## Finite Metric Spaces - Combinatorics, Geometry and Algorithms (2002)

Venue: | In Proceedings of the International Congress of Mathematicians III |

Citations: | 48 - 2 self |

### BibTeX

@INPROCEEDINGS{Linial02finitemetric,

author = {Nathan Linial},

title = {Finite Metric Spaces - Combinatorics, Geometry and Algorithms},

booktitle = {In Proceedings of the International Congress of Mathematicians III},

year = {2002},

pages = {573--586}

}

### Years of Citing Articles

### OpenURL

### Abstract

This article deals only with what might be called the geometrization of combinatorics. Namely, the idea that viewing combinatorial objects from a geometric perspective often yields unexpected insights. Even more concretely, we concentrate on finite metric spaces and their embeddings

### Citations

451 | The Geometry of Graphs and Some of its Algorithmic Applications
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- 1995
(Show Context)
Citation Context ...andomly chosen k-regular graph has expansion > k=10 tends to 1 as the number of vertices n tends to 1. It turns out that the metrics of expander graphs are as far from l 2 as possible. 2 Theorem 3 ([=-=L-=-LR95], see also [Mat97, LM00]). Let G be an n-vertex k-regular -expander graph (k 3, > 0). Then c 2 (G) c log n where c depends only on k and . Metric geometry is by no means a new subject, and ind... |

405 |
Extensions of Lipschitz mappings into a Hilbert space
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Citation Context ...practitioners in these areas often speak about the curse of dimensionality when they refer to this problem. In l 2 there is a basic result that answers this problem. Theorem 10 (Johnson Lindenstrauss =-=[JL-=-84]). Every n-point metric in l 2 can be embedded into l k 2 with distortions1 + where k O( log n 2 ). Here, again, the proof yields an ecient randomized algorithm. Namely, select a random kdimensi... |

330 | Eigenvalues and expanders
- Alon
- 1986
(Show Context)
Citation Context ...he same vectors happen to be also the eigenvectors of Q and all have nonnegative eigenvalues. As another application of this method (also from [LM00]), here is a quick proof of Theorem 3. It is known =-=[Alo8-=-6] that if G is a k-regular -expander graph and A is G's adjacency matrix, then the second eigenvalue of A issksfor somesthat depends on k and , but not on the size of the graph 4 . It is not hard to ... |

253 | On approximating arbitrary metrices by tree metrics
- Bartal
- 1998
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Citation Context ...c = 3. Open Problem 2. Is it true that bw(G) O((G) log n)? It is not hard to see that this bound would be tight for expanders. 4.3 Bartal's method The following general structure theorem of Bartal [B=-=ar98]-=- has numerous algorithmic applications: Theorem 9. For everysnite metric space (X; d) there is a collection of trees fT i j i 2 Ig, each of which has X as its set of leaves, and positive weights fp i ... |

206 |
A lower bound for the smallest eigenvalue of the laplacian. Problems in analysis (Papers dedicated to
- Cheeger
- 1969
(Show Context)
Citation Context ...orem 2, such graphs show that this conjecture, if true, is best possible. Recently, the following was shown: 4 A'ssrst eigenvalue is clearly k. This is the combinatorial analogue of Cheeger's Theorem =-=[Che7-=-0] about the spectrum of the Laplacian. 5 Theorem 7 ([LMN]). Let G be a k-regular graph k 3 with girth g. Then c 2 (G) sp g). Two proofs of this theorem are given in [LMN]. One is based on the notion... |

175 |
Theory and Applications of Distance Geometry
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Citation Context ... > 0). Then c 2 (G) c log n where c depends only on k and . Metric geometry is by no means a new subject, and indeed metrics that embed isometrically into l 2 were characterized long ago (see e.g. [B=-=lu70-=-]). This is a special case of the more recent results. Let ' : X ! l n 2 be an embedding. The condition that distortion(') c can be expressed as a system of linear inequalities in the entries of the ... |

161 |
Lectures in Discrete Geometry
- Matousek
- 2002
(Show Context)
Citation Context ...n a recent meeting (Haifa, March '02), a list of open problems in this area has been collected, see http://www.kam.m.cuni.cz/~matousek/haifaop.ps. More extensive surveys of this area can be found in [=-=Mat02-=-] Chapter 15, and [Ind01]. In view of this description, it should not come as a surprise to the reader that this theory is characterized as being Asymptotic: We are mostly interested in analyzing lar... |

120 | Algorithmic Applications of Low-distortion Geometric Embeddings
- Indyk
- 2001
(Show Context)
Citation Context ...a, March '02), a list of open problems in this area has been collected, see http://www.kam.m.cuni.cz/~matousek/haifaop.ps. More extensive surveys of this area can be found in [Mat02] Chapter 15, and [=-=Ind01-=-]. In view of this description, it should not come as a surprise to the reader that this theory is characterized as being Asymptotic: We are mostly interested in analyzing large,snite metric spaces, ... |

104 | Excluded minors, network decomposition, and multicommodity flow
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- 1993
(Show Context)
Citation Context ...ent vertices are very likely to reside in the same part. Such partitions have proved instrumental in the design of many algorithms. In fact, an important tool in Rao's Theorem 6 was an earlier result =-=[KPR93]-=- about the existence of very sparse partitions for the members of any minor-closed families family of graphs. 5 The mysterious l 1 We know much less about metric embeddings into l 1 , and the attempts... |

92 | Approximating the bandwidth via volume respecting embeddings
- Feige
- 1998
(Show Context)
Citation Context ...ere the minimum is over all 1 : 1 mapss: V ! f1; : : : ; ng. It is NP -hard to compute this parameter, and for many years no decent approximation algorithm was known. However, a recent paper by Feige =-=[Fei00-=-] provides a polylogarithmic approximation for the bandwidth. The statement of his algorithm is simple enough to be recorded here: 1. Compute (a slight modication of) the embedding ' : G ! l 2 that ap... |

66 | On metric Ramsey-type phenomena - Bartal, Linial, et al. - 2004 |

57 |
Some results on convex bodies and Banach spaces
- Dvoretzky
- 1960
(Show Context)
Citation Context ...ped linear theory. See [MS86] for an introduction to thisseld and [BL00] for a comprehensive cover of the nonlinear theory. The grandfather of the linear theory is the celebrated theorem of Dvoretzky =-=[D-=-vo61]. Theorem 1 (Dvoretzky). For every n and > 0, every n-dimensional normed space contains a k =s 2 log n)-dimensional space whose Banach-Mazur distance from l 2 is 1 + . Thus, among embeddings i... |

50 |
Markov chains, Riesz transforms and Lipschitz maps
- Ball
- 1992
(Show Context)
Citation Context ...aplacian. 5 Theorem 7 ([LMN]). Let G be a k-regular graph k 3 with girth g. Then c 2 (G) sp g). Two proofs of this theorem are given in [LMN]. One is based on the notion of Markov Type due to Ball [B=-=al92-=-]. The underlying idea of this proof is that a random walk on a graph with girth g and all vertex degree 3 drifts at a constant speed away from its starting point for time g). On the other hand, in a... |

46 |
distortion and volume preserving embeddings for planar and Euclidean metrics
- Small
- 1999
(Show Context)
Citation Context ... that this bound is attained for complete binary trees. (See [LS] for an elementary proof of this.) 3.2 Planar graphs It turns out that the metrics of planar graphs have good embedding into l 2 . Rao =-=[Rao99]-=- showed: Theorem 6. Every planar graph embeds in l 2 with distortion O( p log n). A recent construction of Newman and Rabinovich [NR02] shows that this bound is tight. 3.3 Graphs of high girth The gir... |

36 | A Lower Bound on the Distortion of Embedding Planar Metrics into Euclidean Space
- Newman, Rabinovich
- 2002
(Show Context)
Citation Context ...the metrics of planar graphs have good embedding into l 2 . Rao [Rao99] showed: Theorem 6. Every planar graph embeds in l 2 with distortion O( p log n). A recent construction of Newman and Rabinovich =-=[NR02]-=- shows that this bound is tight. 3.3 Graphs of high girth The girth of a graph is the length of the shortest cycle in the graph. If you restrict your attention (as we do in this section) to graphs in ... |

29 | On embedding trees into uniformly convex Banach spaces
- Matou˘sek
- 1999
(Show Context)
Citation Context ...m inequality (e.g. [DL97]). It is also not hard to see that every tree metric embeds isometrically into l 1 . They can also be embedded into l 2 with a relatively low distortion. Theorem 5 (Matousek [=-=Mat99-=-]). Every tree on n vertices can be embedded into l 2 with distortions O( p log log n). Bourgain [Bou86] had earlier shown that this bound is attained for complete binary trees. (See [LS] for an eleme... |

19 | On Euclidean embeddings and bandwidth minimization, in
- Dunagan, Vempala
(Show Context)
Citation Context ...ices in G at distance r from x. It's easy to see that bw(G) s(G)) and an interesting feature of Feige's proof is that it shows that bw(G) O((G) log c n). His paper gives c = 3:5 which was later [DV9=-=-=-9] improved to c = 3. Open Problem 2. Is it true that bw(G) O((G) log n)? It is not hard to see that this bound would be tight for expanders. 4.3 Bartal's method The following general structure theor... |

17 | Cuts, trees and `1-embeddings of graphs
- Gupta, Newman, et al.
- 1999
(Show Context)
Citation Context ...r graph embeds into l 1 with distortionsC? Even more daringly, the same can be asked for every minor-closed family of graphs. Some initial success for smaller graph families has been achieved already =-=[GNRS9-=-9]. 5.3 Large Girth Is there an analogue of Theorem 7 for embeddings into l 1 ? Open Problem 7. How small can c 1 (G) be for a a graph G of girth g in which all vertices have degree 3? Specically, ca... |

17 | The euclidean distortion of complete binary trees
- LINIAL, SAKS
- 2003
(Show Context)
Citation Context ...Matousek [Mat99]). Every tree on n vertices can be embedded into l 2 with distortions O( p log log n). Bourgain [Bou86] had earlier shown that this bound is attained for complete binary trees. (See [L=-=S]-=- for an elementary proof of this.) 3.2 Planar graphs It turns out that the metrics of planar graphs have good embedding into l 2 . Rao [Rao99] showed: Theorem 6. Every planar graph embeds in l 2 with ... |

15 |
Deza and Monique Laurent. Geometry of cuts and metrics, volume 15 of Algorithms and Combinatorics
- Marie
- 1997
(Show Context)
Citation Context ...og n. This often applies as well to graphs with arbitrary nonnegative edge lengths. 3.1 Trees The metrics of trees are quite restricted. They can be characterized through a four-term inequality (e.g. =-=[DL97-=-]). It is also not hard to see that every tree metric embeds isometrically into l 1 . They can also be embedded into l 2 with a relatively low distortion. Theorem 5 (Matousek [Mat99]). Every tree on n... |

13 |
On lipschitz embedding of metric spaces in hilbert space
- Bourgain
- 1985
(Show Context)
Citation Context ...ce whose Banach-Mazur distance from l 2 is 1 + . Thus, among embeddings into normed spaces, embeddings into l 2 are the hardest to come by. We begin our story with an important theorem of Bourgain [B=-=ou85-=-]. Theorem 2. Every n-point metric space 1 embeds in l 2 with distortion O(log n). Not only is this a fundamental result, Bourgain's proof of the theorem readily translates into an ecient randomized ... |

13 | Least-distortion Euclidean embeddings of graphs: products of cycles and expanders
- Linial, Magen, et al.
(Show Context)
Citation Context ...e well known, namely, they are the 2 r Walsh functions. The same vectors happen to be also the eigenvectors of Q and all have nonnegative eigenvalues. As another application of this method (also from =-=[LM00-=-]), here is a quick proof of Theorem 3. It is known [Alo86] that if G is a k-regular -expander graph and A is G's adjacency matrix, then the second eigenvalue of A issksfor somesthat depends on k and ... |

9 | On embedding expanders into l p spaces - Matousek - 1997 |

9 |
Combinatorial Optimization: polyhedra and eciency
- Schrijver
(Show Context)
Citation Context ...applications it has to the design of new algorithms. 4.1 Multicommoditysow and Sparsest cuts Flows in networks are a classical subject in discrete optimization and a topic of many investigations (see =-=[Sch02-=-] for a comprehensive coverage). You are given a network i.e., a graph with two specied vertices: The source s and the sink t. Edges have nonnegative capacities. The objective is to ship as much of a ... |

6 |
An O(log k) approximate min-cut max- theorem and approximation algorithm
- Aumann, Rabani
- 1998
(Show Context)
Citation Context ... unit capacities and every pair of vertices form a source-sink pair with a unit demand. It is not hard to see that in this case O( mins(S) log n ). On the other hand, Theorem 8 ([LLR95], see also [A=-=R9-=-8]). In the k-commodity problem max smin (S) log k ): We will be able to review the proof in Section 5. 4.2 Graph bandwidth In this computational problem, we are presented with an n-vertex graph G. It... |

5 |
The metrical interpretation of superre in Banach spaces
- Bourgain
- 1986
(Show Context)
Citation Context ...l 1 . They can also be embedded into l 2 with a relatively low distortion. Theorem 5 (Matousek [Mat99]). Every tree on n vertices can be embedded into l 2 with distortions O( p log log n). Bourgain [B=-=ou86]-=- had earlier shown that this bound is attained for complete binary trees. (See [LS] for an elementary proof of this.) 3.2 Planar graphs It turns out that the metrics of planar graphs have good embeddi... |

5 | More on embedding subspaces of L p in l n r - Schechtman - 1987 |

4 |
On the nonexistence of uniform homeomorphisms between L p -spaces
- En
- 1969
(Show Context)
Citation Context ...;j j ; where the maximum is over all matrices Q so that 1. Q is positive semidenite, and 2. The entries in every row in Q sum to zero. Consider the metric of the r-dimensional cube. As shown by En o [=-=Enf69]-=-, the least distorted embedding of this metric is simply the identity map into l r 2 , which has distortion p r. Oursrst illustration for the power of the quadratic programming method is that we provi... |

3 |
Lower bounds on the distortion of embedding metric spaces in graphs. Discrete Comput. Geom
- Rabinovich, Raz
- 1998
(Show Context)
Citation Context ...ds us to ask: Open Problem 1. How small can c 2 (G) be for a a graph G of girth g in which all vertices have degree 3? The answer lies betweensp g) and O(g). An earlier result of Rabinovich and Raz [=-=RR98-=-] reveals another connection between high girth and distortion. Let ' be a map from a graph of girth g to a graph of smaller Euler characteristic (jEj jV j + 1). Then distortion(') sg). 4 Algorithmic ... |

2 |
Asymptotic theory of normed spaces
- Milman, Schechtman
- 1986
(Show Context)
Citation Context ...be c, if there is a linear map ' : X ! Y with distortion(') c. What we are doing here may very well be described as a search for the metric counterpart of this highly developed linear theory. See [M=-=S86]-=- for an introduction to thisseld and [BL00] for a comprehensive cover of the nonlinear theory. The grandfather of the linear theory is the celebrated theorem of Dvoretzky [Dvo61]. Theorem 1 (Dvoretzky... |

2 | Embedding subspaces of L 1 into l - Talagrand - 1990 |