Approximating the minimum weight triangulation

In <italic>O(n</italic> log <italic>n)</italic> time we compute a triangulation with <italic>O(n)</italic> new points, and no obtuse triangles, that has length within a constant factor of the minimum possible. We also approximate the minimum weight Steiner triangulation using triangulations with no sharp angles. No previous polyonomial time triangulation achieved an approximation factor better than <italic>O</italic>(log <italic>n</italic>).

[1]  David G. Kirkpatrick,et al.  A Note on Delaunay and Optimal Triangulations , 1980, Inf. Process. Lett..

[2]  Peter Gilbert.  New Results on Planar Triangulations. , 1979 .

[3]  David Eppstein,et al.  Edge insertion for optimal triangulations , 1993, Discret. Comput. Geom..

[4]  Hanan Samet,et al.  The Quadtree and Related Hierarchical Data Structures , 1984, CSUR.

[5]  Christos Levcopoulos.  An \Omega(\sqrt(n)) Lower Bound for the Nonoptimality of the Greedy Triangulation , 1987, Inf. Process. Lett..

[6]  Tiow Seng Tan,et al.  A quadratic time algorithm for the minmax length triangulation , 1991, [1991] Proceedings 32nd Annual Symposium of Foundations of Computer Science.

[7]  David A. Plaisted,et al.  A Heuristic Triangulation Algorithm , 1987, J. Algorithms.

[8]  David Eppstein.  The Farthest Point Delaunay Triangulation Minimizes Angles , 1991, Comput. Geom..

[9]  Glenn K. Manacher,et al.  Neither the Greedy Nor the Delaunay Triangulation of a Planar Point Set Approximates the Optimal Triangulation , 1979, Inf. Process. Lett..

[10]  G. Klincsek.  Minimal Triangulations of Polygonal Domains , 1980 .

[11]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[12]  Tiow Seng Tan,et al.  An O(n2log n) time algorithm for the MinMax angle triangulation , 1990, SCG '90.

[13]  Andrzej Lingas,et al.  Fast Algorithms for Greedy Triangulation , 1990, SWAT.

[14]  Charles K. Mesztenyi,et al.  On a Data Structure for Adaptive Finite Element Mesh Refinements , 1980, TOMS.

[15]  Errol L. Lloyd.  On triangulations of a set of points in the plane , 1977, 18th Annual Symposium on Foundations of Computer Science (sfcs 1977).

[16]  Mark Yerry,et al.  A Modified Quadtree Approach To Finite Element Mesh Generation , 1983, IEEE Computer Graphics and Applications.

[17]  David M. Mount,et al.  Globally-Equiangular triangulations of co-circular points in 0(n log n) time , 1988, SCG '88.

[18]  Ding-Zhu Du,et al.  An approach for proving lower bounds: solution of Gilbert-Pollak's conjecture on Steiner ratio , 1990, Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science.

[19]  Robin Sibson,et al.  Locally Equiangular Triangulations , 1978, Comput. J..

[20]  R. B. Simpson,et al.  On optimal interpolation triangle incidences , 1989 .

[21]  Rainald Löhner,et al.  Some useful data structures for the generation of unstructured grids , 1988 .

[22]  D. Eppstein,et al.  MESH GENERATION AND OPTIMAL TRIANGULATION , 1992 .

[23]  李幼升,et al.  Ph , 1989 .

[24]  David Eppstein,et al.  Provably good mesh generation , 1990, Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science.

[25]  Kenneth L. Clarkson,et al.  Approximation algorithms for planar traveling salesman tours and minimum-length triangulations , 1991, SODA '91.