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EMBEDDING METRICS INTO ULTRAMETRICS AND GRAPHS INTO SPANNING TREES WITH CONSTANT AVERAGE DISTORTION∗
"... Abstract. This paper addresses the basic question of how well a tree can approximate distances of a metric space or a graph. Given a graph, the problem of constructing a spanning tree in a graph which strongly preserves distances in the graph is a fundamental problem in network design. We present sc ..."
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( 1/). For the graph setting, we prove that any weighted graph contains a spanning tree with scaling distortion O( 1/). These bounds are tight even for embedding into arbitrary trees. These results imply that the average distortion of the embedding is constant and that the 2 distortion is O( logn
Embedding metrics into ultrametrics and graphs into spanning trees with constant average distortion, 2006. Arxiv
"... This paper addresses the basic question of how well can a tree approximate distances of a metric space or a graph. Given a graph, the problem of constructing a spanning tree in a graph which strongly preserves distances in the graph is a fundamental problem in network design. We present scaling dist ..."
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Cited by 18 (9 self)
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constant average distortion and O ( √ log n) ℓ2distortion. This follows from the following results: we prove that any metric space embeds into an ultrametric with scaling distortion O ( √ 1/ǫ). For the graph setting we prove that any weighted graph contains a spanning tree with scaling distortion O
Embedding ultrametrics into lowdimensional spaces
 In 22nd Annual ACM Symposium on Computational Geometry
, 2006
"... We study the problem of minimumdistortion embedding of ultrametrics into the plane and higher dimensional spaces. Ultrametrics are a natural class of metrics that frequently occur in applications involving hierarchical clustering. Lowdistortion embeddings of ultrametrics into the plane help visual ..."
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Cited by 6 (6 self)
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We study the problem of minimumdistortion embedding of ultrametrics into the plane and higher dimensional spaces. Ultrametrics are a natural class of metrics that frequently occur in applications involving hierarchical clustering. Lowdistortion embeddings of ultrametrics into the plane help
Ultrametric Subsets with . . .
"... It is shown that for every ε ∈ (0, 1), every compact metric space (X, d) has a compact subset S ⊆ X that embeds into an ultrametric space with distortion O(1/ε), and dimH(S) � (1 − ε) dimH(X), where dimH(·) denotes Hausdorff dimension. The above O(1/ε) distortion estimate is shown to be sharp via ..."
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It is shown that for every ε ∈ (0, 1), every compact metric space (X, d) has a compact subset S ⊆ X that embeds into an ultrametric space with distortion O(1/ε), and dimH(S) � (1 − ε) dimH(X), where dimH(·) denotes Hausdorff dimension. The above O(1/ε) distortion estimate is shown to be sharp via
Contour Detection and Hierarchical Image Segmentation
 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE
, 2010
"... This paper investigates two fundamental problems in computer vision: contour detection and image segmentation. We present stateoftheart algorithms for both of these tasks. Our contour detector combines multiple local cues into a globalization framework based on spectral clustering. Our segmentati ..."
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Cited by 383 (23 self)
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segmentation algorithm consists of generic machinery for transforming the output of any contour detector into a hierarchical region tree. In this manner, we reduce the problem of image segmentation to that of contour detection. Extensive experimental evaluation demonstrates that both our contour detection
Lowdistortion Embeddings of General Metrics Into the
"... A lowdistortion embedding between two metric spaces is a mapping which preserves the distances between each pair of points, up to a small factor called distortion. Lowdistortion embeddings have recently found numerous applications in computer science. Most of the known embedding results are ”absol ..."
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. In this paper we show algorithms and hardness results for relative embedding problems. In particular we give: • an algorithm that, given a general metric M, finds an embedding with distortion O( ∆ 3/4 poly(cline(M))), where ∆ is the spread of M • an algorithm that, given a weighted tree metric M, finds
Online, Dynamic, and Distributed Embeddings of Approximate Ultrametrics
"... Abstract. The theoretical computer science community has traditionally used embeddings of finite metrics as a tool in designing approximation algorithms. Recently, however, there has been considerable interest in using metric embeddings in the context of networks to allow network nodes to have more ..."
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Cited by 1 (0 self)
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observation has motivated the study of these special metrics, including strong results about embeddings into trees and ultrametrics. Unfortunately all of the current embeddings require complete knowledge about the network up front, and so are less useful in real networks which change frequently. We give
Results 1  10
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