# Understanding the A* Algorithm (A Star Algorithm)

## 1. Principles of the A* Search Algorithm

The `A*`

algorithm is an extension of the traditional breadth-first search (BFS) or Dijkstra's algorithm. While BFS and Dijkstra's algorithm use a greedy approach based on the cost function `g(v)`

, `A*`

incorporates a heuristic function `h(v)`

to form a greedy approach based on the total cost function `f(v) = g(v) + h(v)`

.

In fact, BFS and Dijkstra's algorithm can be seen as special cases of the `A*`

search algorithm, where `h(v)`

is always zero.

## 2. Efficiency of the A* Algorithm

It is difficult to achieve a perfect (correct & efficient) heuristic function `h(v)`

. However, by incorporating some prior assumptions about the problem at hand, we can introduce a certain bias in the search order, making `A*`

faster than BFS or Dijkstra's algorithm in most cases.

The following animated GIF provides a visual representation:

From this GIF, we can observe the following:

- Due to the bias introduced by the heuristic function,
`A*`

is generally faster than BFS, but only by a small margin (typically searching only half of the nodes). - In the worst-case scenario,
`A*`

performs similarly to BFS.

In conclusion, the `A*`

algorithm combines the benefits of a heuristic function with a greedy search approach, allowing for efficient pathfinding in many scenarios. However, the effectiveness of `A*`

heavily relies on the quality of the heuristic function employed.