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

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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.