We introduce a new graph problem, the token dropping game, and we show how to solve it efficiently in a distributed setting. We use the token dropping game as a tool to design an efficient distributed algorithm for stable orientations and more generally for locally optimal semi-matchings. The prior work by Czygrinow et al. (DISC 2012) finds a stable orientation in $O(Delta^5)$ rounds in graphs of maximum degree $Delta$, while we improve it to $O(Delta^4)$ and also prove a lower bound of $Omega(Delta)$.
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