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]

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 ${\displaystyle \Omega (n^{2})}$,^[2] thus the number of changes to any triangulation is also lower bounded by ${\displaystyle \Omega (n^{2})}$. Finding any triangulation scheme that has a near-quadratic bound on the number of discrete changes is an important open problem.^[1]

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 ${\displaystyle \Omega (n^{2})}$^[4] and ${\displaystyle O(n^{2+\epsilon })}$.^[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 ${\displaystyle O(1)}$.

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

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