Diagnosis determination: decision trees optimizing simultaneously worst and expected testing cost

In several applications of automatic diagnosis and active learning a central problem is the evaluation of a discrete function by adaptively querying the values of its variables until the values read uniquely determine the value of the function. In general reading the value of a variable is done at the expense of some cost (computational or possibly a fee to pay the corresponding experiment). The goal is to design a strategy for evaluating the function incurring little cost (in the worst case or in expectation according to a prior distribution on the possible variables' assignments). Our algorithm builds a strategy (decision tree) which attains a logarithmic approximation simultaneously for the expected and worst cost spent. This is best possible since, under standard complexity assumption, no algorithm can guarantee o(logn) approximation.