Heuristic Functions: Continued

Significance of g(n) and h(n)

g(n): The Actual Cost to Reach Node n

  • \(g(n)\) represents the actual cost of the path from start node to the \(n\). It is the sum of the costs of all edges or actions taken to reach node \(n\) from the start node.

  • Significance to guide the search process: \(g(n)\) keeps track of the cost incurred so far in the search process. It ensures that the search process or algorithm does not revisit or expand nodes with suboptimal paths (i.e., paths with higher costs than the best known path to that node)

  • Significance in avoiding redundant computations: By keeping track of the actual cost \(g(n)\), the algorithm avoids exploring paths that would require more cost than those already explored. This allows the algorithm to focus on the most promising paths reducing the overall search space.

When algorithm A* expands a node, it has already computed the exact cost to reach that node (i.e. g(n)), ensuring that it is expanding the least costly node in terms of the the actual travel cost.

h(n): The Heuristic Estimate of the Cost to the Goal Node

  • \(h(n)\) provides an estimate of the cost to reach the goal node from \(n\). It is an approximation of the remaining cost to the goal.

  • Significance in guiding search towards the goal: \(h(n)\) helps prioritize nodes that are lilely to lead to the goal more quickly. By providing an estimate cost \(h(n)\) directs the search towards nodes that seem to be closer to the goal node, speeding up the process and reducing the search space.

  • Efficiency: A well designed heuristic function can significantly improve the efficiency of the search by reducing the number of nodes that the algorithm needs to explore/expand. In the best case scenario, if \(h(n)\) is perfect (i.e. it exactly matches the true remaining cost), the algorithm will perform optimally, visiting only the nodes on the optimal path.

In case of algorithm A*, if the heuristic is weak (e.g. if it is always 0), then A* degenerates to uniform cost search, which is less efficient, but still guarantees optimal solution.

Applications of Heuristic Functions

  • Pathfinding algorithms (like A*, Dijkstra’s algorithm) use heuristic functions to obtain the shortest path between two points

  • Game AI (like chess) use heuristic functions to provide potential to choose most strategic steps or moves

  • Optimization problems (like TSP) use heuristic functions to find near-optimal solutions

  • Constraint satisfaction problems

Challenges and Limitations of Heuristic Functions

  • Design complexity

    • One of the biggest challenges is to design an effective heuristic

    • A heuristic must accurately estimate the cost to the goal without being too conservative or to aggressive

    • A poorly designed heuristic can lead to inefficient searches or to suboptimal solutions

  • Problem-specific nature

    • A heuristic function is designed for specific problem which puts a limit to its generalization

    • A heuristic that works well for a particular problem may not be applicable to another problem

  • Computational overhead

    • While heuristics reduce search space, calculating a heuristic at each step can introduce computational overhead

    • If the cost of computing the heuristic outweighs its benefits, it may not imrpove the overall performance

  • Risk of suboptimal solutions

    • Inadmissible heuristics, while faster, risk leading to suboptimal solutions

    • AI Applications that require precision must carefully consider the trade-off between speed and accuracy

Weighted Heuristic Function

The concept of 'weighting' a heuristic refers to the idea of adjusting or scaling the heuristic value to influence how much the heuristic will guide the search compared to other factors like the actual cost of the path.

Evaluation function with weighted heuristic is defined as:

\[f(n) = g(n) + w \cdot h(n)\]

where:

  • \(w\), is a constant that can increase or decrease the influence of the heuristic on the search process.

By adjusting the weight \(w\), we can influence the balance between exploration (minimizing the actual path cost) and exploitation (minimizing the actual distance). A higher weight value tends to make the search more greedy and faster, but potentially less optimal, while a smaller weight encourages a more balance search strategy.

When \(w > 1\), i.e. the weight of the heuristic is increased, the algorithm places more emphasis on the heuristic part of the cost function, which represents the estimated distance to the goal. This can make the algorithm greedier, as it focuses more on nodes that seem to lead more directly to the goal. This may lead to faster exploration of the search space, potentially reaching the goal quicker, since the algorithm is focusing on the most promising paths based on the heuristic.

This will enable the search process to reduce the need to explore less promising branches of the search tree and hence will result in the reduction of exploration of nodes which will thus reduce the overall computational costs (in terms of time and money).

Note
 Increasing the weight of the heuristic might lead to a solution faster, but it might not be the best one in terms of total cost. A* with heavily weighted heuristic may not find the least cost path but may find a "good-enough" path quickly.

A weighted heuristic can reduce computational cost by:

  • Focusing on the search on more promising paths

  • Reducing the number of nodes expanded

  • Speeding up the search process

Note

Experimental evidence suggests that search efficiency is often enhanced by allowing the value of \(w\) to vary inversely with the depth of node in the search tree.

At shallow depth, the search relies on the heuristic component, while at greater depth, the search becomes increasingly breadth first, to ensure that some path to a goal will eventually be found.

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