Shallow-Low-Light Trees, and Tight Lower Bounds for Euclidean Spanners

Abstract

We show that for every n-point metric space M and positive integer k, there exists a spanning tree T with unweighted diameter O(k) and weight w(T) = O(k ldr n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1/k</sup> ) <b xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ldr</b> w(MST(M)), and a spanning tree T' with weight w(T') = O(k) <b xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ldr</b> w(MST(M)) and unweighted diameter O(k ldr n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1/k</sup> ). Moreover, there is a designated point rt such that for every other point v, both dist <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</sub> (rt, v) and dist <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</sub> (rt, v) are at most (1 + epsiv) <b xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ldr</b> dist <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</sub> (rt,v), for an arbitrarily small constant epsiv > 0. We prove that the above tradeoffs are tight up to constant factors in the entire range of parameters. Furthermore, our lower bounds apply to a basic one-dimensional Euclidean space. Finally, our lower bounds for the particular case of unweighted diameter O(log n) settle a long-standing open problem in Computational Geometry.