Properties
Like breadth-first search, A* is complete and will always find a solution if one exists.
If the heuristic function is admissible, meaning that it never overestimates the actual minimal cost of reaching the goal, then A* is itself admissible (or optimal) if we do not use a closed set. If a closed set is used, then must also be monotonic (or consistent) for A* to be optimal. This means that for any pair of adjacent nodes and, where denotes the length of the edge between them, we must have:
This ensures that for any path from the initial node to :
where denotes the length of a path, and is the path extended to include . In other words, it is impossible to decrease (total distance so far + estimated remaining distance) by extending a path to include a neighboring node. (This is analogous to the restriction to nonnegative edge weights in Dijkstra's algorithm.) Monotonicity implies admissibility when the heuristic estimate at any goal node itself is zero, since (letting be a shortest path from any node to the nearest goal ):
A* is also optimally efficient for any heuristic, meaning that no optimal algorithm employing the same heuristic will expand fewer nodes than A*, except when there are multiple partial solutions where exactly predicts the cost of the optimal path. Even in this case, for each graph there exists some order of breaking ties in the priority queue such that A* examines the fewest possible nodes.
Read more about this topic: A* Search Algorithm
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