Bandwidth and Low Dimensional Embedding

We design an algorithm to embed graph metrics into `p with dimension and distortion bothdependent only upon the bandwidth of the graph. In particular we show that any graph ofbandwidth k embeds with distortion polynomial in k into O(log k) dimensional `p, 1 ≤ p ≤ ∞.Prior to our result the only known embedding with distortion independent of n was into highdimensional `1 and had distortion exponential in k. Our low dimensional embedding is basedon a general method for reducing dimension in an `p embedding, satisfying certain conditions,to the intrinsic dimension of the point set, without substantially increasing the distortion. Wealso study the extension to graphs of bounded tree-bandwidth. ∗Institute of Computer Science, Hebrew University of Jerusalem, Israel. Email: [email protected]. The workwas done in part while the author was at the Center for the Mathematics of Information, Caltech, and the Institutefor Pure and Applied Mathematics, UCLA.†Department of Computer Science, University of California, Los Angeles. Email: [email protected].‡Department of Computer Science, University of California, Los Angeles. Email: [email protected].§Princeton University and Center for Computational Intractability. Email: [email protected].