Abstract
Given a set S of n points in the plane, we define a Manhattan Network on S as a rectilinear network G with the property that for every pair of points in S, the network G contains the shortest rectilinear path between them. A Minimum Manhattan Network on S is a Manhattan network of minimum possible length. A Manhattan network can be thought of as a graph G = (V; E), where the vertex set V corresponds to points from S and a set of steiner points S′, and the edges in E correspond to horizontal or vertical line segments connecting points in S ∪ S′. A Manhattan network can also be thought of as a 1-spanner (for the L 1-metric) for the points in S.
Let R be an algorithm that produces a rectangulation of a staircase polygon in time R(n) of weight at most r times the optimal. We design an O(n log n + R(n)) time algorithm which, given a set S of n points in the plane, produces a Manhattan network on S with total weight at most 4r times that of a minimum Manhattan network. Using known rectangulation algorithms, this gives us an O(n 3)-time algorithm with approximation factor four, and an O(n log n)-time algorithm with approximation factor eight.
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© 1999 Springer-Verlag Berlin Heidelberg
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Gudmundsson, J., Levcopoulos, C., Narasimhan, G. (1999). Approximating Minimum Manhattan Networks. In: Hochbaum, D.S., Jansen, K., Rolim, J.D.P., Sinclair, A. (eds) Randomization, Approximation, and Combinatorial Optimization. Algorithms and Techniques. RANDOM APPROX 1999 1999. Lecture Notes in Computer Science, vol 1671. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-48413-4_4
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DOI: https://doi.org/10.1007/978-3-540-48413-4_4
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