Haifa University - Dept. of CS Theory Seminar

Speaker : Shai Solomon, Ben-Gurion University.

Title: "Shallow, Low, and Light Trees, and Tight Lower Bounds for Euclidean Spanners".

Date: Thursday, June 5.

Place: Haifa U. Education Building, room 566.

Time: 12:15 - 13:30

Abstract:

We show that for every n-point metric space M and positive integer k=O(log n), there exists a spanning tree T with unweighted diameter O(k) and weight w(T) = O(k \cdot n^{1/k}) * w(MST(M)), and a spanning tree T' with weight w(T') = O(k) * w(MST(M)) and unweighted diameter O(k * n^{1/k}). Moreover, there is a designated point rt such that for every other point v$, both dist_T(rt,v) and dist_{T'}(rt,v) are at most (1+\eps) * dist_M(rt,v), for an arbitrarily small constant \epsilon > 0. These trees also achieve an optimal maximum degree. Furthermore, we demonstrate that these trees can be constructed efficiently. We prove that these tradeoffs between unweighted diameter and weight are **tight up to constant factors** in the entire range of parameters. Moreover, our lower bounds apply to a basic one-dimensional Euclidean space. Our lower bounds for the particular case of unweighted diameter O(log n) are of independent interest, settling a long-standing open problem in Computational Geometry. In STOC'95 Arya et al. devised a construction of Euclidean Spanners with unweighted diameter O(log n) and weight O(log n) * w(MST(M)). In SODA'05 Agarwal et al. showed that this result is tight up to a factor of O(log log n). We close this gap and show that the result of Arya et al. is tight up to constant factors. Finally, our upper bounds imply improved approximation algorithms for the **minimum h-hop spanning tree** and **bounded diameter minimum spanning tree** problems for metric spaces. Joint work with Yefim Dinitz and Michael Elkin.