I'm looking for a sorting algorithm that minimizes comparisons instead of time complexity.
The algorithm shouldn't compare any two elements for which the relation between them can be derived from comparisons previously made. For example, given three elements $A$, $B$, and $C$ such that $A<B<C$, the algorithm shouldn't attempt to compare $A$ to $C$ if it already compared $A$ to $B$ and $B$ to $C$. (Of course, if the algorithm compared $A$ to $B$ and then $A$ to $C$, it would then also need to compare $B$ to $C$. That's okay.)
Since sorting algorithms have been studied for decades, I'm guessing there already exists an algorithm for this.
My use case is allowing a user to sort things by preference by showing two of those things side-by-side.