Subexponential Algorithms for Rectilinear Steiner Tree and Arborescence Problems

A rectilinear Steiner tree for a set T of points in the plane is a tree which connects T using horizontal and vertical lines, In the Rectilinear Steiner Tree problem, input is a set T of n points in the Euclidean plane (R) and the goal is to find an rectilinear Steiner tree for T of smallest possible total length. A rectilinear Steiner arborecence for a set T of points and root r ∈ T is a rectilinear Steiner tree S for T such that the path in S from r to any point t ∈ T is a shortest path. In the Rectilinear Steiner Arborescence problem the input is a set T of n points in R, and a root r ∈ T , the task is to find an rectilinear Steiner arborescence for T , rooted at r of smallest possible total length. In this paper, we give the first subexponential time algorithms for both problems. Our algorithms are deterministic and run in 2O( √ n logn) time.