Robotic Sorting on the Grid.

Inspired by robotic applications, we study the problem of sorting a set of items over a physical domain with mobile agents. Given m mobile robots and a grid where each cell contains a single element, the objective is to design algorithms that allow robots to cooperatively sort the elements over the grid in the minimum time. We assume a synchronous model where robots do not communicate, can carry up to c elements, and can move between adjacent cells in one unit of time (grab and release time is negligible). First, we show that any algorithm requires at least units of time to sort an n-element line (an n × 1 grid) and present an algorithm that sorts the elements in time. Then, we show that any n × n-grid requires at least time and present an algorithm that completes in time. Our algorithms have an equivalent competitive ratio to Shear Sort [Isaacd et al., Proc ICPP 1986] with only m = n agents (compared to the n2 processors required by Shear Sort). Finally, we present experimental results that show the capacity has very little impact on the overall runtime and that a simplification of the algorithm leads to better results.