Explain what is A* Search?

A* (A-star) search is a widely used graph traversal and pathfinding algorithm in computer science and artificial intelligence. It is designed to find the shortest path between a start node and a goal node in a weighted graph. The algorithm is known for its efficiency and optimality, making it suitable for various applications such as robotics, video games, and network routing.

Key Concepts of A* Search

1. Informed Search: A* is an informed search algorithm, meaning it uses heuristics to guide its search process. This allows it to explore the most promising paths first, improving efficiency compared to uninformed search algorithms.

2. Evaluation Function: The core of A* is its evaluation function $$ f(n) $$, which is defined as:
   $$
   f(n) = g(n) + h(n)
   $$

   • $$ g(n) $$: The cost from the start node to the current node $$ n $$.
   • $$ h(n) $$: The heuristic estimate of the cost from node $$ n $$ to the goal node. This heuristic must be admissible, meaning it never overestimates the true cost to reach the goal.

3. Optimality and Completeness: A* is both optimal and complete. It guarantees finding the shortest path if one exists, provided the heuristic function is admissible and consistent.

How A* Search Works

1. Initialization: The algorithm starts by placing the start node in an open list, which contains nodes to be evaluated. A closed list is also maintained to keep track of nodes that have already been evaluated.

2. Node Selection: At each step, the node with the lowest $$ f(n) $$ value is selected from the open list for expansion.

3. Expansion: The selected node is moved to the closed list, and its neighbors are evaluated. For each neighbor:

   • If the neighbor is not in the open list, it is added with its $$ f $$-score.
   • If the neighbor is already in the open list, its $$ g $$-score and $$ f $$-score are updated if the new path is cheaper.

4. Termination: The process continues until the goal node is selected for expansion, indicating that the shortest path has been found. If the open list becomes empty without finding the goal, it means there is no path from the start to the goal.

Pseudocode for A* Algorithm

Here is a simplified pseudocode for the A* algorithm:

function AStar(start, goal):
    openSet := {start}
    closedSet := {}
    cameFrom := an empty map
    gScore[start] := 0
    fScore[start] := heuristic_estimate(start, goal)

    while openSet is not empty:
        current := node in openSet with lowest fScore
        if current equals goal:
            return reconstruct_path(cameFrom, current)

        openSet.remove(current)
        closedSet.add(current)

        for each neighbor of current:
            if neighbor in clo...