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## Approximation Algorithms for Low-Distortion Embeddings Into Low-Dimensional Spaces

### Citations

634 | Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis. - Kruskal - 1964 |

521 | The geometry of graphs and some of its algorithmic applications
- Linial, London, et al.
- 1995
(Show Context)
Citation Context ...us, the problems are fundamentally different, which raises new interesting issues. Despite the differences, we mention two combinatorial results that are relevant in our context. The first one is the =-=[LLR94]-=- adaptation of Bourgain’s construction [Bou85] that enables embedding of an arbitrary metric into l O(log2 n) 2 with maximum multiplicative distortion O(log n). It should be noted, however, that for t... |

440 | Nonmetric multidimensional scaling: A numerical method. - Kruskal - 1964 |

397 | The analysis of proximities: multidimensional scaling with an unknown distance function, - Shepard - 1962 |

328 |
On Lipschitz embeddings of finite metric spaces in Hilbert space
- Bourgain
- 1985
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Citation Context ...which raises new interesting issues. Despite the differences, we mention two combinatorial results that are relevant in our context. The first one is the [LLR94] adaptation of Bourgain’s construction =-=[Bou85]-=- that enables embedding of an arbitrary metric into l O(log2 n) 2 with maximum multiplicative distortion O(log n). It should be noted, however, that for the applications mentioned earlier, the most in... |

126 | Algorithmic applications of low-distortion geometric embeddings.
- Indyk
- 2001
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Citation Context ... long a subject of extensive mathematical studies. During the last few years, such embeddings found multiple and diverse uses in computer science as well; many such applications have been surveyed in =-=[Ind01]-=-. However, the problems addressed in this paper are fundamentally different from those investigated in the aforementioned literature. In a nutshell, our problems are algorithmic, as opposed to combina... |

103 | Approximating the bandwidth via volume respecting embeddings - FEIGE |

91 | On the approximability of numerical taxonomy (fitting distances by tree metrics), - Agarwala, Bafna, et al. - 1999 |

82 | A robust model for finding optimal evolutionary trees
- Farach, Kannan, et al.
- 1995
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Citation Context ...path that led to formation of the genetic sequences. Motivated by these applications M. Farach-Colton and S. Kannan show how to find an ultrametric T with minimum possible maximum additive distortion =-=[FCKW93]-=-. There is also a 3-approximation algorithm for the case of embedding arbitrary metrics into weighted tree metrics to minimize the maximum additive two-sided error [ABFC+96]. [Dha04] recently gave an ... |

76 |
Drei Satze uber die n-dimensionale euklidische
- Borsuk
- 1933
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Citation Context ...neral, we show that an n-point spherical metric can be embedded with distortion O( √ n), and this bound is optimal in the worst case. (The lower bound is shown by resorting to the Borsuk-Ulam theorem =-=[Bor33]-=-, which roughly states that any continuous mapping from S2 into the plane maps two antipodes of S2 into the same point.) For the algorithmic problem of embedding M into the plane, we give a 3.512-appr... |

46 | K-ary Clustering with Optimal Leaf Ordering for Gene Expression Data - Bar-Joseph, Demaine, et al. - 2002 |

31 |
Uber die zusammenziehenden und Lipschitzchen Transformationen.
- Kirszbraun
- 1934
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Citation Context ... f : X → R2 has distortion Ω(√n). Proof. Let X ⊂ S2 be a set of n points, such that X is a O(1/ √ n)-net of S2, and let f : X → R2 be a nonexpanding embedding. Since S2 ⊂ R3, by Kirszbraun’s Theorem (=-=[Kir34]-=-, see also [LN]), we obtain that f can be extended to a non-expanding mapping f ′ : S2 → R2. Also, by the Borsuk-Ulam Theorem, it follows that there exist antipodals p, q ∈ S2, such that f ′(p) = f ′(... |

30 | Bi-Lipschitz embeddings into low dimensional Euclidean spaces
- Matousek
- 1990
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Citation Context ...or computing the distortion is based on the analogous algorithm for the bandwidth problem by Saxe [Sax80]. 2 A c-approximation algorithm We start by stating an algorithmic version of a fact proved in =-=[Mat90]-=-. Lemma 2.1. Any shortest path metric over an unweighted graph G = (V,E) can be embedded into a line with distortion at most 2n− 1 in time O(|V |+ |E|). Proof. Let T be a spanning tree of the graph. W... |

27 |
Low distortion maps between point sets.
- Kenyon, Rabani, et al.
- 2004
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Citation Context ...e computed using the l1 norm [B0̆3]. Bădoiu, Indyk and Rabinovich [BIR03] gave a weaklyquasi-polynomial time algorithm for the same problem in the l2 norm. Very recently, Kenyon, Rabani and Sinclair =-=[KRS04]-=- gave exact algorithms for minimum (multiplicative) distortion embeddings of metrics onto simpler metrics (e.g., line metrics). Their algorithms work as long as the minimum distortion is small, e.g., ... |

27 |
On the distortion required for embedding finite metric spaces into normed spaces
- Matousek
- 1996
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Citation Context ...paces happen to be lowdimensional. Similarly, any metric can be embedded into d-dimensional Euclidean space with multiplicative distortion O(min[n 2 d log3/2 n, n]) and no better than Ω(n1/b(d+1)/2c) =-=[Mat96]-=-. However, the worst-case guarantees are rather large for small d, especially for the case d = 1 that we consider here. Previous Work on the Algorithmic Problem. To our knowledge there have been few a... |

20 | Approximation algorithm for embedding metrics into a two-dimensional space - BĂDOIU |

20 |
Dynamic-programming algorithms for recognizing small-bandwidth graphs in polynomial time
- Saxe
- 1980
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Citation Context ...in the case of exact algorithms the situation is quite different. In particular, our exact algorithm for computing the distortion is based on the analogous algorithm for the bandwidth problem by Saxe =-=[Sax80]-=-. 2 A c-approximation algorithm We start by stating an algorithmic version of a fact proved in [Mat90]. Lemma 2.1. Any shortest path metric over an unweighted graph G = (V,E) can be embedded into a li... |

10 | Improved bandwidth approximation for trees - Gupta - 2000 |

10 | Fitting points on the real line and its application to rh mapping - Hastad, Ivansson, et al. - 1461 |

5 |
Approximate algorithms for embedding metrics into lowdimensional spaces
- BĂDOIU, INDYK, et al.
- 2003
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Citation Context ...oiu extended the algorithm to the 2-dimensional plane with maximum two-sided additive error when the distances in the target plane are computed using the l1 norm [B0̆3]. Bădoiu, Indyk and Rabinovich =-=[BIR03]-=- gave a weaklyquasi-polynomial time algorithm for the same problem in the l2 norm. Very recently, Kenyon, Rabani and Sinclair [KRS04] gave exact algorithms for minimum (multiplicative) distortion embe... |

5 | Approximating additive distortion of embeddings into line metrics
- Dhamdhere
- 2004
(Show Context)
Citation Context ...itive distortion [FCKW93]. There is also a 3-approximation algorithm for the case of embedding arbitrary metrics into weighted tree metrics to minimize the maximum additive two-sided error [ABFC+96]. =-=[Dha04]-=- recently gave an O(log1/p n)- approximation for embedding arbitrary n-point metrics into the line to minimize the `p norm of the two-sided error vector | |f(u)− f(v)| − D(u, v)|. Distortion vs Bandwi... |

5 |
Absolute lipschitz extendability. Comptes Rendus de l’Académie des
- Lee, Naor
(Show Context)
Citation Context ...stortion Ω(√n). Proof. Let X ⊂ S2 be a set of n points, such that X is a O(1/ √ n)-net of S2, and let f : X → R2 be a nonexpanding embedding. Since S2 ⊂ R3, by Kirszbraun’s Theorem ([Kir34], see also =-=[LN]-=-), we obtain that f can be extended to a non-expanding mapping f ′ : S2 → R2. Also, by the Borsuk-Ulam Theorem, it follows that there exist antipodals p, q ∈ S2, such that f ′(p) = f ′(q). Since X is ... |

4 | Computational aspects of radiation hybrid. Doctoral Dissertation - Ivansson - 2000 |