带每替换最大延迟的分页技术

中文摘要 AI 辅助

经典分页将每次缺页与立即替换绑定，本文研究缺页可等待时算法结构的变化。在提出的每替换最大延迟模型中，加载待处理页面的代价为1个移动单位加上其最久未处理请求的时长，且会清除该页面的所有待处理请求；等价地，瞬时持有率为待处理页面数而非待处理请求数，经典竞争层次结构在该模型下依然成立。对于缓存大小k，本文给出确定性的(5k+3)-竞争阈值-LRU算法，以及针对无知对手的随机5H_k-竞争算法；经典下界实例给出匹配的Ω(k)和Ω(H_k)阶。该随机算法使用与缓存无关的时间窗口生成普通分页序列，且序列在随机选择前固定，随后将影子分页算法投影到非主动物理替换上。离线场景则与经典情况不同：本文给出带一个空洞的精确O(nk)动态规划、固定空洞数的精确配置动态规划，以及无需固定空洞数的确定性非主动多项式时间5-近似算法；但在物理延迟问题中，最远未来使用的 victim 选择在仅3个页面时就可能次优。

英文摘要

Classical paging serves every miss immediately. We study paging with per-replacement maximum delay, where loading a pending page costs one movement plus the age of its oldest outstanding request and clears that page's entire episode. Equivalently, the holding rate is the number of pending pages. For cache size $k$, threshold LRU is strictly $(5k+3)$-competitive, while a randomized algorithm is strictly $5H_k$-competitive against an oblivious adversary; classical constructions give matching $Ω(k)$ and $Ω(H_k)$ orders. Offline, we obtain an exact $O(nk)$ dynamic program with one hole, an exact configuration algorithm for any fixed number of holes, and a nonproactive polynomial-time $5$-approximation in general. Physical farthest-next-use can nevertheless fail with only three pages. For page-dependent fetch costs with spread $ρ=w_{\max}/w_{\min}$, the exact fixed-number-of-holes algorithms persist. We obtain a $(3ρ+2)$-approximation and online guarantees with multiplicative factors $O(ρk)$ deterministically and $O(ρH_k)$ randomly against an oblivious adversary, plus any additive term inherited from the corresponding classical guarantee. A strict $Ω(\sqrtρ)$ randomized lower bound already holds for one cache slot. Thus maximum delay preserves the unit-cost competitive hierarchy but disrupts classical offline structure and makes weighted timing spread-sensitive.

发表机构

• Southern University of Science and Technology（南方科技大学）

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