Adaptive intersection and t-threshold problems

Consider the problem of computing the intersection of k sorted sets whose sizes sum to n. In the comparison model, we prove a new lower bound which depends on the non-deterministic complexity of the instance, and implies that the algorithm of Demaine, López-Ortiz and Munro [1] is optimal in this “adaptive” sense (for k much smaller than n). We extend the lower bound and the algorithm to the t-Threshold Problem, which consists in finding the elements which are in at least t of the k sets. For the t-threshold Problem, the analysis of our algorithm gives a complexity in O(log k) times the lower bound. These problems are motivated by boolean queries in text database systems.