A* Algorithm (Cambridge International A-Level Computer Science)

A* algorithm

What is the A* algorithm?

• The A (A-star) algorithm is a pathfinding algorithm that builds upon Dijkstra’s algorithm

• It introduces a heuristic function to improve efficiency

• It is used to find the shortest path from a starting node to a goal node in a graph

• While Dijkstra’s algorithm considers only the actual cost from the start node:

  • A* enhances this by estimating the remaining distance to the goal
  • This makes it more goal-oriented and often more efficient

How A* improves on Dijkstra’s

• To avoid inefficient detours, A* introduces a heuristic function, called h(x)

• It estimates the straight-line distance (Euclidean distance) from the current node to the goal

• A* combines both cost and estimate using the formula:

  • f(x) = g(x) + h(x)

    • g(x) = the actual cost from the start node to the current node
    • h(x) = the heuristic estimate to the goal node
    • f(x) = the total estimated cost of the cheapest solution through node x

• By choosing nodes with the lowest f(x) value, A* aims to explore paths that are both cheap and move closer to the goal

Heuristics in A*

• The heuristic function h(x) should never overestimate the true cost to the goal
• This ensures A* remains optimally efficient
• The closer h(x) is to the true cost, the fewer nodes A* needs to explore
• If h(x) increases, it may indicate the algorithm is moving away from the goal, helping it backtrack or choose a better path
• Below is an illustration of the A* search:

Figure 2: Performing the A Search*

A* vs Dijkstra’s

Programming A* algorithm

How do you write an A* algorithm?

Pseudocode

• graph[min] assumes you're storing adjacency lists
• h[node] is your heuristic table
• g[] and f[] are maps (dictionaries) for real cost and estimated total cost
• cost(min, neighbour) should be defined or assumed available

Python

Example usage:

Output:

• A possible shortest path from 'A' to 'E', guided by the heuristic values