Heuristic Functions

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 A heuristic function is a function that estimates the distance or cost to reach a goal state from a given state in a search problem. It is used in informed search algorithms to guide the search towards more promising paths in the search space.

A good heuristic function should be admissible, meaning it never overestimates the actual cost of reaching the goal state. It should also be consistent, meaning that the estimated cost of reaching a successor state is less than or equal to the estimated cost of reaching the current state plus the actual cost of transitioning between them.

Some common heuristic functions include:

  1. Manhattan distance: This is a heuristic used in grid-based search problems such as the 8-puzzle or 15-puzzle. It estimates the distance between two cells as the sum of the absolute differences of their row and column indices.

  2. Euclidean distance: This heuristic estimates the straight-line distance between two points in continuous space. It is commonly used in pathfinding problems.

  3. Maximum cardinality search: This heuristic estimates the maximum number of nodes that can be reached from the current node without revisiting any previously visited nodes. It is used in some tree and graph search problems.

  4. Pattern database: This heuristic precomputes the exact cost of reaching the goal state from all possible substates of the search problem, and uses this information to estimate the cost of reaching the goal state from a given state.

  5. Neural networks: In some cases, heuristic functions can be learned from data using machine learning techniques such as neural networks. These methods can be used to learn a heuristic function that is tailored to a specific problem instance.

The choice of heuristic function depends on the specific problem being solved and the characteristics of the search space. A good heuristic function can significantly improve the efficiency of an informed search algorithm, while a poor heuristic can lead to inefficient or even incorrect solutions.

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