On Selecting the k Largest with Restricted Quadratic Queries

It is well known that to find the k largest elements (and their ranking) of n real numbers, n + (k-1)log n + O(1) comparisons are both necessary and sufficient. A natural question is what happens if comparisons are replaced by some other type of query? We generalize Yao's result by showing that the lower bound still holds when queries of the form Σ i a i x i <0 and (Σ i a i x i )(Σ i b i x i )<0 are used.