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71
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 66 (5 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
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 36 (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.
Randomized 3D Geographic Routing
"... Abstract—We reconsider the problem of geographic routing in wireless ad hoc networks. We are interested in local, memoryless routing algorithms, i.e. each network node bases its routing decision solely on its local view of the network, nodes do not store any message state, and the message itself can ..."
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Cited by 29 (0 self)
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Abstract—We reconsider the problem of geographic routing in wireless ad hoc networks. We are interested in local, memoryless routing algorithms, i.e. each network node bases its routing decision solely on its local view of the network, nodes do not store any message state, and the message itself can only carry information about O(1) nodes. In geographic routing schemes, each network node is assumed to know the coordinates of itself and all adjacent nodes, and each message carries the coordinates of its target. Whereas many of the aspects of geographic routing have already been solved for 2D networks, little is known about higherdimensional networks. It has been shown only recently that there is in fact no local memoryless routing algorithm for 3D networks that delivers messages deterministically. In this paper, we show that a cubic routing stretch constitutes a lower bound for any local memoryless routing algorithm, and propose and analyze several randomized geographic routing algorithms which work well for 3D network topologies. For unit ball graphs, we present a technique to locally capture the surface of holes in the network, which leads to 3D routing algorithms similar to the greedyfacegreedy approach for 2D networks. I.
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 20 (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).
Greedy Routing with Bounded Stretch
"... Abstract—Greedy routing is a novel routing paradigm where messages are always forwarded to the neighbor that is closest to the destination. Our main result is a polynomialtime algorithm that embeds combinatorial unit disk graphs (CUDGs – a CUDG is a UDG without any geometric information) into O(log ..."
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Cited by 18 (0 self)
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Abstract—Greedy routing is a novel routing paradigm where messages are always forwarded to the neighbor that is closest to the destination. Our main result is a polynomialtime algorithm that embeds combinatorial unit disk graphs (CUDGs – a CUDG is a UDG without any geometric information) into O(log 2 n)dimensional space, permitting greedy routing with constant stretch. To the best of our knowledge, this is the first greedy embedding with stretch guarantees for this class of networks. Our main technical contribution involves extracting, in polynomial time, a constant number of isometric and balanced tree separators from a given CUDG. We do this by extending the celebrated LiptonTarjan separator theorem for planar graphs to CUDGs. Our techniques extend to other classes of graphs; for example, for general graphs, we obtain an O(log n)stretch greedy embedding into O(log 2 n)dimensional space. The greedy embeddings constructed by our algorithm can also be viewed as a constantstretch compact routing scheme in which each node is assigned an O(log 3 n)bit label. To the best of our knowledge, this result yields the best known stretchspace tradeoff for compact routing on CUDGs. Extensive simulations on random wireless networks indicate that the average routing overhead is about 10%; only few routes have a stretch above 1.5. I.
A doubling dimension threshold Θ(log log n) for augmented graph navigability
 In 14th European Symposium on Algorithm (ESA), LNCS 4168
, 2006
"... Abstract. In his seminal work, Kleinberg showed how to augment meshes using random edges, so that they become navigable; that is, greedy routing computes paths of polylogarithmic expected length between any pairs of nodes. This yields the crucial question of determining wether such an augmentation i ..."
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Cited by 17 (7 self)
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Abstract. In his seminal work, Kleinberg showed how to augment meshes using random edges, so that they become navigable; that is, greedy routing computes paths of polylogarithmic expected length between any pairs of nodes. This yields the crucial question of determining wether such an augmentation is possible for all graphs. In this paper, we answer negatively to this question by exhibiting a threshold on the doubling dimension, above which an infinite family of graphs cannot be augmented to become navigable whatever the distribution of random edges is. Precisely, it was known that graphs of doubling dimension at most O(log log n) are navigable. We show that for doubling dimension ≫ log log n, an infinite family of graphs cannot be augmented to become navigable. Finally, we complete our result by studying the special case of square meshes, that we prove to always be augmentable to become navigable.
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 14 (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
Improved algorithms for fully dynamic geometric spanners and geometric routing
 In ACM Symposium on Discrete Algorithms
, 2008
"... For a set S of points in R d, a tspanner is a sparse graph on the points of S such that between any pair of points there is a path in the spanner whose total length is at most t times the Euclidean distance between the points. In this paper, we show how to construct a (1 + ε)spanner with O(n/ε d) ..."
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Cited by 13 (8 self)
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For a set S of points in R d, a tspanner is a sparse graph on the points of S such that between any pair of points there is a path in the spanner whose total length is at most t times the Euclidean distance between the points. In this paper, we show how to construct a (1 + ε)spanner with O(n/ε d) edges and maximum degree O(1/ε d) in time O(n log n). A spanner with similar properties was previously presented in [6, 8]. However, using our new construction (coupled with several other innovations) we obtain new results for two fundamental problems for constant doubling dimension metrics: The first result is an essentially optimal compact routing scheme. In particular, we show how to perform routing with a stretch of 1 + ɛ, where the label size is ⌈log n ⌉ and the size of the table stored at each point is only O(log n/ε d). This routing problem was first considered by Peleg and Hassin [11], who presented a routing scheme in the plane. Later, Chan et al. [6] and Abraham et al. [1] considered this problem for doubling dimension metric spaces. Abraham et al. [1] were the first to present a (1 + ɛ) routing scheme where the label size depends solely on the number of points. In their scheme labels are of size of ⌈log n⌉, and each point stores a table of size O(log 2 n/ε d). In our routing scheme, we achieve routing tables of size O(log n/ε d), which is essentially the same size as a label (up to the factor of 1/ε d). The second and main result of this paper is the first fully dynamic geometric spanner with polylogarithmic update time for both insertions and deletions. We present an algorithm that allows points to be inserted into and deleted from S with an amortized update time of O(log 3 n).
Combinatorial algorithms for nearest neighbors, nearduplicates and smallworld design
 In Proceedings of the 20th Annual ACMSIAM Symposium on Discrete Algorithms, SODA’09
, 2009
"... We study the so called combinatorial framework for algorithmic problems in similarity spaces. Namely, the input dataset is represented by a comparison oracle that given three points x, y, y ′ answers whether y or y ′ is closer to x. We assume that the similarity order of the dataset satisfies the fo ..."
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Cited by 11 (1 self)
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We study the so called combinatorial framework for algorithmic problems in similarity spaces. Namely, the input dataset is represented by a comparison oracle that given three points x, y, y ′ answers whether y or y ′ 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. Though the oracle gives much less information compared to the standard general metric space model where distance values are given, one can still design very efficient algorithms for various fundamental computational tasks. For nearest neighbor search we present deterministic and exact algorithm with almost linear time and space complexity of preprocessing, and nearlogarithmic time complexity of search. Then, for nearduplicate detection we present the first known deterministic algorithm that requires just nearlinear time + time proportional to the size of output. Finally, we show that for any dataset satisfying the disorder inequality a visibility graph can be constructed: all outdegrees are nearlogarithmic and greedy routing deterministically converges to the nearest neighbor of a target in logarithmic number of steps. The later result is the first known workaround for Navarro’s impossibility of generalizing Delaunay graphs. The technical contribution of the paper consists of handling “false positives ” in data structures and an algorithmic technique upasidedownfilter.