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Minimum Weight Triangulation

Minimum Weight Triangulation. Plamen Ivanov Problem 1 from The Open Problems Project http://maven.smith.edu/~orourke/TOPP/P1.html#gj-cigtn-79. The MWT. A minimum weight triangulation of P is a triangulation of P which has a minimum sum of the Euclidean lengths of its edges. Application:

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Minimum Weight Triangulation

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  1. Minimum Weight Triangulation PlamenIvanov Problem 1 from The Open Problems Project http://maven.smith.edu/~orourke/TOPP/P1.html#gj-cigtn-79

  2. The MWT • A minimum weight triangulation of P is a triangulation of P which has a minimum sum of the Euclidean lengths of its edges. • Application: • Optimal Triangulation • Stock cutting, • finite element analysis, • terrain modeling, • numerical approximation

  3. Example Triangulations Figure taken from: Mulzer, W., and Rote, G. 2008. Minimum-weight triangulation is NP-hard. J. ACM 55, 2, Article 11 (May 2008)

  4. The Question • Can a minimum weight triangulation of a planar point set---one minimizing the total edge length--be found in polynomial time?

  5. Approaches • Minimum Weight Steiner Triangulation: • David Eppstein: shows that by using a Quadtree and O(n) Steiner points a MWST can be approximated. Additionally the MWT has weight of θ(n) times that of the MWST. • Running time O(n log n + k) , where k is the number of Steiner points.

  6. A Quadtree

  7. Approaches • Minimum Weight Steiner Triangulation: • The Quadtree: • partitions the space which has the input set of points • each point is alone in a square • Steiner points: • add a Steiner point at each location where the convex hull of the points crosses a line segment of the quadtree. • include as Steiner points all corners of quadtreesquares

  8. Approaches • Greedy Approximations: • Christos Levcopoulos and DragoKrznaric: show a quasi-greedy approach that approximates the optimal by a tight factor of . Running time is O(n log n). • Matthew T. Dickerson, Scott A. McElfresh, and Mark H. Montague: give O(n2) running times for several algorithms for general cases

  9. Polynomial-Time Solvable Cases • For an n-vertex convex polygon, the problem can be solved in O(n3) using dynamic programming. • P.D. Gilbert. New results in planar triangulations. Master Thesis, University of Illinois,Urbana, 1979. • G. Klincsek. Minimal triangulations of polygonal domains. Annals Discrete Math. 9 (1980) 121-123. • When a given point set lies on a constant number of nested convex hulls. • E. Anagnostou and D. Corneil. Polynomial-time instances of the minimum weight triangulation problem. Comput. Geom. 3 (1993) 247-259

  10. Solution • Wolfgang Mulzer and Günter Rote `06: • Showed that the MWT is an NP-Hard problem. • Used a polynomial time reduction from Positive Planar 1-in-3 SAT problem to MWT. • 1-in-3 SAT is a known NP-Hard problem.

  11. Further Work • It is not known whether the MWT problem is in NP, since it is not known how to compare sums of Euclidean lengths in polynomial time.

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