additive spanners, or spanners with sublinear additive distortion must either be very dense, slow to construct, or have very weak guarantees on distortion. 1
C.: Sparse spanners vs. compact routing
 In: SPAA
, 2011
"... Routing with multiplicative stretch 3 (which means that the path used by the routing scheme can be up to three times longer than a shortest path) can be done with routing tables of ˜ Θ ( √ n) bits 1 per node. The space lower bound is due to the existence of dense graphs with large girth. Dense grap ..."
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Cited by 3 (2 self)
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Routing with multiplicative stretch 3 (which means that the path used by the routing scheme can be up to three times longer than a shortest path) can be done with routing tables of ˜ Θ ( √ n) bits 1 per node. The space lower bound is due to the existence of dense graphs with large girth. Dense graphs can be sparsified to subgraphs, called spanners, with various stretch guarantees. There are spanners with additive stretch guarantees (some even have constant additive stretch) but only very few additive routing schemes are known. In this paper, we give reasons why routing in unweighted graphs with additive stretch is difficult in the form of space lower bounds for general graphs and for planar graphs. We prove that any routing scheme using routing tables of size µ bits per node and addresses of polylogarithmic length has additive stretch ˜ Ω ( p n/µ) for general graphs, and ˜ Ω ( √ n/µ) for planar graphs, respectively. Routing with tables of size Õ(n1/3) thus requires a polynomial additive stretch of ˜Ω(n 1/3), whereas spanners with average degree O(n 1/3) and constant additive stretch exist for all graphs. Spanners, however sparse they are, do not tell us how to route. These bounds provide the first separation of sparse spanner problems and compact routing problems. On the positive side, we give an almost tight upper bound: we present the first nontrivial compact routing scheme with o(lg 2 n)bit addresses, additive stretch Õ(n1/3), and table size Õ(n1/3) bits for all graphs with linear local treewidth such as planar, boundedgenus, and apexminorfree graphs. C.G. is also member of “l’Institut Universitaire de France”. He is also supported by ANR projects “ALADDIN”, and the
Efficient Distributed Source Detection with Limited Bandwidth
"... Given a simple graph G = (V, E) and a set of sources S ⊆ V, denote for each node v ∈ V by L (∞) v the lexicographically ordered list of distance/source pairs (d(s, v), s), where s ∈ S. For integers d, k ∈ N∪{∞}, we consider the source detection, or (S, d, k)detection task, requiring each node v to ..."
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Given a simple graph G = (V, E) and a set of sources S ⊆ V, denote for each node v ∈ V by L (∞) v the lexicographically ordered list of distance/source pairs (d(s, v), s), where s ∈ S. For integers d, k ∈ N∪{∞}, we consider the source detection, or (S, d, k)detection task, requiring each node v to learn the (if for all of them d(s, v) ≤ d) or all entries (d(s, v), s) ∈ L (∞) v satisfying that d(s, v) ≤ d (otherwise). Solutions to this problem provide natural generalizations of concurrent breadthfirst search (BFS) tree constructions. For example, the special case of k = ∞ requires each source s ∈ S to build a complete BFS tree rooted at s, whereas the special case of d = ∞ and S = V requires constructing a partial BFS tree comprising at least k nodes from every node in V. In this work, we give a simple, nearoptimal solution for the source detection task in the CONGEST model, where messages contain at most O(log n) bits, running in d + k rounds. We demonstrate its utility for various routing problems, exact and approximate diameter computation, and spanner construction. For those problems, we obtain algorithms in the CONGEST model that are faster and in some cases much simpler than previous solutions. first k entries of L (∞)