Euclidean Spanners in High Dimensions

• Main Authors: Har-Peled, Sariel (Author), Indyk, Piotr (Contributor), Sidiropoulos, Anastasios (Author)
• Other Authors: Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory (Contributor), Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science (Contributor)
• Format: Article
• Language: English
• Published: Society for Industrial and Applied Mathematics, 2014-05-15T17:22:50Z.
• Subjects: Article
• Online Access: Get fulltext

 A classical result in metric geometry asserts that any n-point metric admits a linear-size spanner of dilation O(log n) [PS89]. More generally, for any c > 1, any metric space admits a spanner of size O(n[superscript 1+1/c]), and dilation at most c. This bound is tight assuming the well-known girth conjecture of Erdős [Erd63]. We show that for a metric induced by a set of n points in high-dimensional Euclidean space, it is possible to obtain improved dilation/size trade-offs. More specifically, we show that any n-point Euclidean metric admits a near-linear size spanner of dilation O(√log n). Using the LSH scheme of Andoni and Indyk [AI06] we further show that for any c > 1, there exist spanners of size roughly O(n[superscript1+1/c[superscript 2]]) and dilation O(c). Finally, we also exhibit super-linear lower bounds on the size of spanners with constant dilation.