Optimizing Both Checking and Update Costs in Random Walk Search

2026-09-16

AI Summary

本文提出了一种优化随机游走搜索中更新与检查成本的方法，通过使用平均化游走策略，在保持期望检查次数的同时减少了总搜索成本。

Abstract

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.

Problem

• Random Walks
• Cost Optimization
• Marked State
• Checking Cost
• Update Cost

Innovation

• averaged walk
• Markov chains
• coupling with Kac's lemma
• optimizing costs