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

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Venue: | Combinatorica |

Citations: | 450 - 19 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}

}

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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