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15
Distance Estimation and Object Location via Rings of Neighbors
 In 24 th Annual ACM Symposium on Principles of Distributed Computing (PODC
, 2005
"... We consider four problems on distance estimation and object location which share the common flavor of capturing global information via informative node labels: lowstretch routing schemes [47], distance labeling [24], searchable small worlds [30], and triangulationbased distance estimation [33]. Fo ..."
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Cited by 64 (4 self)
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We consider four problems on distance estimation and object location which share the common flavor of capturing global information via informative node labels: lowstretch routing schemes [47], distance labeling [24], searchable small worlds [30], and triangulationbased distance estimation [33]. Focusing on metrics of low doubling dimension, we approach these problems with a common technique called rings of neighbors, which refers to a sparse distributed data structure that underlies all our constructions. Apart from improving the previously known bounds for these problems, our contributions include extending Kleinberg’s small world model to doubling metrics, and a short proof of the main result in Chan et al. [14]. Doubling dimension is a notion of dimensionality for general metrics that has recently become a useful algorithmic concept in the theoretical computer science literature. 1
Routing in networks with low doubling dimension
 In 26 th International Conference on Distributed Computing Systems (ICDCS). IEEE Computer
, 2006
"... This paper studies compact routing schemes for networks with low doubling dimension. Two variants are explored, nameindependent routing and labeled routing. The key results obtained for this model are the following. First, we provide the first nameindependent solution. Specifically, we achieve con ..."
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Cited by 63 (8 self)
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This paper studies compact routing schemes for networks with low doubling dimension. Two variants are explored, nameindependent routing and labeled routing. The key results obtained for this model are the following. First, we provide the first nameindependent solution. Specifically, we achieve constant stretch and polylogarithmic storage. Second, we obtain the first truly scalefree solutions, namely, the network’s aspect ratio is not a factor in the stretch. Scalefree schemes are given for three problem models: nameindependent routing on graphs, labeled routing on metric spaces, and labeled routing on graphs. Third, we prove a lower bound requiring linear storage for stretch < 3 schemes. This has the important ramification of separating for the first time the nameindependent problem model from the labeled model for these networks, since compact stretch1+ε labeled schemes are known to be possible.
Object Location Using Path Separators
, 2006
"... We study a novel separator property called kpath separable. Roughly speaking, a kpath separable graph can be recursively separated into smaller components by sequentially removing k shortest paths. Our main result is that every minor free weighted graph is kpath separable. We then show that kpat ..."
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Cited by 35 (11 self)
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We study a novel separator property called kpath separable. Roughly speaking, a kpath separable graph can be recursively separated into smaller components by sequentially removing k shortest paths. Our main result is that every minor free weighted graph is kpath separable. We then show that kpath separable graphs can be used to solve several object location problems: (1) a smallworldization with an average polylogarithmic number of hops; (2) an (1 + ε)approximate distance labeling scheme with O(log n) space labels; (3) a stretch(1 + ε) compact routing scheme with tables of polylogarithmic space; (4) an (1+ε)approximate distance oracle with O(n log n) space and O(log n) query time. Our results generalizes to much wider classes of weighted graphs, namely to boundeddimension isometric sparable graphs.
Optimalstretch nameindependent compact routing in doubling metrics
 In PODC
, 2006
"... We consider the problem of nameindependent routing in doubling metrics. A doubling metric is a metric space whose doubling dimension is a constant, where the doubling dimension of a metric space is the least value α such that any ball of radius r can be covered by at most 2 α balls of radius r/2. G ..."
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Cited by 19 (2 self)
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We consider the problem of nameindependent routing in doubling metrics. A doubling metric is a metric space whose doubling dimension is a constant, where the doubling dimension of a metric space is the least value α such that any ball of radius r can be covered by at most 2 α balls of radius r/2. Given any δ> 0 and a weighted undirected network G whose shortest path metric d is a doubling metric with doubling dimension α, we present a nameindependent routing scheme for G with (9+δ)stretch, (2+ 1 δ)O(α) (log ∆) 2 (log n)bit routing information at each node, and packet headers of size O(log n), where ∆ is the ratio of the largest to the smallest shortest path distance in G. In addition, we prove that for any ǫ ∈ (0, 8), there is a doubling metric network G with n nodes, doubling dimension α ≤ 6 − log ǫ, and ∆ = O(2 1/ǫ n) such that any nameindependent routing scheme on G with routing information at each node of size o(n (ǫ/60)2)bits has stretch larger than 9 − ǫ. Therefore assuming that ∆ is bounded by a polynomial on n, our algorithm basically achieves optimal stretch for nameindependent routing in doubling metrics with packet header size and routing information at each node both bounded by a polylogarithmic function of n.
On spacestretch tradeoffs: upper bounds
 In SPAA
, 2006
"... One of the fundamental tradeoffs in compact routing schemes is between the space used to store the routing table on each node and the stretch factor of the routing scheme – the maximum ratio over all pairs between the cost of the route induced by the scheme and the cost of a minimum cost path betwe ..."
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Cited by 18 (8 self)
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One of the fundamental tradeoffs in compact routing schemes is between the space used to store the routing table on each node and the stretch factor of the routing scheme – the maximum ratio over all pairs between the cost of the route induced by the scheme and the cost of a minimum cost path between the same pair. All previous routing schemes required storage that is dependent on the diameter of the network. We present a new scalefree routing scheme, whose storage and header sizes are independent of the aspect ratio of the network. Our scheme is based on a decomposition into sparse and dense neighborhoods. Given an undirected network with arbitrary weights and n arbitrary node names, for any integer k ≥ 1 we present the first scalefree routing scheme with asymptotically optimal spacestretch tradeoff that does not require edge weights to be polynomially bounded. The scheme uses e O(n 1/k) space routing table at each node, and routes along paths of asymptotically optimal linear stretch O(k).
Compact Routing in PowerLaw Graphs
"... Abstract. We adapt the compact routing scheme by Thorup and Zwick to optimize it for powerlaw graphs. We analyze our adapted routing scheme based on the theory of unweighted random powerlaw graphs with fixed expected degree sequence by Aiello, Chung, and Lu. Our result is the first theoretical bou ..."
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Cited by 15 (3 self)
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Abstract. We adapt the compact routing scheme by Thorup and Zwick to optimize it for powerlaw graphs. We analyze our adapted routing scheme based on the theory of unweighted random powerlaw graphs with fixed expected degree sequence by Aiello, Chung, and Lu. Our result is the first theoretical bound coupled to the parameter of the powerlaw graph model for a compact routing scheme. In particular, we prove that, for stretch 3, instead of routing tables with Õ(n 1/2) bits as in the general scheme by Thorup and Zwick, expected sizes of O(n γ log n) bits are sufficient, and that all the routing tables can be constructed at once in expected time O(n 1+γ log n), with γ = τ−2 + ε, where τ ∈ (2, 3) 2τ−3 is the powerlaw exponent and ε> 0. Both bounds also hold with probability at least 1 − 1/n (independent of ε). The routing scheme is a labeled scheme, requiring a stretch5 handshaking step and using addresses and message headers with O(log n log log n) bits, with probability at least 1−o(1). We further demonstrate the effectiveness of our scheme by simulations on realworld graphs as well as synthetic powerlaw graphs. With the same techniques as for the compact routing scheme, we also adapt the approximate distance oracle by Thorup and Zwick for stretch 3 and obtain a new upper bound of expected Õ(n1+γ) for space and preprocessing. 1
StrongDiameter Decompositions of Minor Free Graphs
, 2007
"... We provide the first sparse covers and probabilistic partitions for graphs excluding a fixed minor that have strong diameter bounds; i.e. each set of the cover/partition has a small diameter as an induced subgraph. Using these results we provide improved distributed nameindependent routing schemes ..."
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Cited by 10 (3 self)
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We provide the first sparse covers and probabilistic partitions for graphs excluding a fixed minor that have strong diameter bounds; i.e. each set of the cover/partition has a small diameter as an induced subgraph. Using these results we provide improved distributed nameindependent routing schemes. Specifically, given a graph excluding a minor on r vertices and a parameter ρ> 0 we obtain the flowing results: (1) a polynomial algorithm that constructs a set of clusters such that each cluster has a strongdiameter of O(r 2 ρ) and each vertex belongs to 2 O(r) r! clusters; (2) a nameindependent routing scheme with a stretch of O(r 2) and tables of size 2 O(r) r! log 4 n bits; (3) a randomized algorithm that partitions the graph such that each cluster has strongdiameter O(r6 r ρ) and the probability an edge (u, v) is cut is O(r d(u, v)/ρ).
Logic, Graphs, and Algorithms
, 2007
"... Algorithmic meta theorems are algorithmic results that apply to whole families of combinatorial problems, instead of just specific problems. These families are usually defined in terms of logic and graph theory. An archetypal algorithmic meta theorem is Courcelle’s Theorem [9], which states that all ..."
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Cited by 6 (0 self)
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Algorithmic meta theorems are algorithmic results that apply to whole families of combinatorial problems, instead of just specific problems. These families are usually defined in terms of logic and graph theory. An archetypal algorithmic meta theorem is Courcelle’s Theorem [9], which states that all graph properties definable in monadic secondorder logic can be decided in linear time on graphs of bounded tree width. This article is an introduction into the theory underlying such meta theorems and a survey of the most important results in this area. 1
Collective Tree Spanners in Graphs with Bounded Parameters
 ALGORITHMICA
, 2006
"... In this paper we study collective additive tree spanners for special families of graphs including planar graphs, graphs with bounded genus, graphs with bounded treewidth, graphs with bounded cliquewidth, and graphs with bounded chordality. We say that a graph G = (V, E) admits a system of μ colle ..."
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Cited by 4 (4 self)
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In this paper we study collective additive tree spanners for special families of graphs including planar graphs, graphs with bounded genus, graphs with bounded treewidth, graphs with bounded cliquewidth, and graphs with bounded chordality. We say that a graph G = (V, E) admits a system of μ collective additive tree rspanners if there is a system T (G) of at most μ spanning trees of G such that for any two vertices x,y of G a spanning tree T ∈ T (G) exists such that dT (x, y) ≤ dG(x, y) + r. We describe a general method for constructing a “small” system of collective additive tree rspanners with small values of r for “well ” decomposable graphs, and as a byproduct show (among other results) that any weighted planar graph admits a system of O ( √ n) collective additive tree 0spanners, any weighted graph with treewidth at most k − 1 admits a system of k log 2 n collective additive tree 0spanners, any weighted graph with cliquewidth at most k admits a system of k log 3/2 n collective additive tree (2w)spanners, and any weighted graph with size of largest induced cycle at most c admits a system of log 2 n collective additive tree (2⌊c/2⌋w)spanners and a system of 4 log 2 n collective additive tree (2(⌊c/3⌋+1)w)spanners (here, w is the maximum edge weight in G). The latter result is refined for weighted weakly chordal graphs: any such graph admits a system of 4 log 2 n collective additive tree (2w)spanners. Furthermore, based on this collection of trees, we derive a compact and efficient routing scheme for those families of graphs. Results of this paper were partially presented at the ISAAC’05 conference [14].
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