Query Minimization Under Stochastic Uncertainty

We study problems with stochastic uncertainty data on intervals for which the precise value can be queried by paying a cost. The goal is to devise an adaptive decision tree to find a correct solution to the problem in consideration while minimizing the expected total query cost. We show that sorting in this scenario can be performed in polynomial time, while finding the data item with minimum value seems to be hard. This contradicts intuition, since the minimum problem is easier both in the online setting with adversarial inputs and in the offline verification setting. However, the stochastic assumption can be leveraged to beat both deterministic and randomized approximation lower bounds for the online setting. Although some literature has been devoted to minimizing query/probing costs when solving uncertainty problems with stochastic input, none of them have considered the setting we describe. Our approach is closer to the study of query-competitive algorithms, and it gives a better perspective on the impact of the stochastic assumption.