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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.
Local global tradeoffs in metric embeddings
 In Proceedings of the FortyEighth Annual IEEE Symposium on Foundations of Computer Science
, 2007
"... Suppose that every k points in a n point metric space X are Ddistortion embeddable into ℓ1. We give upper and lower bounds on the distortion required to embed the entire space X into ℓ1. This is a natural mathematical question and is also motivated by the study of relaxations obtained by liftandp ..."
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Cited by 10 (1 self)
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Suppose that every k points in a n point metric space X are Ddistortion embeddable into ℓ1. We give upper and lower bounds on the distortion required to embed the entire space X into ℓ1. This is a natural mathematical question and is also motivated by the study of relaxations obtained by liftandproject methods for graph partitioning problems. In this setting, we show that X can be embedded into ℓ1 with distortion O(D×log(n/k)). Moreover, we give a lower bound showing that this result is tight if D is bounded away from 1. For D = 1+δ we give a lower bound of Ω(log(n/k) / log(1/δ)); and for D = 1, we give a lower bound of Ω(log n/(log k + log log n)). Our bounds significantly improve on the results of Arora, Lovász, Newman, Rabani, Rabinovich and Vempala, who initiated a study of these questions.
Compact Routing with Slack in Low Doubling Dimension ABSTRACT
"... We consider the problem of compact routing with slack in networks of low doubling dimension. Namely, we seek nameindependent routing schemes with (1 + ɛ) stretch and polylogarithmic storage at each node: since existing lower bound precludes such a scheme, we relax our guarantees to allow for (i) a s ..."
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Cited by 6 (1 self)
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We consider the problem of compact routing with slack in networks of low doubling dimension. Namely, we seek nameindependent routing schemes with (1 + ɛ) stretch and polylogarithmic storage at each node: since existing lower bound precludes such a scheme, we relax our guarantees to allow for (i) a small fraction of nodes to have large storage, say size of O(n log n) bits, or (ii) a small fraction of sourcedestination pairs to have larger, but still constant, stretch. In this paper, given any constant ɛ ∈ (0, 1), any δ ∈ Θ(1 / polylog n) and any connected edgeweighted undirected graph G with doubling dimension α ∈ O(log log n) andarbitrary node names, we present
Additive Spanners and (α, β)Spanners
"... An (α, β)spanner of an unweighted graph G is a subgraph H that distorts distances in G up to a multiplicative factor of α and an additive term β. It is well known that any graph contains a (multiplicative) (2k − 1, 0)spanner of size O(n 1+1/k) and an (additive) (1, 2)spanner of size O(n 3/2). How ..."
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Cited by 6 (2 self)
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An (α, β)spanner of an unweighted graph G is a subgraph H that distorts distances in G up to a multiplicative factor of α and an additive term β. It is well known that any graph contains a (multiplicative) (2k − 1, 0)spanner of size O(n 1+1/k) and an (additive) (1, 2)spanner of size O(n 3/2). However no other additive spanners are known to exist. In this paper we develop a couple of new techniques for constructing (α, β)spanners. Our first result is an additive (1, 6)spanner of size O(n 4/3). The construction algorithm can be understood as an economical agent that assigns costs and values to paths in the graph, purchasing affordable paths and ignoring expensive ones, which are intuitively wellapproximated by paths already purchased. We show that this path buying algorithm can be parameterized in different ways to yield other sparsenessdistortion tradeoffs. Our second result addresses the problem of which (α, β)spanners can be computed efficiently, ideally in linear time. We show that for any k, a (k, k − 1)spanner with size O(kn 1+1/k) can be found in linear time, and further, that in a distributed network the algorithm terminates in a constant number of rounds. Previous spanner constructions with similar performance had roughly twice the multiplicative distortion.
Algorithms and Models for Problems in Networking
, 2010
"... Many interesting theoretical problems arise from computer networks. In this thesis we will consider three of them: algorithms and data structures for problems involving distances in networks (in particular compact routing schemes, distance labels, and distance oracles), algorithms for wireless capac ..."
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Many interesting theoretical problems arise from computer networks. In this thesis we will consider three of them: algorithms and data structures for problems involving distances in networks (in particular compact routing schemes, distance labels, and distance oracles), algorithms for wireless capacity and scheduling problems, and algorithms for optimizing iBGP overlays in autonomous systems on the Internet. While at first glance these problems may seem extremely different, they are similar in that they all attempt to look at a previously studied networking problem in new, more realistic frameworks. In other words, they are all as much about new models for old problems as they are about new algorithms. In this thesis we will define these models, design algorithms for them, and prove hardness and impossibility results for these three types of problems. viAcknowledgments This thesis would have been impossible without the guidance of my advisor, Anupam Gupta. While we may not have written many papers together, he has been an invaluable mentor who always has good ideas and interesting thoughts. I was