Decision Trees for the efficient evaluation of discrete functions: worst case and expected case analysis

We study the problem of evaluating a discrete function by adaptively querying the values of its variables until they uniquely determine the value of the function. Each variable has a cost which has to be paid in order to read its value. This problem arises in several applications, among them, automatic diagnosis design and active learning. We consider two goals: the minimization of the expected cost paid for evaluating the function and the minimization of the worst case cost. A strategy for evaluating the function can be represented by a decision tree. We provide an algorithm which builds a decision tree whose expected cost is an $O(\log n)$ approximation of the minimum possible expected cost. This is the best possible approximation achievable, under the assumption that ${\cal P} \neq {\cal NP}$, and significantly improves on the previous best known result which is an $O(\log 1/p_{\min})$ approximation, where $p_{\min}$ is the minimum probability among the objects. We also show how to obtain $O(\log n)$ approximation, in fact best possible, w.r.t the worst case minimization. Therefore, we also address the issue regarding the existence of a trade-off between the minimization of the worst cost and the expected cost. We present a polynomial time procedure that given a parameter $\rho >0$ and two decision trees $D_W$ and $D_E$, the former with worst cost $W$ and the latter with expected cost $E$, produces a decision tree $D$ with worst testing cost at most $(1+\rho)W$ and expected cost at most $(1+1/\rho)E$. For the relevant case of uniform costs, the bound is improved to $(1+\rho)W$ and $(1+2/(\rho^2 +2 \rho))E$. An interesting consequence of this result is the existence of a decision tree that provides simultaneously an $O(\log n)$ approximation for both criteria.