Optimal Euclidean Tree Covers

A (1 + ε)-stretch tree cover of a metric space is a collection of trees, where every pair of points has a (1 + ε)-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 + ε)-stretch tree cover with O_d(ε^−d · log(1/ε)) 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_εd^/ε) + n · ε^−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 + ε)-stretch tree cover with O_d(ε^−d+1 · log(1/ε)) trees, which is optimal (up to the log(1/ε) 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 + ε)-stretch Steiner tree cover (that may use Steiner points) with O_d(ε^(−d+1)/² · log(1/ε)) 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.