Approximating Minimum Manhattan Networks

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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 SS′. 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.