优化随机游走搜索中的检查与更新成本

中文摘要 AI 辅助

随机游走是搜索问题的标准工具，在该问题中，状态可被局部更新并测试其是否被标记。当更新状态与检查其是否被标记具有不同成本时，两种经典策略分别优化成本的不同部分：每步后检查在更新次数上最优，而仅在混合后重复检查在检查次数上最优。对于单个标记状态 $m$ 以及从平稳分布 $\pi$ 出发的游走，Dohotaru 和 Høyer 指出，对于按固定长度块检查的游走，两种保证可同时实现；他们的论证通过量子游走勾勒，并观察到他们不知道经典证明。我们给出一个简短且自包含的经典证明，适用于任意不可约马尔可夫链。该算法将原始转移矩阵 $P$ 替换为平均游走 $ \overline P_\tau = \frac{1}{\tau}\sum_{k=1}^{\tau} P^k, $ 其中 $\tau$ 的量级为 $\pi(m)HT(m)$。利用与原始游走的耦合及 Kac 引理，我们直接证明平均游走在期望上以 $O(1/\pi(m))$ 次检查命中标记状态。由此产生的搜索成本在期望上为 \\[ S + O(HT(m))U + O(1/\pi(m))C \\]，其中 $S$、$U$ 和 $C$ 分别表示设置、更新和检查成本。

英文摘要

Random walks are a standard tool for search problems in which a state can be updated locally and tested for being marked. When updating the state and checking whether it is marked have different costs, two classical strategies optimize different parts of the cost: checking after every step is optimal in the number of updates, while repeatedly checking only after mixing is optimal in the number of checks. For a single marked state $m$ and a walk started from its stationary distribution $π$, Dohotaru and Høyer stated that both guarantees can be matched simultaneously, for a walk that checks after blocks of a fixed length; their argument is sketched through quantum walks, and they observe that they know of no classical proof. We give a short and self-contained classical proof of such a tradeoff, for arbitrary irreducible Markov chains. The algorithm replaces the original transition matrix $P$ by the averaged walk $ \overline P_τ= \frac{1}τ\sum_{k=1}^τ P^k, $ where $τ$ is of order $π(m)HT(m)$. Using a coupling with the original walk and Kac's lemma, we prove directly that the averaged walk hits the marked state in $O(1/π(m))$ checks in expectation. The resulting search cost is \[ S + O(HT(m))U + O(1/π(m))C \] in expectation, where $S$, $U$, and $C$ denote setup, update, and checking costs.

发表机构

• Université Paris Cité（巴黎西岱大学）
• CNRS, IRIF（法国国家科学研究中心，IRIF研究所）
• Centre for Quantum Information and Communication, École polytechnique de Bruxelles, Université libre de Bruxelles（布鲁塞尔自由大学，布鲁塞尔理工学院量子信息与通信中心）

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