Phillip Keldenich

Oswin Aichholzer, Joseph Dorfer, Sándor P. Fekete et al.

We give an overview of the 2026 Computational Geometry Challenge targeting the problem of finding a Central Triangulation under Parallel Flip Operations in triangulations of point sets. A flip is the parallel exchange of a set of edges in a triangulation with opposing diagonals of the convex quadrilaterals containing them. The challenge objective was, given a set of triangulations of a fixed point set, to determine a central triangulation with respect to parallel flip distances. More precisely, this asks for a triangulation that minimizes the sum of flip distances to all elements of the input

Sándor P. Fekete, Phillip Keldenich, Dominik Krupke et al.

We give an overview of the 2021 Computational Geometry Challenge, which targeted the problem of optimally coordinating a set of robots by computing a family of collision-free trajectories for a set set S of n pixel-shaped objects from a given start configuration into a desired target configuration.

Erik D. Demaine, Sándor P. Fekete, Phillip Keldenich et al.

We present a number of breakthroughs for coordinated motion planning, in which the objective is to reconfigure a swarm of labeled convex objects by a combination of parallel, continuous, collision-free translations into a given target arrangement. Problems of this type can be traced back to the classic work of Schwartz and Sharir (1983), who gave a method for deciding the existence of a coordinated motion for a set of disks between obstacles; their approach is polynomial in the complexity of the obstacles, but exponential in the number of disks. Other previous work has largely focused on {\em sequential} schedules, in which one robot moves at a time. We provide constant-factor approximation algorithms for minimizing the execution time of a coordinated, {\em parallel} motion plan for a swarm of robots in the absence of obstacles, provided some amount of separability. Our algorithm achieves {\em constant stretch factor}: If all robots are at most $d$ units from their respective starting positions, the total duration of the overall schedule is $O(d)$. Extensions include unlabeled robots and different classes of robots. We also prove that finding a plan with minimal execution time is NP-hard, even for a grid arrangement without any stationary obstacles. On the other hand, we show that for densely packed disks that cannot be well separated, a stretch factor $Ω(N^{1/4})$ may be required. On the positive side, we establish a stretch factor of $O(N^{1/2})$ even in this case.