Optimal Euclidean Tree Covers

A (1+e)-stretch tree cover of a metric space is a collection of trees, where every pair of points has a (1+e)-stretch path in one of the trees. The celebrated Dumbbell Theorem [Arya et al. STOC'95] states that any set of n points in d-dimensional Euclidean space admits a (1+e)-stretch tree cover with O_d(e^{-d} ⋅ log(1/e)) trees, where the O_d notation suppresses terms that depend solely on the dimension d. The running time of their construction is O_d(n log n ⋅ log(1/e)/e^d + n ⋅ e^{-2d}). Since the same point may occur in multiple levels of the tree, the maximum degree of a point in the tree cover may be as large as Ω(log Φ), where Φ is the aspect ratio of the input point set. In this work we present a (1+e)-stretch tree cover with O_d(e^{-d+1} ⋅ log(1/e)) trees, which is optimal (up to the log(1/e) factor). Moreover, the maximum degree of points in any tree is an absolute constant for any d. As a direct corollary, we obtain an optimal {routing scheme} in low-dimensional Euclidean spaces. We also present a (1+e)-stretch Steiner tree cover (that may use Steiner points) with O_d(e^{(-d+1)/2} ⋅ log(1/e)) trees, which too is optimal. The running time of our two constructions is linear in the number of edges in the respective tree covers, ignoring an additive O_d(n log n) term; this improves over the running time underlying the Dumbbell Theorem.