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We describe an algorithm that takes as input n points in the plane and aparameter {\epsilon}, and produces as output an embedded planar graph havingthe given points as a subset of its vertices in which the graph distances are a(1 + {\epsilon})-approximation to the geometric distances between the givenpoints. For point sets in which the Delaunay triangulation has bounded sharpestangle, our algorithm's output has O(n) vertices, its weight is O(1) times theminimum spanning tree weight, and the algorithm's running time is bounded byO(n \sqrt{log log n}). We use this result in a similarly fast deterministicapproximation scheme for the traveling salesperson problem.

Near-Linear-Time Deterministic Plane Steiner Spanners and TSP Approximation for Well-Spaced Point Sets