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New and Improved Bounds for Markov Paging

Chirag Pabbaraju, Ali Vakilian

TL;DR

This work studies online paging under a Markov request model, focusing on how online policies fare against the optimal online policy. It shows the dominating distribution algorithm achieves a tight $2$-competitive bound against $\mathrm{OPT}$, improving upon the classic $4$-competitive guarantee, and proves a $1.5907$ lower bound for the class of dominating distribution algorithms. The authors also tighten the associated analysis and derive a sharper bound for the median deterministic algorithm to $4$-competitive, while identifying and addressing sources of looseness via an enhanced charging scheme. Finally, the paper develops learning-theoretic results showing that a learned Markov chain from samples yields near-optimal competitiveness with high probability, enabling data-driven, learning-augmented paging approaches.

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.

New and Improved Bounds for Markov Paging

TL;DR

This work studies online paging under a Markov request model, focusing on how online policies fare against the optimal online policy. It shows the dominating distribution algorithm achieves a tight -competitive bound against , improving upon the classic -competitive guarantee, and proves a lower bound for the class of dominating distribution algorithms. The authors also tighten the associated analysis and derive a sharper bound for the median deterministic algorithm to -competitive, while identifying and addressing sources of looseness via an enhanced charging scheme. Finally, the paper develops learning-theoretic results showing that a learned Markov chain from samples yields near-optimal competitiveness with high probability, enabling data-driven, learning-augmented paging approaches.

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 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 -competitive against . We substantially improve their analysis and show that the dominating distribution algorithm is in fact -competitive against . We also show a lower bound of -competitiveness for this algorithm -- to the best of our knowledge, no such lower bound was previously known.

Paper Structure

This paper contains 15 sections, 12 theorems, 63 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Other Related Work
  3. Background and Setup
  4. Markov Paging
  5. Dominating Distribution Algorithm
  6. 4-competitive Analysis of lund1999paging
  7. Sources of Looseness in the Analysis
  8. A Conservative Approach to Handling Doubly-Charged Pages
  9. Inexhaustive Clearing of Charges upon a Page Request
  10. No Accounting for Uncharged, Non-first-timer Pages
  11. Tightening the Analysis to 2-competitiveness
  12. Lower bound for Dominating Distribution Algorithms
  13. Warm-up: A 1.5-competitive Lower Bound
  14. A Slightly More General Analysis that Gives a Better Lower Bound
  15. Learning the Markov Chain from Samples

Key Result

Theorem 1

The dominating distribution algorithm suffers at most 2 times more cache misses in expectation compared to the optimal online algorithm in the Markov Paging model.

Figures (5)

  • Figure 1: Charging Scheme
  • Figure 2: Request for $q$ (a singly charged page) at $t_2$ constitutes a unit cache miss for $\mathrm{OPT}$, but the division by 2 undercounts this (and every other singly-charged cache miss in $\mathrm{OPT}$).
  • Figure 3: Here, $\beta(t_1)=1$, and $\mathrm{OPT}$ suffers a cache miss at $t_2$. However, $p$ still holds a charge on $s$ at time $T$ because it is requested at $t_3 > T$. Thus, this charge is included in $\mathcal{O}$, canceling out the contribution due to the cache miss at $t_2$.
  • Figure 4: At $t_3$, $q$ does not have any charges on it, but still causes a cache miss for $\mathrm{OPT}$.
  • Figure 5: Updated Charging Scheme

Theorems & Definitions (27)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4: Theorems 1, 2 in karlin1992markov
  • Theorem 5: Theorem 3.4 in lund1999paging
  • Claim 2.1
  • proof
  • Lemma 3.1: Lemma 2.5 in lund1999paging
  • proof
  • Corollary 3.2
  • ...and 17 more