Abstract:
A linear arrangement of an n-vertex graph is a one-to-one mapping of its vertices to the integers f1; : : : ; ng. The bandwidth of a linear arrangement is the maximum difference between mapped values of adjacent vertices. The problem of finding a linear arrangement with smallest possible bandwidth in NP-hard. We present a randomized algorithm that runs in nearly linear time and outputs a linear arrangement whose bandwidth is within a polylogarithmic multiplicative factor of optimal. Our algorithm is based on a new notion, called volume respecting embeddings, which is a natural extension of small distortion embeddings of Bourgain and of Linial, London and Rabinovich. 1 Introduction We consider the problem of minimizing the bandwidth of an undirected connected graph G(V; E), where n = jV j and m = jEj. One needs to find a linear arrangement of the vertices, namely, a one-to-one mapping f : V \Gamma! f1; 2; : : : ng, for which the bandwidth, i.e. max (i;j)2E jf(i) \Gamma f(j)j, i...
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