I am talking about frontier A* where you discard all nodes that have already been visited. This requires the heuristic to be consistent. As opposed to the breadth-first heuristic search which revisits nodes.
Ah. Yes, in frontier A* you definitely need a consistent heuristic, although I'd be surprised if frontier A* is the most popular form of the algorithm used, particularly for grid search. You really only want it in a very large implicit domain where the closed list would be too large to fit in memory. If the problem space is small enough then the traditional form of A* with an explicit closed list is much easier to think about.
A nitpick: "breadth-first heuristic search" is an algorithm developed by Zhou and Hansen and, while it's related to A* (in that it uses an admissible heuristic to prune the search space), it's not actually a variant of A*. (It's not a best-first search algorithm, since it doesn't expand nodes in increasing order of f cost.)
A nitpick: "breadth-first heuristic search" is an algorithm developed by Zhou and Hansen and, while it's related to A* (in that it uses an admissible heuristic to prune the search space), it's not actually a variant of A*. (It's not a best-first search algorithm, since it doesn't expand nodes in increasing order of f cost.)