[Paper Review] Knowledge State Algorithms: Randomization with Limited Information

Wolfgang Bein, Lawrence L. Larmore+1 more|ArXiv.org|Jan 23, 2007

Optimization and Search Problems10 references3 citations

TL;DR

This paper introduces the knowledge state model, a framework for designing competitive randomized online algorithms with limited memory. It applies the method to the paging problem, presenting optimally competitive algorithms for k=2 and k=3 with minimal bookmark usage, achieving H₂ and H₃ competitiveness respectively.

ABSTRACT

We introduce the concept of knowledge states; many well-known algorithms can be viewed as knowledge state algorithms. The knowledge state approach can be used to to construct competitive randomized online algorithms and study the tradeoff between competitiveness and memory. A knowledge state simply states conditional obligations of an adversary, by fixing a work function, and gives a distribution for the algorithm. When a knowledge state algorithm receives a request, it then calculates one or more "subsequent" knowledge states, together with a probability of transition to each. The algorithm then uses randomization to select one of those subsequents to be the new knowledge state. We apply the method to the paging problem. We present optimally competitive algorithm for paging for the cases where the cache sizes are k=2 and k=3. These algorithms use only a very limited number of bookmarks.

Motivation & Objective

• To address the trade-off between memory usage and competitiveness in randomized online algorithms.
• To develop a formal framework—knowledge state algorithms—that enables systematic construction of such algorithms.
• To achieve optimal competitiveness (Hₖ) for the paging problem with minimal state representation, particularly for small k.
• To explore whether an order O(k) bookmark algorithm can achieve optimal competitiveness for the k-cache problem.
• To investigate breaking the 2-competitive barrier for the 2-server problem in general metric spaces.

Proposed method

• The knowledge state model uses conditional obligations via a work function and defines a distribution over future states to guide randomization.
• At each request, the algorithm computes one or more subsequent knowledge states with associated transition probabilities.
• The algorithm selects the next state via randomization based on these probabilities, forming a Markov-like process over knowledge states.
• The method employs potential functions (Φ) to formally analyze and prove competitiveness ratios using a mixed model of online computation.
• Competitiveness is proven by verifying that the expected cost plus change in potential is bounded by the optimal offline cost.
• The approach is applied to the paging problem by defining knowledge states based on page sets and request patterns, with explicit constructions for k=2 and k=3.

Experimental results

Research questions

• RQ1 Can a knowledge state model be used to design randomized online algorithms with provably optimal competitiveness and bounded memory?
• RQ2 For the k-cache problem, does an order O(k) bookmark algorithm exist that achieves Hₖ competitiveness?
• RQ3 Can the 2-competitive barrier for the 2-server problem be broken in general metric spaces using this framework?
• RQ4 What is the minimal memory requirement (in terms of number of knowledge states) to achieve optimal competitiveness for small k?
• RQ5 Can the knowledge state method be systematically applied to other online problems beyond paging and the server problem?

Key findings

• An order 2 knowledge state algorithm is constructed that is H₂-competitive for the 2-cache problem, using only one bookmark.
• An order 3 knowledge state algorithm is developed that is H₃-competitive for the 3-cache problem, improving memory efficiency over prior methods like EQUITABLE.
• For the 2-server problem in the class M₂,₄ (metric spaces with distances 1 or 2), a computer-generated order 2 knowledge state algorithm achieves 7/4-competitiveness.
• The theoretical lower bound for competitiveness of any order 2 knowledge state algorithm in M₂,₄ is 173+√137 / 112 ≈ 1.76, indicating room for improvement.
• An order 3 knowledge state algorithm for M₂,₄ is found to be 19/12-competitive, and this is proven to be optimal, as no randomized algorithm in this space can achieve better competitiveness.
• All knowledge states and transition probabilities in the order 3 algorithm are expressible using rational numbers, ensuring computational feasibility.