The paper proposes distributed algorithms for the uniform partitioning of a bounded region using a swarm of N asynchronous, oblivious, opaque, and luminous mobile robots. The robots have no global agreement on coordinate axes and do not know the total number of robots N.
For a rectangular region, the algorithm runs in O(N) epochs and uses 4 colors. It first brings all the robots to the boundary of the rectangle, then the monitor robots count the number of robots on the two longest sides and redistribute the robots if needed. Finally, the robots move to their respective final positions.
For a square region, the algorithm also runs in O(N) epochs and uses 7 colors. It follows a similar strategy as the rectangular case, but handles additional cases where the robots are distributed unevenly among the sides.
For a circular region, the algorithm runs in O(N^2) epochs and uses 9 colors. It forms eligible clusters of robots and the head of each cluster moves towards the other cluster to achieve uniform partitioning.
The algorithms ensure collision-free movement of the robots and that each partition contains exactly one robot. The time complexity analysis shows that the proposed algorithms are efficient.
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