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#### David Liedtke, M.Sc.

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## 2021

Exploration and Convex Hull Construction in the Three-Dimensional Hybrid Model

D.J. Liedtke, Masterarbeit, 2021

Motivated by the prospect of computing agents that explore unknown environments and construct convex hulls on the nanoscale, we investigate the capabilities and limitations of a single deterministic finite automaton robot in the three-dimensional hybrid model for programmable matter. In this model, active robots move on a set of passive tiles, called configuration, with the geometric shape of rhombic dodecahedra on the adjacency graph of the face-centered cubic sphere-packing. We show that the exploration problem is equally hard in the hybrid model and in three-dimensional mazes, in which tiles have the shape of cubes and are positioned at the vertices of $\mathbb{Z}^3$. Thereby, a single robot with a constant number of pebbles cannot solve this problem in the hybrid model on arbitrary configurations. We provide algorithms for a robot with two pebbles that solve the exploration problem in the subclass of compact configurations of size $n$ in $\O(n^3)$ rounds. Further, we investigate the robot's capabilities of detection and hull construction in terms of restricted orientation convexity. We show that a robot without any pebble can detect strong $\O$-convexity in $\O(n)$ rounds, but cannot detect weak $\O$-convexity, not even if provided with a single pebble. Assuming that a robot can construct tiles from scratch and deconstruct previously constructed tiles, we show that the strong $\O$-hull of any given configuration of size $n$ can be constructed in $\O(n^4)$ rounds, even if the robot cannot distinguish constructed from native tiles.

## 2018

Influence of Stationary Robots on Continuous Robot Formation Problems

D.J. Liedtke, Bachelorarbeit, 2018

We consider a group of $n$ autonomous mobile robots of which $m$ are stationary thus cannot move. Robots are represented by points in the Euclidean plane. They have no memory, do not communicate or share a common coordinate system and they move solely based on the positioning of other robots within their limited viewing range of 1. The goal is to gather the robots inside of the convex hull of all stationary robots. A variant of this problem, the general gathering problem, has been studied in various different time models. In this work, we consider a continuous time model, where robots continuously observe their neighbors, compute the next target of movement and move with a speed limit of 1 at any time. Regarding the robots' local strategy, we only study contracting algorithms in which every robot that is positioned on the border of the convex hull of all robots moves into this hull. We present a time bound of $\mathcal{O}(nd)$ for any general contracting algorithms in a configuration with only a single stationary robot. For configurations with more stationary robots, we prove that robots converge against the convex hull of all stationary robots and that no upper bound on the runtime exists. For the specific contracting algorithms Go-To-The-Left, Go-On-Bisector and Go-To-The-Middle, we provide linear time bounds.

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