Global-input-local-output p-time algorithms

Are there polynomial-time algorithms whose input is global but output is local in nature? What I have in mind is a problem instead of an algorithm. It’s the satisfiability (SAT) problem. Each clause is global information, because it’s surrounded by a host of satisfiable assignments, or rather their inverses, in the search space. But the goal, a satisfying assignment, is very local, i.e., one point in the search space. I want example problems and solutions that might shed light on SAT.