Given a set of entities each holding a Boolean state, the Density Classification Task (DCT) asks them to converge to the most represented state. Given a directed graph of entities where each node synchronously updates to the local majority among its in-neighbors, we characterize in terms of three forbidden patterns when it solves DCT, and show that discovering these patterns is complete for NP and PSPACE.

翻译：给定一组各自持有布尔状态的实体，密度分类任务要求它们收敛到出现频率最高的状态。在有向图表示的实体网络中，当每个节点同步更新为其入邻居的局部多数状态时，我们通过三种禁止模式刻画了其解决密度分类任务的条件，并证明发现这些模式对于NP和PSPACE是完全的。