Question

a) Give a set of broad conditions under which A* search reduces to BFS. (A* is A* search, which is a search function like BFS, DFS UCS, etc.)

b) Does A* always return an optimal solution? Explain.

This is what I have for the whole question.

Answer 1

a) The conditions under which A* reduces to BFS are as follows :-

1. The heuristic function returns a constant value. If the heuristic function returns a constant value then all subsequent paths will add the same cost to the total cost.

2. The cost of all actions are equal.

If both conditions are satisfied, A* reduces to a mere BFS search.

b) A* will return an optimal solution only if the heuristic function used is admissible. A heuristic function is said to be admissible if it never overestimates the cost of reaching a goal.

If the heuristic is admissible, then when the algorithm terminates its search, then it has found a path from start point to the goal point such that its actual cost is less than estimated cost of any other path from the start to the goal. Since our heuristic was admissible, it was underestimating the cost. So if the actual cost of solution path (found by A*) is is less than the estimated cost of other paths by the heuristic function, it is bound to be lower than the actual cost of that path because cost found by the heuristic function is always less than or equal to the actual cost.

Let f(n) be the cost of solution path found by A*. Let a(n) be actual cost of any arbitrary path, and h(n) be the estimated cost of that path by the heuristic function.

So, as heuristic is admissible, h(n)<=a(n).

But as f(n) is solution of A*, it must be less than h(n). f(n)<=h(n).

Hence, f(n)<=a(n).

Therefore, A* has found optimal path.

However, if heuristic is not admissible, then A* may return non-optimal solution. Suppose, h(n) overestimates the cost. And we have to go from point A to B. Let the optimal path go through a point C. Suppose the cost evaluated by the heuristic says it takes million miles to go from A to B via a point C (since it is overestimating). Since no two points are that far from each other on earth, A* will never consider going through point C as it will find a path having less cost than that. So, in this case it will not return optimal path.