Kinetic triangulation

A kinetic triangulation data structure is a kinetic data structure that maintains a triangulation of a set of moving points. Maintaining a kinetic triangulation is important for applications that involve motion planning, such as video games, virtual reality, dynamic simulations and robotics.^[1]

Choosing a triangulation scheme[edit]

The efficiency of a kinetic data structure is defined based on the ratio of the number of internal events to external events, thus good runtime bounds can sometimes be obtained by choosing to use a triangulation scheme that generates a small number of external events. For simple affine motion of the points, the number of discrete changes to the convex hull is estimated by $\Omega (n^{2})$ ,^[2] thus the number of changes to any triangulation is also lower bounded by $\Omega (n^{2})$ . Finding any triangulation scheme that has a near-quadratic bound on the number of discrete changes is an important open problem.^[1]

Delaunay triangulation[edit]

The Delaunay triangulation seems like a natural candidate, but a tight worst-case analysis of the number of discrete changes that will occur to the Delaunay triangulation (external events) was considered an open problem until 2015;^[3] it has now been bounded to be between $\Omega (n^{2})$ ^[4] and $O(n^{2+\epsilon })$ .^[5]

There is a kinetic data structure that efficiently maintains the Delaunay triangulation of a set of moving points,^[6] in which the ratio of the total number of events to the number of external events is $O(1)$ .

Other triangulations[edit]

Kaplan et al. developed a randomized triangulation scheme that experiences an expected number of $O(n^{2}\beta _{s+2}(n)\log ^{2}n)$ external events, where $s$ is the maximum number of times each triple of points can become collinear, $\beta _{s+2}(q)={\frac {\lambda _{s+2}(q)}{q}}$ , and $\lambda _{s+2}(q)$ is the maximum length of a Davenport-Schinzel sequence of order s + 2 on n symbols.^[1]

Pseudo-triangulations[edit]

There is a kinetic data structure (due to Agarwal et al.) which maintains a pseudo-triangulation in $O(n^{2}2^{\sqrt {\log n\log \log n}})$ events total.^[7] All events are external and require $O(\lg n)$ time to process.

References[edit]

^ ^a ^b ^c Kaplan, Haim; Rubin, Natan; Sharir, Micha (June 2010). A Kinetic Triangulation Scheme for Moving Points in The Plane (PDF). SCG. ACM. Retrieved May 19, 2012.
^ Sharir, M.; Agarwal, P. K. (1995). Davenport-Schinzel sequences and their geometric applications. New York: Cambridge University Press.
^ Demaine, E.D.; Mitchell, J. S. B.; O’Rourke, J. "The Open Problems Project". Retrieved May 19, 2012.
^ Agarwal, Pankaj K.; Basch, Julien; de Berg, Mark; Guibas, Leonidas J.; Hershberger, John (June 1999). Lower bounds for kinetic planar subdivisions. SCG. ACM. pp. 247–254. doi:10.1145/304893.304961.
^ Rubin, Natan (June 2015). "On Kinetic Delaunay Triangulations: A Near-Quadratic Bound for Unit Speed Motions". J ACM. ACM. doi:10.1145/2746228. S2CID 2493978.
^ Gerhard Albers, Leonidas J. Guibas, Joseph S. B. Mitchell, and Thomas Roos. Voronoi diagrams of moving points. Int. J. Comput. Geometry Appl., 8(3):365{380, 1998.
^ Pankaj K. Agarwal, Julien Basch, Leonidas J. Guibas, John Hershberger, and Li Zhang. Deformable free-space tilings for kinetic collision detection. I. J. Robotic Res., 21(3):179{198, 2002. [1]

[micha-1] Kaplan, Haim; Rubin, Natan; Sharir, Micha (June 2010). A Kinetic Triangulation Scheme for Moving Points in The Plane (PDF). SCG. ACM. Retrieved May 19, 2012.

[convex_hull-2] Sharir, M.; Agarwal, P. K. (1995). Davenport-Schinzel sequences and their geometric applications. New York: Cambridge University Press.

[delaunay_hardness-3] Demaine, E.D.; Mitchell, J. S. B.; O’Rourke, J. "The Open Problems Project". Retrieved May 19, 2012.

[lower_bound-4] Agarwal, Pankaj K.; Basch, Julien; de Berg, Mark; Guibas, Leonidas J.; Hershberger, John (June 1999). Lower bounds for kinetic planar subdivisions. SCG. ACM. pp. 247–254. doi:10.1145/304893.304961.

[delaunay_bounded-5] Rubin, Natan (June 2015). "On Kinetic Delaunay Triangulations: A Near-Quadratic Bound for Unit Speed Motions". J ACM. ACM. doi:10.1145/2746228. S2CID 2493978.

[6] Gerhard Albers, Leonidas J. Guibas, Joseph S. B. Mitchell, and Thomas Roos. Voronoi diagrams of moving points. Int. J. Comput. Geometry Appl., 8(3):365{380, 1998.

[7] Pankaj K. Agarwal, Julien Basch, Leonidas J. Guibas, John Hershberger, and Li Zhang. Deformable free-space tilings for kinetic collision detection. I. J. Robotic Res., 21(3):179{198, 2002. [1]

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