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Knowledge State Algorithms

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, Hk=∑ki=11iHk=∑i=1k1i. 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.

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