Kinetic and Stationary Point-Set Embeddability for Plane Graphs

We investigate a kinetic version of point-set embeddability. Given a plane graph $G(V,E) where |V| = n$, and a set P of n moving points where the trajectory of each point is an algebraic function of constant maximum degree s, we maintain a point-set embedding of G on P with at most three bends per edge during the motion. This requires reassigning the mapping of vertices to points from time to time. Our kinetic algorithm uses linear size, O(nlogn) preprocessing time, and processes $O(n^2 β_2s+2 (n)logn)$ events, each in$O(log^2 n)$ time. Here, $β_s (n) = λ_s (n)/ n$ is an extremely slow-growing function and $λ_s (n)$ is the maximum length of Davenport-Schinzel sequences of order s on n symbols