The A* algorithm is a __graph search algorithm__ that finds the shortest path between two nodes in a graph. It works by maintaining a priority queue of nodes, ordered by their estimated distance to the goal. The algorithm then repeatedly pops the node with the lowest estimated distance from the queue and expands it, adding its children to the queue. This process continues until the goal node is reached, or the queue is empty.

The A* algorithm uses a __heuristic function__ to estimate the distance to the goal from a given node. A heuristic function is a function that estimates the cost of reaching the goal based on some domain knowledge or intuition. It helps A* prioritize the nodes that are more likely to lead to the goal, reducing the number of nodes that need to be explored.

The runtime complexity of the A* algorithm depends on the heuristic function used and the structure of the graph. In general, the runtime complexity is proportional to the number of nodes expanded by the algorithm, which can vary depending on how accurate and consistent the heuristic function is. An accurate heuristic function is one that closely approximates the actual cost to reach the goal, and a consistent heuristic function is one that satisfies the triangle inequality for any three nodes in the graph. These properties guarantee that A* always expands the most promising node first, and that it always finds the optimal solution if it exists.

In terms of input size n, where n is the number of nodes in the graph, the worst-case runtime complexity of A* is O(n), which happens when A* has to examine every node in the graph once. This can occur when the heuristic function is very bad or when there is no path to the goal. However, in practice, A* can perform much better than this with a good heuristic function, often finding the shortest path in a time that is O(log n) or O(n^c), where c < 1, depending on how well the heuristic function approximates the actual cost. For example, if A* uses a perfect heuristic function that always returns the exact cost to reach the goal, then it will find the shortest path in O(n) time, since it will only expand one node per level of depth. However, such a perfect heuristic function may not exist or may be too expensive to compute for some problems.
spam fails_task lang_mismatch pii not_appropriate hate_speech sexual_content quality toxicity humor helpfulness creativity violence