Bounded geometries, fractals, and low-distortion embeddings

Gupta, Anupam; Krauthgamer, Robert; Lee, J.R.

doi:10.1109/sfcs.2003.1238226

Cited by 283 publications

(425 citation statements)

References 39 publications

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“…Since the construction is folklore (see e.g. [22,9,8]), we will only give a brief overview, demonstrating that the embedding can be indeed computed in an on-line fashion.…”

Section: Embedding General Metrics Into Ultrametrics and Into Pmentioning

confidence: 99%

Online Embeddings

Indyk

Magen

Sidiropoulos

et al. 2010

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques

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We initiate the study of on-line metric embeddings. In such an embedding we are given a sequence of n points X = x 1 , . . . , x n one by one, from a metric space M = (X, D). Our goal is to compute a low-distortion embedding of M into some host space, which has to be constructed in an on-line fashion, so that the image of each x i depends only on x 1 , . . . , x i . We prove several results translating existing embeddings to the on-line setting, for the case of embedding into p spaces, and into distributions over ultrametrics.

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“…Since the construction is folklore (see e.g. [22,9,8]), we will only give a brief overview, demonstrating that the embedding can be indeed computed in an on-line fashion.…”

Section: Embedding General Metrics Into Ultrametrics and Into Pmentioning

confidence: 99%

Online Embeddings

Indyk

Magen

Sidiropoulos

et al. 2010

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques

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“…This result is particularly applicable to "growth-bounded" graphs [8,21] which are graphs for which the number of nodes within distance r from any node grows much smaller than ∆ r (typically, as a polynomial in r). Thus, even for graphs with exponential growth (the number of nodes at a distance r can be poly(∆, 2 r )), we still obtain an O(log ∆) bound on the frugality.…”

Section: Lemmamentioning

confidence: 99%

The Randomized Coloring Procedure with Symmetry-Breaking

Pemmaraju

Srinivasan

2008

Automata, Languages and Programming

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Abstract. A basic randomized coloring procedure has been used in probabilistic proofs to obtain remarkably strong results on graph coloring. These results include the asymptotic version of the List Coloring Conjecture due to Kahn, the extensions of Brooks' Theorem to sparse graphs due to Kim and Johansson, and Luby's fast parallel and distributed algorithms for graph coloring. The most challenging aspect of a typical probabilistic proof is showing adequate concentration bounds for key random variables. In this paper, we present a simple symmetry-breaking augmentation to the randomized coloring procedure that works well in conjunction with Azuma's Martingale Inequality to easily yield the requisite concentration bounds. We use this approach to obtain a number of results in two areas: frugal coloring and weighted equitable coloring. A β-frugal coloring of a graph G is a proper vertex-coloring of G in which no color appears more than β times in any neighborhood. Let G = (V, E) be a vertex-weighted graph with weight function w : V → [0, 1] and let W = v∈V w(v). A weighted equitable coloring of G is a proper k-coloring such that the total weight of every color class is "large", i.e., "not much smaller" than W/k; this notion is useful in obtaining tail bounds for sums of dependent random variables.

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“…| for a constant k 1 ; and in [14], a metric has bounded doubling dimension k 2 if B(v, 2r) is contained in the union of at most k 2 balls with radius r for a constant k; in [17,13], a metric has upper bounded growth rate growth rate k 3 if for every p ∈ V and every r ≥ 1, |B(p, r)| ≤ ρr k3 , for a constant ρ and k 3 . A few sensor network papers [24,29] consider a model when the growth rate is both upper and lower bounded, i.e., ρ − r k4 ≤ |B(p, r)| ≤ ρ + r k4 for a constant k 4 , where ρ − ≤ ρ + are two constants.…”

Section: Preliminariesmentioning

confidence: 99%

“…It is not hard to see that M growth ⊆ M expansion ⊆ M doubling ⊆ M + growth . See [14,13] for more discussions. In terms of the results in this paper the detailed definitions actually matter.…”

Section: Preliminariesmentioning

confidence: 99%

Resilient and Low Stretch Routing through Embedding into Tree Metrics

Gao

Zhou

2011

Lecture Notes in Computer Science

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Abstract. Given a network, the simplest routing scheme is probably routing on a spanning tree. This method however does not provide good stretch -the route between two nodes can be much longer than their shortest distance, nor does it give good resilience -one node failure may disconnect quadratically many pairs. In this paper we use two trees to achieve both constant stretch and good resilience. Given a metric (e.g., as the shortest path metric of a given communication network), we build two hierarchical well-separated trees using randomization such that for any two nodes u, v, the shorter path of the two paths in the two respective trees gives a constant stretch of the metric distance of u, v, and the removal of any node only disconnect the routes between O(1/n) fraction of all pairs. Both bounds are in expectation and hold true as long as the metric follows certain geometric growth rate (the number of nodes within distance r is a polynomial function of r), which holds for many realistic network settings such as wireless ad hoc networks and Internet backbone graphs. The algorithms have been implemented and tested on real data.

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Bounded geometries, fractals, and low-distortion embeddings

Cited by 283 publications

References 39 publications

Online Embeddings

Online Embeddings

The Randomized Coloring Procedure with Symmetry-Breaking

Resilient and Low Stretch Routing through Embedding into Tree Metrics

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