This paper focuses on competitive function evaluation in the context of computing with priced information. A function f is given together with a cost cx for each variable x of f. The cost cx has to be paid to read the value of x. The problem is to design algorithms that query the values of the variables sequentially in order to compute the function while trying to minimize the total cost incurred. Competitive analysis is employed to evaluate the performance of the algorithms. We describe a novel approach for devising efficient algorithms in this setting. We apply our approach to several classes of functions which have been studied in the literature of computing with priced information. In all cases considered, our approach provides algorithms that achieve better bounds than the best known algorithm for the same class of functions.More precisely, for the class of monotone boolean functions, we give a polynomial time algorithm with extremal competitiveness (k+l - √ min(k,l)) where k (l) denotes the minimum number of variables that one must read, in the worst case, in order to prove that the function under consideration evaluates to 1 (0). This dramatically improves upon the best known result which is an exponential time 2 max(k, l)-competitive algorithm. For the subclass of monotone boolean functions known as Threshold Trees we further improve our bounds and give a polynomial time algorithm with extremal competitive ratio 1.618 max(k, l).We then apply our methodology to classes of non-boolean functions. We consider the case of the so called Game Trees. We improve upon previously published results for this class of functions providing a polynomial time algorithm with extremal competitive ratio 1.5 γ(f), where γ(f) is a lower bound on the extremal competitive ratio of any deterministic algorithm.Finally, we consider the case when f is the function min (minimum). In this case, we are able to determine the optimal competitiveness for the problem. In fact we provide an algorithm with an (n-2)-competitive ratio, which matches the known lower bound.

A new strategy for querying priced information

CICALESE, Ferdinando;
2005

Abstract

This paper focuses on competitive function evaluation in the context of computing with priced information. A function f is given together with a cost cx for each variable x of f. The cost cx has to be paid to read the value of x. The problem is to design algorithms that query the values of the variables sequentially in order to compute the function while trying to minimize the total cost incurred. Competitive analysis is employed to evaluate the performance of the algorithms. We describe a novel approach for devising efficient algorithms in this setting. We apply our approach to several classes of functions which have been studied in the literature of computing with priced information. In all cases considered, our approach provides algorithms that achieve better bounds than the best known algorithm for the same class of functions.More precisely, for the class of monotone boolean functions, we give a polynomial time algorithm with extremal competitiveness (k+l - √ min(k,l)) where k (l) denotes the minimum number of variables that one must read, in the worst case, in order to prove that the function under consideration evaluates to 1 (0). This dramatically improves upon the best known result which is an exponential time 2 max(k, l)-competitive algorithm. For the subclass of monotone boolean functions known as Threshold Trees we further improve our bounds and give a polynomial time algorithm with extremal competitive ratio 1.618 max(k, l).We then apply our methodology to classes of non-boolean functions. We consider the case of the so called Game Trees. We improve upon previously published results for this class of functions providing a polynomial time algorithm with extremal competitive ratio 1.5 γ(f), where γ(f) is a lower bound on the extremal competitive ratio of any deterministic algorithm.Finally, we consider the case when f is the function min (minimum). In this case, we are able to determine the optimal competitiveness for the problem. In fact we provide an algorithm with an (n-2)-competitive ratio, which matches the known lower bound.
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11386/3108941
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