On metric ramsey-type phenomena

Bartal, Yair; Linial, Nathan; Mendel, Manor

doi:10.1145/780606.780610

Cited by 61 publications

(161 citation statements)

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“…(Recall that a metric space (X, d X ) is said to embed into Hilbert space with distortion α if there exists a mapping f : X → L 2 such that for every x, y ∈ X, we have d X (x, y) ≤ f (x) − f (y) 2 ≤ αd X (x, y)). This problem has since been investigated by several authors, motivated in part by the discovery of its applications to online algorithms-we refer to [5] for a discussion of the history and applications of the metric Ramsey problem.…”

Section: Introductionmentioning

confidence: 99%

“…The most recent work on the metric Ramsey problem is due to Bartal, Linial, Mendel and Naor [5], who obtained various nearly optimal upper and lower bounds in several contexts. Among the results in [5] is the following theorem which deals with the case of large distortion: For every ε ∈ (0, 1), any n-point metric space has a subset of size n 1−ε which embeds into an ultrametric space with distortion O(log(2/ε)/ε) (recall that an ultrametric space (X, d X ) is a metric space satisfying d X (x, y) ≤ max{d X (x, z), d X (y, z)} for every x, y, z ∈ X). Since ultrametric spaces embed isometrically into Hilbert space, this is indeed a metric Ramsey theorem.…”

Section: Introductionmentioning

confidence: 99%

“…Since ultrametric spaces embed isometrically into Hilbert space, this is indeed a metric Ramsey theorem. Moreover, it was shown in [5] that this result is optimal up to the log(2/ε) factor, i.e. there exist arbitrarily large n-point metric spaces, every subset of which of size n 1−ε incurs distortion (1/ε) in any embedding into Hilbert space.…”

Section: Introductionmentioning

confidence: 99%

“…In the four years that elapsed since our work on [5] there has been remarkable development in the structure theory of finite metric spaces. In particular, the theory of random partitions of metric spaces has been considerably refined, and was shown to have numerous applications in mathematics and computer science (see for example [17,25,24,1] and the references therein).…”

Section: Introductionmentioning

confidence: 99%

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Ramsey partitions and proximity data structures

Mendel¹,

Naor²

2007

J. Eur. Math. Soc.

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174

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This paper addresses two problems lying at the intersection of geometric analysis and theoretical computer science: The non-linear isomorphic Dvoretzky theorem and the design of good approximate distance oracles for large distortion. We introduce the notion of Ramsey partitions of a finite metric space, and show that the existence of good Ramsey partitions implies a solution to the metric Ramsey problem for large distortion (also known as the non-linear version of the isomorphic Dvoretzky theorem, as introduced by Bourgain, Figiel, and Milman in [8]). We then proceed to construct optimal Ramsey partitions, and use them to show that for every ε ∈ (0, 1), every n-point metric space has a subset of size n 1−ε which embeds into Hilbert space with distortion O(1/ε). This result is best possible and improves part of the metric Ramsey theorem of Bartal, Linial, Mendel and Naor [5], in addition to considerably simplifying its proof. We use our new Ramsey partitions to design approximate distance oracles with a universal constant query time, closing a gap left open by Thorup and Zwick in [32]. Namely, we show that for every n-point metric space X, and k ≥ 1, there exists an O(k)-approximate distance oracle whose storage requirement is O(n 1+1/k), and whose query time is a universal constant. We also discuss applications of Ramsey partitions to various other geometric data structure problems, such as the design of efficient data structures for approximate ranking.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Ramsey partitions and proximity data structures

Mendel¹,

Naor²

2007

J. Eur. Math. Soc.

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174

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“…The OCO problem is a special case of the classic metrical task system problem, where both the metric space and the cost functions can be arbitrary. The optimal deterministic competitive ratio for a general metrical task system is 2n − 1, where n is the number of points in the metric [23], and the optimal randomized competitive ratio is Ω(log n/ log log n) [20,21], and O(log 2 n log log n) [30].…”

Section: Looking Backwardmentioning

confidence: 99%