Rectilinear Steiner Trees in Narrow Strips
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A rectilinear Steiner tree for a set P of points in ℝ^2 is a tree that connects the points in P using horizontal and vertical line segments. The goal of Minimal Rectilinear Steiner Tree is to find a rectilinear Steiner tree with minimal total length. We investigate how the complexity of Minimal Rectilinear Steiner Tree for point sets P inside the strip (-∞,+∞)× [0,δ] depends on the strip width δ. We obtain two main results. 1) We present an algorithm with running time n^O(√(δ)) for sparse point sets, that is, point sets where each 1×δ rectangle inside the strip contains O(1) points. 2) 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 2^O(δ√(δ)) n.
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