Subexponential Algorithms for Rectilinear Steiner Tree and Arborescence Problems

Abstract

A rectilinear Steiner tree for a set K of points in the plane is a tree that connects k using horizontal and vertical lines. In the R ectilinear S teiner T ree problem, the input is a set K ={ z 1 , z 2 ,…, z n } of n points in the Euclidean plane (R 2 ), and the goal is to find a rectilinear Steiner tree for k of smallest possible total length. A rectilinear Steiner arborescence for a set k of points and a root r ∈ K is a rectilinear Steiner tree T for K such that the path in T from r to any point z ∈ K is a shortest path. In the R ectilinear S teiner A rborescence problem, the input is a set K of n points in R 2 , and a root r ∈ K , and the task is to find a rectilinear Steiner arborescence for K , rooted at r of smallest possible total length. In this article, we design deterministic algorithms for these problems that run in 2 O (√ n log n ) time.