Rectilinear Steiner Trees in Narrow Strips

A rectilinear Steiner tree for a set P of points in R2 is a tree that connects the points in P using horizontal and vertical line segments. The goal of Minimum Rectilinear Steiner Tree is to find a rectilinear Steiner tree with minimal total length. We investigate how the complexity of Minimum Rectilinear Steiner Tree for point sets P inside the strip (−∞, +∞) × [0, δ] depends on the strip width δ. We obtain two main results. We present an algorithm with running time nO( √ δ) for sparse point sets, that is, point sets where each 1 × δ rectangle inside the strip contains O(1) points. For random point sets, where the points are chosen randomly inside a rectangle of height δ and expected width n, we present an algorithm that is fixed-parameter tractable with respect to δ and linear in n. It has an expected running time of 2O(δ √ δ)n. 2012 ACM Subject Classification Theory of computation → Design and analysis of algorithms

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