Studies on Stochastic Dynamic Programming Based on Parametric Multi-stage Estimation

我们研究了一个最佳的重建方法的LRU堆栈的实现。LRU堆栈是一个线性列表，其中元素以最近最少使用的顺序存储。如果使用数组实现，每次都将访问的元素移到前面，总成本将非常大。Barriga和Ayani提出了一种有效的方法，其中元素的移动被延迟，直到违反了上行/下行访问模式。然而，当访问模式不规则时，该方法是无效的。因此，我们提出了我们的LRU堆栈的实现，其中数组和链表混合使用。利用Barriga和Ayani提出的延迟更新技术，可以考虑一种有效的堆栈重构方法。接下来，我们制定的预期成本与剩余的n个请求的动态规划。通过对方程组的分析，我们可以得到一个最佳的叠加重建时间，以及一些单调的结果。我们对两种不同的访问模式，即请求的均匀几何分布和截断几何分布进行了同样的分析。特别是，如果请求是均匀分布的，那么我们必须等待重建，直到最大访问索引超过5 N/7，其中N是元素的总数。

We investigate an optimal reconstruction method for an implementation of LRU stacks. The LRU stack is a linear list in which elements are stored in the least recently used order. If it were implemented by using an array and the accessed element were moved to the front for each time, the total cost would be very large. Barriga and Ayani proposed an effective method where the moving of elements is delayed until ascending/descending access pattern is violated. However, this method is not effective when the access pattern is irregular. So we present our implementation of an LRU stack, where an array and a linked list are mixedly used. Then an effective way of reconstructing the stack can be considered by using the lazy update technique proposed by Barriga and Ayani. Next we formulate the expected costs with remaining n requests by dynamic programming. Analyzing the equations, we can obtain an optimal reconstruction timing of the stack, and some monotone results. We make the same analysis of different two types of access patterns, that is, the uniform and the truncated geometric distributions of requests. In particular, if requests are uniformly distributed, it turns out that we have to wait the reconstruction until the maximum accessed index exceeds 5N/7, where N is the total number of elements.