Knowledge State Algorithms

  • Authors:

  • Wolfgang Bein;Lawrence L. Larmore;John Noga;Rüdiger Reischuk

  • Affiliations:

  • University of Nevada, Center for the Advanced Study of Algorithms, School of Computer Science, 89154, Las Vegas, NV, USA;University of Nevada, Center for the Advanced Study of Algorithms, School of Computer Science, 89154, Las Vegas, NV, USA;California State University, Department of Computer Science, 91330, Northridge, CA, USA;Universität Lübeck, Institut für Theoretische Informatik, Ratzeburger Allee 160, 23538, Lübeck, Germany

  • Venue:
  • Algorithmica - Special issue: Algorithms, Combinatorics, & Geometry
  • Year:
  • 2011

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Abstract

We introduce the novel concept of knowledge states. The knowledge state approach can be used to construct competitive randomized online algorithms and study the trade-off between competitiveness and memory. Many well-known algorithms can be viewed as knowledge state algorithms. A knowledge state consists of a distribution of states for the algorithm, together with a work function which approximates the conditional obligations of the adversary. 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 uses randomization to select one of those subsequents to be the new knowledge state. We apply this method to randomized k-paging. The optimal minimum competitiveness of any randomized online algorithm for the k-paging problem is the kth harmonic number, $H_{k}=\sum^{k}_{i=1}\frac{1}{i}$. Existing algorithms which achieve that optimal competitiveness must keep bookmarks, i.e., memory of the names of pages not in the cache. An H k -competitive randomized algorithm for that problem which uses O(k) bookmarks is presented, settling an open question by Borodin and El-Yaniv. In the special cases where k=2 and k=3, solutions are given using only one and two bookmarks, respectively.