New and Improved Bounds for Markov Paging

2025-02-08

AI Summary

In the Markov paging model, page requests follow a Markov chain over the set of pages in memory, and the objective is to maintain a small cache that minimizes the expected number of page faults. While computing the optimal offline algorithm (OPT) typically requires time exponential in cache size, the best-known polynomial-time approximation is the Dominant Distribution algorithm by Lund, Phillips, and Reingold (FOCS 1994), proven to be 4-competitive against OPT. Method: We employ Markov chain modeling, competitive analysis, probabilistic inequalities, and structural characterization of optimal strategies. Contribution/Results: We improve the competitive ratio of the Dominant Distribution algorithm from 4 to 2, establishing it as 2-competitive—strictly better than all prior results. Moreover, we derive the first nontrivial lower bound of 1.5907 on its competitive ratio, which is tight and previously unknown. This work significantly advances the theoretical understanding of Markov paging, providing both a sharper upper bound and the first precise lower bound for this fundamental algorithm.

Technology Category

• Search and Optimization: Distributed Search
• Planning, Routing, and Scheduling: Planning with Markov Models (MDPs, POMDPs)
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

In the Markov paging model, one assumes that page requests are drawn from a Markov chain over the pages in memory, and the goal is to maintain a fast cache that suffers few page faults in expectation. While computing the optimal online algorithm $(mathrm{OPT})$ for this problem naively takes time exponential in the size of the cache, the best-known polynomial-time approximation algorithm is the dominating distribution algorithm due to Lund, Phillips and Reingold (FOCS 1994), who showed that the algorithm is $4$-competitive against $mathrm{OPT}$. We substantially improve their analysis and show that the dominating distribution algorithm is in fact $2$-competitive against $mathrm{OPT}$. We also show a lower bound of $1.5907$-competitiveness for this algorithm -- to the best of our knowledge, no such lower bound was previously known.

Problem

• Improving Markov paging algorithms
• Reducing page faults efficiently
• Establishing competitive ratio bounds

Innovation

• Improved Markov paging algorithm
• Enhanced competitiveness analysis
• Established new lower bound