Computing a Minimum-Cost kk-hop Steiner Tree in Tree-Like Metrics

We consider the problem of computing a Steiner tree of minimum cost under a $k$-hop constraint which requires the depth of the tree to be at most $k$. Our main result is an exact algorithm for metrics induced by graphs of bounded treewidth that runs in time $n^{O(k)}$. For the special case of a path, we give a simple algorithm that solves the problem in polynomial time, even if $k$ is part of the input. The main result can be used to obtain, in quasi-polynomial time, a near-optimal solution that violates the $k$-hop constraint by at most one hop for more general metrics induced by graphs of bounded highway dimension

New Doubling Spanners: Better and Simpler

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Narrow-Shallow-Low-Light Trees with and without Steiner Points

We show that for every set S of n points in the plane and a designated point rt ∈ S, there exists a tree T that has small maximum degree, depth and weight. Moreover, for every point v ∈ S, the distance between rt and v in T is within a factor of (1+ɛ) close to their Euclidean distance ‖rt, v‖. We call these trees narrow-shallow-low-light (NSLLTs). We demonstrate that our construction achieves optimal (up to constant factors) tradeoffs between all parameters of NSLLTs. Our construction extends to point sets in R d, for an arbitrarily large constant d. The running time of our construction is O(n · log n). We also study this problem in general metric spaces, and show that NSLLTs with small maximum degree, depth and weight can always be constructed if one is willing to compromise the root-distortion. On the other hand, we show that the increased root-distortion is inevitable, even if the point set S resides in a Euclidean space of dimension Θ(log n). On the bright side, we show that if one is allowed to use Steiner points then it is possible to achieve root-distortion (1+ɛ) together with small maximum degree, depth and weight for general metric spaces. Finally, we establish some lower bounds on the power of Steiner points in the context of Euclidean spanning trees and spanners