Asymptotic theory of finite dimensional normed spaces
Asymptotic theory of finite dimensional normed spaces
Small distortion and volume preserving embeddings for planar and Euclidean metrics
SCG '99 Proceedings of the fifteenth annual symposium on Computational geometry
Approximating the bandwidth via volume respecting embeddings
Journal of Computer and System Sciences - 30th annual ACM symposium on theory of computing
Lectures on Discrete Geometry
Random Projection: A New Approach to VLSI Layout
FOCS '98 Proceedings of the 39th Annual Symposium on Foundations of Computer Science
Probabilistic approximation of metric spaces and its algorithmic applications
FOCS '96 Proceedings of the 37th Annual Symposium on Foundations of Computer Science
Metric Embeddings—Beyond One-Dimensional Distortion
Discrete & Computational Geometry
Euclidean distortion and the sparsest cut
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
On distance scales, embeddings, and efficient relaxations of the cut cone
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Volume in general metric spaces
ESA'10 Proceedings of the 18th annual European conference on Algorithms: Part II
Hardness results for approximating the bandwidth
Journal of Computer and System Sciences
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In [Rao 1999], it is shown that every n-point Euclidean metric with polynomial aspect ratio admits a Euclidean embedding with k-dimensional distortion at most O(√log n log k), a result which is tight for constant values of k. We show that this holds without any assumption on the aspect ratio, and give an improved bound of O(√log n (log k)1/4). Our main result is an upper bound of O(√log n log log n) independent of the value of k, nearly resolving the main open questions of [Dunagan-Vempala 2001] and [Krauthgamer-Linial-Magen 2004]. The best previous bound was O(log n), and our bound is nearly tight, as even the 2-dimensional volume distortion of an n-vertex path is Ω(√log n).