## The geometry of graphs and some of its algorithmic applications (1995)

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Venue: | Combinatorica |

Citations: | 454 - 18 self |

### BibTeX

@ARTICLE{Linial95thegeometry,

author = {Nathan Linial and Eran London and Yuri Rabinovich},

title = {The geometry of graphs and some of its algorithmic applications},

journal = {Combinatorica},

year = {1995},

volume = {15},

pages = {577--591}

}

### Years of Citing Articles

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### Abstract

In this paper we explore some implications of view-ing graphs as geometric objects. This approach of-fers a new perspective on a number of graph-theoretic and algorithmic problems. There are several ways to model graphs geometrically and our main concern here is with geometric representations that respect the met-ric of the (possibly weighted) graph. Given a graph G we map its vertices to a normed space in an attempt to (i) Keep down the dimension of the host space and (ii) Guarantee a small distortion, i.e., make sure that distances between vertices in G closely match the dis-tances between their geometric images. In this paper we develop efficient algorithms for em-bedding graphs low-dimensionally with a small distor-tion. Further algorithmic applications include: 0 A simple, unified approach to a number of prob-lems on multicommodity flows, including the Leighton-Rae Theorem [29] and some of its ex-tensions. 0 For graphs embeddable in low-dimensional spaces with a small distortion, we can find low-diameter decompositions (in the sense of [4] and [34]). The parameters of the decomposition depend only on the dimension and the distortion and not on the size of the graph. 0 In graphs embedded this way, small balanced separators can be found efficiently. Faithful low-dimensional representations of statisti-cal data allow for meaningful and efficient cluster-ing, which is one of the most basic tasks in pattern-recognition. For the (mostly heuristic) methods used

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Citation Context ...ing, this short discussion leaves out large amounts of relevant work, for example, on embedding graphs in particular families of graphs such as d-dimensional lattices, cubes, squashed cubes etc. (see =-=[27]-=-, [62]). Particularly relevant are notions of dimension that emerge from such considerations, see, e.g., chapter 5 in [8]. The possibility of embedding graphs in spaces other than Euclidean and spheri... |

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Citation Context ... topological models, (ii) adjacency models, or (iii) metric models. The topological approach is mainly concerned with graph planarity and embeddability of graphs on other 2-dimensional manifolds (see =-=[34]-=- for a recent survey). It also deals with 3-dimensional embeddings of a graph, mostly in the context of knot theory. (See Welsh's book [61] for some of these developments.) Particularly fascinating is... |

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Citation Context ...s only the relation of adjacency/nonadjacency of vertices, but not necessarily the actual distance. A prime example for this approach is the Koebe--Andreev-- Thurston Theorem (see [36], [3], [4], and =-=[60]-=-) that every planar graph is the contact graph of openly disjoint planar discs. Higher--dimensional results in the same vein appear in [21], [22], [49], and [55]. Another noteworthy adjacency model ca... |

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Citation Context ...at of positive/negative curvature in geometry still awaits a satisfactory explanation. Low--dimensional models for finite metric spaces have previously been studied mostly by functional analysts (see =-=[5]-=-, [9], [12], [19], [28], [31], [32], and [48]). Study of graph metrics has also led to the notion of spanners (see [2], [51], and [58]) and hop--sets [15]. Local--global considerations, which are 3 co... |

3 |
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(Show Context)
Citation Context ...f that depends only on the dimension and the distortion at which the graph is embeddable, not on the number of vertices. For many algorithmic applications of low--diameter decompositions, see [7] and =-=[16]-=-. Graphs embeddable in a d-dimensional normed space with a small distortion have balanced separators of only O(d \Delta n 1\Gamma 1 d ) vertices. If the embedding is given, such separators can be foun... |

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Citation Context ...y still awaits a satisfactory explanation. Low--dimensional models for finite metric spaces have previously been studied mostly by functional analysts (see [5], [9], [12], [19], [28], [31], [32], and =-=[48]-=-). Study of graph metrics has also led to the notion of spanners (see [2], [51], and [58]) and hop--sets [15]. Local--global considerations, which are 3 commonplace in geometry, arose for graphs as we... |

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