An 𝒪(3.82^k) Time FPT Algorithm for Convex Flip Distance
Let P be a convex polygon in the plane, and let T be a triangulation of P. An edge e in T is called a diagonal if it is shared by two triangles in T. A flip of a diagonal e is the operation of removing e and adding the opposite diagonal of the resulting quadrilateral to obtain a new triangulation of P from T. The flip distance between two triangulations of P is the minimum number of flips needed to transform one triangulation into the other. The Convex Flip Distance problem asks if the flip distance between two given triangulations of P is at most k, for some given parameter k ∈ ℕ.
We present an FPT algorithm for the Convex Flip Distance problem that runs in time 𝒪(3.82^k) and uses polynomial space, where k is the number of flips. This algorithm significantly improves the previous best FPT algorithms for the problem.
Flip distance
Rotation distance
Triangulations
Exact algorithms
Parameterized complexity
Theory of computation~Computational geometry
Theory of computation~Parameterized complexity and exact algorithms
44:1-44:14
Regular Paper
https://arxiv.org/abs/2209.13134
Haohong
Li
Haohong Li
Department of Computer Science, Lafayette College, Easton, PA, USA
Ge
Xia
Ge Xia
Department of Computer Science, Lafayette College, Easton, PA, USA
10.4230/LIPIcs.STACS.2023.44
O. Aichholzer, F. Hurtado, and M. Noy. A lower bound on the number of triangulations of planar point sets. Computational Geometry: Theory and Applications, 29(2):135-145, 2004.
O. Aichholzer, W. Mulzer, and A. Pilz. Flip distance between triangulations of a simple polygon is NP-complete. Discrete & Computational Geometry, 54(2):368-389, 2015.
A. Bonnin and J.-M. Pallo. A shortest path metric on unlabeled binary trees. Pattern Recognition Letters, 13(6):411-415, 1992.
P. Bose and F. Hurtado. Flips in planar graphs. Computational Geometry: Theory and Applications, 42(1):60-80, 2009.
M. B. Calvo and S. Kelk. An improved kernel for the flip distance problem on simple convex polygons. In Meng He and Don Sheehy, editors, Proceedings of the 33rd Canadian Conference on Computational Geometry, CCCG 2021, pages 195-199, 2021.
Y.-J. Chen, J.-M. Chang, and Y.-L. Wang. An efficient algorithm for estimating rotation distance between two binary trees. International Journal of Computer Mathematics, 82:1095-1106, 2005.
S. Cleary and K. St. John. Rotation distance is fixed-parameter tractable. Information Processing Letters, 109(16):918-922, 2009.
S. Cleary and K. St. John. A linear-time approximation algorithm for rotation distance. J. Graph Algorithms Appl., 14:385-390, 2010.
K. Culik and D. Wood. A note on some tree similarity measures. Information Processing Letters, 15(1):39-42, 1982.
R. Downey and M. Fellows. Parameterized Complexity. Springer, New York, 1999.
Q. Feng, S. Li, X. Meng, and J. Wang. An improved fpt algorithm for the flip distance problem. Information and Computation, 281, 2021.
S. Fordham and S. Cleary. Minimal length elements of thompson’s groups f(p). Geometriae Dedicata, 141:163-180, 2007.
S. Hanke, T. Ottmann, and S. Schuierer. The edge-flipping distance of triangulations. Journal of Universal Computer Science, 2(8):570-579, 1996.
F. Hurtado, M. Noy, and J. Urrutia. Flipping edges in triangulations. Discrete & Computational Geometry, 22(3):333-346, 1999.
A. B. Kahn. Topological sorting of large networks. Commun. ACM, 5(11):558-562, November 1962.
I. Kanj, E. Sedgwick, and G. Xia. Computing the flip distance between triangulations. Discret. Comput. Geom., 58(2):313-344, 2017.
I. Kanj and G. Xia. Flip Distance Is in FPT Time O(n+ k ⋅ c^k). In proceedings of STACS, volume 30 of LIPIcs, pages 500-512, 2015.
C. Lawson. Transforming triangulations. Discrete Mathematics, 3(4):365-372, 1972.
M. Li and L. Zhang. Better approximation of diagonal-flip transformation and rotation transformation. In Proceedings of the 4th Annual International Conference on Computing and Combinatorics, COCOON '98, pages 85-94, 1998.
A. Lubiw and V. Pathak. Flip distance between two triangulations of a point set is NP-complete. Computational Geometry: Theory and Applications, 49:17-23, 2015.
J. Lucas. An improved kernel size for rotation distance in binary trees. Information Processing Letters, 110(12):481-484, 2010.
F. Luccio and L. Pagli. On the upper bound on the rotation distance of binary trees. Information Processing Letters, 31(2):57-60, 1989.
R. Niedermeier. Invitation to Fixed-Parameter Algorithms. Oxford University Press, USA, 2006.
J. Pallo. On the rotation distance in the lattice of binary trees. Information Processing Letters, 25(6):369-373, 1987.
A. Pilz. Flip distance between triangulations of a planar point set is APX-hard. Computational Geometry: Theory and Applications, 47(5):589-604, 2014.
L. Pournin. The diameter of associahedra. Advances in Mathematics, 259:13-42, 2014.
D. Sleator, R. Tarjan, and W. Thurston. Rotation distance, triangulations, and hyperbolic geometry. Journal of the American Mathematical Society, 1:647-681, 1988.
R. P. Stanley. Enumerative Combinatorics, volume 2. Cambridge University Press, 1999.
Haohong Li and Ge Xia
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