We apply the power-of-two-choices paradigm to a random walk on a graph: rather than moving to a uniform random neighbour at each step, a controller is allowed to choose from two independent uniform random neighbours. We prove that this allows the controller to significantly accelerate the hitting and cover times in several natural graph classes. In particular, we show that the cover time becomes linear in the number n of vertices on discrete tori and bounded degree trees, of order $${\mathcal O}(n\log \log n)$$ on bounded degree expanders, and of order $${\mathcal O}(n{(\log \log n)^2})$$ on the Erdős–Rényi random graph in a certain sparsely connected regime. We also consider the algorithmic question of computing an optimal strategy and prove a dichotomy in efficiency between computing strategies for hitting and cover times.

中文翻译：

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随机游走的两种选择的力量

我们将二次选择的幂范式应用于图上的随机游走：不是在每一步移动到一个统一的随机邻居，而是允许控制器从两个独立的统一随机邻居中进行选择。我们证明这允许控制器在几个自然图类中显着加快命中和覆盖时间。特别是，我们表明覆盖时间在数量上变得线性n离散圆环和有界度树上的顶点的顺序$${\mathcal O}(n\log \log n)$$在有界度扩展器上，并且是有序的$${\mathcal O}(n{(\log \log n)^2})$$在某个稀疏连接状态下的 Erdős-Rényi 随机图上。我们还考虑了计算最优策略的算法问题，并证明了命中和覆盖时间计算策略之间效率的二分法。