Goormaghtigh conjecture

In computer science, specifically in algorithms related to pathfinding, a heuristic function is said to be admissible if it never overestimates the cost of reaching the goal, i.e. the cost it estimates to reach the goal is not higher than the lowest possible cost from the current point in the path. An admissible heuristic is also known as an optimistic heuristic.

Search algorithms

An admissible heuristic is used to estimate the cost of reaching the goal state in an informed search algorithm. In order for a heuristic to be admissible to the search problem, the estimated cost must always be lower than or equal to the actual cost of reaching the goal state. The search algorithm uses the admissible heuristic to find an estimated optimal path to the goal state from the current node. For example, in A* search the evaluation function (where $n$ is the current node) is:

$f(n)=g(n)+h(n)$

where

$f(n)$ = the evaluation function.

$g(n)$ = the cost from the start node to the current node

$h(n)$ = estimated cost from current node to goal.

$h(n)$ is calculated using the heuristic function. With a non-admissible heuristic, the A* algorithm could overlook the optimal solution to a search problem due to an overestimation in $f(n)$.

Formulation

$n$ is a node

$h$ is a heuristic

$h(n)$ is cost indicated by $h$ to reach a goal from $n$

$C(n)$ is the actual cost to reach a goal from n

$h$ is admissible if

$\forall n,h(n)\leq C(n)$

Construction

An admissible heuristic can be derived from a relaxed version of the problem, or by information from pattern databases that store exact solutions to subproblems of the problem, or by using inductive learning methods.

Examples

Two different examples of admissible heuristics apply to the fifteen puzzle problem:

• Hamming distance
• Manhattan distance

The Hamming distance is the total number of misplaced tiles. It is clear that this heuristic is admissible since the total number of moves to order the tiles correctly is at least the number of misplaced tiles (each tile not in place must be moved at least once). The cost (number of moves) to the goal (an ordered puzzle) is at least the Hamming distance of the puzzle.

The Manhattan distance of a puzzle is defined as:

$h(n)=\sum _{alltiles}distance(tile,correctposition)$

The Manhattan distance is an admissible heuristic because every tile will have to be moved at least the amount of spots in between itself and its correct position. Consider the puzzle below:

43	61	30	81
72	123	93	144
153	132	14	54
24	101	111

The subscripts show the Manhattan distance for each tile. The total Manhattan distance for the shown puzzle is:

$h(n)=3+1+0+1+2+3+3+4+3+2+4+4+4+1+1=36$

Notes

While all consistent heuristics are admissible, not all admissible heuristics are consistent.

For tree search problems, if an admissible heuristic is used, the A* search algorithm will never return a suboptimal goal node.

References

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See also

• Heuristic function
• Search algorithm