The geometry of graphs and some of its algorithmic applications - Foundations of Computer Science, 1994 Proceedings., 35th Annual Symposium on

In this paper we explore some implications of viewing graphs as geometric objects. This approach offers 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 metric 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 distances between their geometric images. In this paper we develop efficient algorithms for embedding graphs low-dimensionally with a small distortion. Further algorithmic applications include: 0 A simple, unified approach to a number of problems on multicommodity flows, including the Leighton-Rae Theorem [29] and some of its extensions. 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. arators can be found efficiently. 0 In graphs embedded this way, small balanced sepFaithful low-dimensional representations of statistical data allow for meaningful and efficient clustering, which is one of the most basic tasks in patternrecognition. For the (mostly heuristic) methods used *Supported in part by grants from the Israeli Academy of Sciences, the Binational Science Foundation Israel-USA and the Niedersachsen-Isriel Research Program. in the practice of pattern-recognition, see Duda and Hart [15], especially chapter 6.