Knowledge State Algorithms: Randomization with Limited Information

TL;DR

This paper introduces knowledge state algorithms, a framework for designing randomized online algorithms with limited memory, and applies it to develop optimal paging algorithms for small cache sizes.

Contribution

It presents the concept of knowledge states for constructing competitive randomized online algorithms and provides optimal paging algorithms for cache sizes 2 and 3.

Findings

Developed the knowledge state framework for online algorithms.

Designed optimal paging algorithms for cache sizes 2 and 3.

Achieved algorithms with limited memory and competitive performance.

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.