Asynchronous Filling by Luminous Robots

In this work we investigate this problem in the asynchronous model by luminous robots. In this model a light is attached to the robots, which serves as externally visible bits stored in that light encoded by a color. First, we present an algorithm solving the asynchronous Filling problem with robots having $1$ hop visibility range, $O(\log\Delta)$ bits of persistent storage, and $\Delta+3$ colors, where $\Delta$ is the maximum degree of the graph. Then we show, how the number of colors can be reduced to $O(1)$ at the cost of the running time. After this we show, how the running time can be improved by robots with visibility range of $2$ hops, $O(\log \Delta)$ bits of persistent memory, and $\Delta + 3$ colors. We show, that in the fully synchronous case, the running time of this algorithm is $O(n)$. We show how to extend our asynchronous solution to the $k$-Door case, $k\geq 2$, by using $\Delta + k + 3$ colors. Finally, we present simulation results.