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23
Optimal scalefree compact routing schemes in doubling networks
 In ACMSIAM symposium on Discrete algorithms
, 2007
"... We consider compact routing schemes in networks of low doubling dimension, where the doubling dimension is the least value α such that any ball in the network can be covered by at most 2 α balls of half radius. There are two variants of routing scheme design: (i) labeled (namedependent) routing, wh ..."
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Cited by 9 (2 self)
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We consider compact routing schemes in networks of low doubling dimension, where the doubling dimension is the least value α such that any ball in the network can be covered by at most 2 α balls of half radius. There are two variants of routing scheme design: (i) labeled (namedependent) routing, where the designer is allowed to rename the nodes so that the names (labels) can contain additional routing information, e.g. topological information; and (ii) nameindependent routing, which works on top of the arbitrary original node names in the network, i.e. the node names are independent of the routing scheme. In this paper, given any constant ǫ ∈ (0, 1), and an nnode weighted network of low doubling dimension α ∈ O(loglog n), we present • A (1+ǫ)stretch labeled compact routing scheme with ⌈log n⌉bit routing labels, O(log 2 � n/log log n)bit packet headers, andbit routing information at each node; ( 1 ǫ)O(α) log 3 n • A (9 + ǫ)stretch nameindependent compact routing scheme with O(log 2 � n/log log n)bit packet headers, andbit routing information at each node. ( 1 ǫ)O(α) log 3 n In addition, we also prove a lower bound: any nameindependent routing scheme with o(n (ǫ/60)2) bits of storage at each node has stretch no less than 9 −ǫ, for any ǫ ∈ (0, 8). Therefore our nameindependent routing scheme achieves asymptotically optimal stretch with polylogarithmic storage at each node and packet headers. Note that both schemes are scalefree in the sense that their space requirements do not depend on the normalized diameter ∆ of the network. We also present a simpler nonscalefree (9 + ǫ)stretch nameindependent compact routing scheme with improved space requirements if ∆ is polynomial in n. 1
Approximating the distortion
 In APPROXRANDOM
, 2005
"... Abstract. Kenyon et al. (STOC 04) compute the distortion between onedimensional finite point sets when the distortion is small; Papadimitriou and Safra (SODA 05) show that the problem is NPhard to approximate within a factor of 3, albeit in 3 dimensions. We solve an open problem in these two paper ..."
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Cited by 8 (0 self)
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Abstract. Kenyon et al. (STOC 04) compute the distortion between onedimensional finite point sets when the distortion is small; Papadimitriou and Safra (SODA 05) show that the problem is NPhard to approximate within a factor of 3, albeit in 3 dimensions. We solve an open problem in these two papers by demonstrating that, when the distortion is large, it is hard to approximate within large factors, even for 1dimensional point sets. We also introduce additive distortion, and show that it can be easily approximated within a factor of two. 1
The Inframetric Model for the Internet
, 2007
"... Abstract—A large amount of algorithms has recently been designed for the Internet under the assumption that the distance defined by the roundtrip delay (RTT) is a metric. Moreover, many of these algorithms (e.g., overlay network construction, routing scheme design, sparse spanner construction) rely ..."
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Cited by 8 (3 self)
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Abstract—A large amount of algorithms has recently been designed for the Internet under the assumption that the distance defined by the roundtrip delay (RTT) is a metric. Moreover, many of these algorithms (e.g., overlay network construction, routing scheme design, sparse spanner construction) rely on the assumption that the metric has bounded ball growth or bounded doubling dimension. This paper analyzes the validity of these assumptions and proposes a tractable model matching experimental observations. On the one hand, based on Skitter data collected by CAIDA and King matrices of Meridian and P2PSim projects, we verify that the ball growth of the Internet, as well as its doubling dimension, can actually be quite large. Nevertheless, we observed that the doubling dimension is much smaller when restricting the measures to balls of large enough radius. Moreover, by computing the number of balls of radius r required to cover balls of radius R> r, we observed that this number grows with R much slower than what is predicted by a large doubling dimension. On the other hand, based on data collected on the PlanetLab platform by the AllSitesPings project, we confirm that the triangle inequality does not hold for a significant fraction of the nodes. Nevertheless, we demonstrate that RTT measures satisfy a weak version of the triangle inequality: there exists a small constant ρ such that for any triple u, v, w, we have RTT(u,v) ≤ ρ ·max{RTT(u,w), RTT(w,v)}. (Smaller bounds on ρ can even be obtained when the triple u, v, w is skewed). We call inframetric a distance function satisfying this latter inequality. Inframetrics subsume standard metrics and ultrametrics. Based on inframetrics and on our observations concerning the doubling dimension, we propose an analytical model for Internet RTT latencies. This model is tuned by a small set of parameters concerning the violation of the triangle inequality and the geometrical dimension of the network. We demonstrate the tractability of our model by designing a simple and efficient compact routing scheme with low stretch. Precisely, the scheme has constant multiplicative stretch and logarithmic additive stretch. I.
Additive Spanners and Distance and Routing Labeling Schemes for Hyperbolic Graphs
"... δHyperbolic metric spaces have been defined by M. Gromov in 1987 via a simple 4point condition: for any four points u, v, w, x, the two larger of the distance sums d(u, v) + d(w, x), d(u, w) + d(v, x),d(u, x) + d(v, w) differ by at most 2δ. They play an important role in geometric group theory, g ..."
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Cited by 4 (0 self)
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δHyperbolic metric spaces have been defined by M. Gromov in 1987 via a simple 4point condition: for any four points u, v, w, x, the two larger of the distance sums d(u, v) + d(w, x), d(u, w) + d(v, x),d(u, x) + d(v, w) differ by at most 2δ. They play an important role in geometric group theory, geometry of negatively curved spaces, and have recently become of interest in several domains of computer science, including algorithms and networking. For example, (a) it has been shown empirically that the internet topology embeds with better accuracy into a hyperbolic space than into an Euclidean space of comparable dimension, (b) every connected finite graph has an embedding in the hyperbolic plane so that the greedy routing based on the virtual coordinates obtained from this embedding is guaranteed to work. A connected graph G = (V, E) equipped with standard graph metric dG is δhyperbolic if the metric space (V, dG) is δhyperbolic. In this paper, using our Layering Partition technique, we provide a simpler construction of distance approximating trees of δhyperbolic graphs on n vertices with an additive error O(δ log n) and show that every nvertex δhyperbolic graph has an additive O(δ log n)spanner with at most O(δn) edges. As a consequence, we show that the family of δhyperbolic graphs with n vertices enjoys an O(δ log n)additive routing labeling scheme with O(δ log 2 n) bit labels and O(log δ) time routing protocol, and an easier constructable O(δ log n)additive distance labeling scheme with O(log 2 n) bit labels and constant time distance decoder.
On triangulation of simple networks
 In 19th Annual ACM Symposium on Parallel Algorithms and Architectures
, 2007
"... Network triangulation is a method for estimating distances between nodes in the network, by letting every node measure its distance to a few beacon nodes, and deducing the distance between every two nodes x, y by using their measurements to their common beacons and applying the triangle inequality. ..."
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Cited by 2 (1 self)
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Network triangulation is a method for estimating distances between nodes in the network, by letting every node measure its distance to a few beacon nodes, and deducing the distance between every two nodes x, y by using their measurements to their common beacons and applying the triangle inequality. Kleinberg, Slivkins and Wexler [FOCS 2004] initiated a theoretical study of triangulation in metric spaces, and Slivkins [PODC 2005] subsequently showed that metrics of bounded doubling dimension admit a triangulation that approximates arbitrarily well all pairwise distances using only O(log n) beacons per point, where n is the number of points in the network. He then asked whether this term is necessary (for doubling metrics). We provide the first lower bounds on the number of beacons required for a triangulation in some specific simple networks. In particular, these bounds (i) answer Slivkins ’ question positively, even for onedimensional metrics, and (ii) prove that, up to constants, Slivkins ’ triangulation achieves an optimal number of beacons (as a function of the approximation guarantee and the doubling dimension).
Metric Clustering via Consistent Labeling
, 2008
"... We design approximation algorithms for a number of fundamental optimization problems in metric spaces, namely computing separating and padded decompositions, sparse covers, and metric triangulations. Our work is the first to emphasize relative guarantees that compare the produced solution to the opt ..."
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Cited by 2 (2 self)
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We design approximation algorithms for a number of fundamental optimization problems in metric spaces, namely computing separating and padded decompositions, sparse covers, and metric triangulations. Our work is the first to emphasize relative guarantees that compare the produced solution to the optimal one for the input at hand. By contrast, the extensive previous work on these topics has sought absolute bounds that hold for every possible metric space (or for a family of metrics). While absolute bounds typically translate to relative ones, our algorithms provide significantly better relative guarantees, using a rather different algorithm. Our technical approach is to cast a number of metric clustering problems that have been well studied—but almost always as disparate problems—into a common modeling and algorithmic framework, which we call the consistent labeling problem. Having identified the common features of all of these problems, we provide a family of linear programming relaxations and simple randomized rounding procedures that achieve provably good approximation guarantees.
Meridian: A Lightweight Framework for Network Location without Virtual Coordinates
 In Proc. of ACM SIGCOMM
, 2005
"... Selecting nodes based on their position in the network is a basic building block for many distributed systems. This paper describes a peertopeer overlay network for performing positionbased node selection. Our system, Meridian, provides a lightweight, accurate and scalable framework for keeping t ..."
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Selecting nodes based on their position in the network is a basic building block for many distributed systems. This paper describes a peertopeer overlay network for performing positionbased node selection. Our system, Meridian, provides a lightweight, accurate and scalable framework for keeping track of location information for participating nodes. The framework consists of an overlay network structured around multiresolution rings, query routing with direct measurements, and gossip protocols for dissemination. We show how this framework can be used to address three commonly encountered problems in largescale distributed systems without having to compute absolute coordinates; namely, closest node discovery, central leader election, and locating nodes that satisfy target latency constraints. We show analytically that the framework is scalable with logarithmic convergence when Internet latencies are modeled as a growthconstrained metric, a lowdimensional Euclidian metric, or a metric of low doubling dimension. Large scale simulations, based on latency measurements from 6.25 million nodepairs, and an implementation deployed on PlanetLab both show that the framework is accurate and effective. 1
Fast, precise and dynamic distance queries
"... We present an approximate distance oracle for a point set S with n points and doubling dimension λ. For every ε> 0, the oracle supports (1 + ε)approximate distance queries in (universal) constant time, occupies space [ε −O(λ) + 2 O(λ log λ)]n, and can be constructed in [2 O(λ) log 3 n+ε −O(λ) +2 O( ..."
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We present an approximate distance oracle for a point set S with n points and doubling dimension λ. For every ε> 0, the oracle supports (1 + ε)approximate distance queries in (universal) constant time, occupies space [ε −O(λ) + 2 O(λ log λ)]n, and can be constructed in [2 O(λ) log 3 n+ε −O(λ) +2 O(λ log λ)]n expected time. This improves upon the best previously known constructions, presented by HarPeled and Mendel [13]. Furthermore, the oracle can be made fully dynamic with expected O(1) query time and only 2O(λ) log n+ε−O(λ) O(λ log λ) +2 update time. This is the first fully dynamic (1 + ε)distance oracle. 1
Research Statement  Aleksandrs Slivkins
"... ery incomplete. In the simplest setting, the data comes from a small number of random probes. The following central question emerges: how to make the best use of such data? Typically it is assumed that we can (at a higher cost) make more measurements, and we are interested in a tradeoff between the ..."
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ery incomplete. In the simplest setting, the data comes from a small number of random probes. The following central question emerges: how to make the best use of such data? Typically it is assumed that we can (at a higher cost) make more measurements, and we are interested in a tradeoff between the number of measurements and the accuracy of the resulting solution. In more complicated settings we may have more control over what measurements we actually make, which leads to another question: how to use this control intelligently given the data available so far? I explored these issues in several contexts related to estimating network distances: metric embeddings, triangulationbased distance estimation and network failure detection. Metric embeddings. The past decade has seen many significant and elegant results in the theory of metric embeddings. The goal here is to minimize distortion, which intuitively is the maximal factor by which distances are expanded or shrunk. Embedding techn
A Framework for Network LocationAware Node Selection
"... We introduce a lightweight, scalable and accurate framework for performing node selection based on network location. The framework, called Meridian, consists of an overlay network structured around multiresolution rings, gossip protocols for ring maintenance, and query routing with direct measureme ..."
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We introduce a lightweight, scalable and accurate framework for performing node selection based on network location. The framework, called Meridian, consists of an overlay network structured around multiresolution rings, gossip protocols for ring maintenance, and query routing with direct measurements to satisfy user specified latency constraints. We show how this framework can be used to address three commonly encountered problems, namely, closest node discovery, central leader election, and locating nodes that satisfy target latency constraints in largescale distributed systems without having to compute absolute coordinates. We also present the Meridian Query Language, a domain specific language for users to construct custom node selection queries based on their specific network location requirements. To facilitate adoption of Meridian, we have deployed a service called ClosestNode.com that provides a DNS to Meridian gateway for oblivious clients to initiate Meridian lookups. We show analytically that the framework is scalable with logarithmic convergence when Internet latencies are modeled as a growthconstrained metric, a lowdimensional Euclidean metric, or a metric of low doubling dimension. Large scale simulations, based on latency measurements from 6.25 million nodepairs as well as an implementation deployed on PlanetLab show that the framework is accurate and effective.