WalkSAT and GSAT are well-known and simple local search algorithms for solving the Boolean satisfiability problem. The pseudocode for the GSAT algorithm is copied from the question Implementing the GSAT algorithm - How to select which literal to flip? and presented below.
procedure GSAT(A,Max_Tries,Max_Flips) A: is a CNF formula for i:=1 to Max_Tries do S <- instantiation of variables for j:=1 to Max_Iter do if A satisfiable by S then return S endif V <- the variable whose flip yield the most important raise in the number of satisfied clauses; S <- S with V flipped; endfor endfor return the best instantiation found end GSAT
Here we flip the variable that maximizes the number of satisfied clauses. How is this done efficiently? The naive method is to flip every variable, and for each step through all clauses and calculate how many of them get satisfied. Even if a clause could be queried for satisfiability in constant time, the naive method would still run in $O(VC)$ time, where $V$ is the number of variables and $C$ the number of clauses. I'm sure we can do better, hence the question:
Many local search algorithms flip the variable's assignment that maximizes the number of satisfied clauses. In practice, with what data structures is this operation supported efficiently?
This is something I feel like textbooks often omit. One example is even the famous Russell & Norvig book.