¿Qué heurística es admisible?

Which heuristic is admissible?

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.

In the study of path-finding problems in artificial intelligence, a heuristic function is said to be consistent, or monotone, if its estimate is always less than or equal to the estimated distance from any neighbouring vertex to the goal, plus the cost of reaching that neighbour.

The heuristic function h(n) is called admissible if h(n) is never larger than h*(n), namely h(n) is always less or equal to true cheapest cost from n to the goal. A* is admissible if it uses an admissible heuristic, and h(goal) = 0. (h(n) is smaller than h*(n)), then A* is guaranteed to find an optimal solution.

If the heuristic function is admissible, meaning that it never overestimates the actual cost to get to the goal, A* is guaranteed to return a least-cost path from start to goal. The algorithm described so far gives us only the length of the shortest path.

A heuristic is admissible if it never overestimates the true cost to a nearest goal. A heuristic is consistent if, when going from neighboring nodes a to b, the heuristic difference/step cost never overestimates the actual step cost.

(c) Prove that if a heuristic is consistent, it must be admissible. We can prove that consistency implies admissibility through induction. Recall that consistency is defined such that h(n) ≤ c(n, n + 1) + h(n + 1). Base Case: We begin by considering the n − 1th node in any path where n denotes the goal state.

Several commonly used heuristic search methods include hill climbing methods, the best-first search, the A* algorithm, simulated-annealing, and genetic algorithms (Russell and Norvig 2003). A classic example of applying heuristic search is the traveling salesman problem (Russell and Norvig 2003).

(Search) Is the max of two admissible heuristics also admissible? Answer: Yes, the max of two admissible heuristics is itself admissible, because each of the two heuristics is guaranteed to underestimate the distance from the given node to the goal, and so therefore must their max.

In an AND-OR graph AO* algorithm [1] is an efficient method to explore a solution path. AO* algorithm works mainly based on two phases. First phase will find a heuristic value for nodes and arcs in a particular level. The changes in the values of nodes will be propagated back in the next phase.

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 = the evaluation function. = estimated cost from current node to goal. is calculated using the heuristic function.

If h1 and h2 are admissible, then h3 = h1 + h2 is in general NOT admissible although this could happen in special cases (i.e., the null heuristic is admissible and it can be added to another heuristic arbitrary many times without violating admissibility). This is very easy to see.

In the case of BFS, the heuristic evaluation function equals to for each node , which is obviously less than the actual cost (thus underestimated ). This heuristic would cause the BFS algorithm to search literally all the possible paths and eventually find the optimal solution (i.e., the shortest path to the goal).

A heuristic function simply determines the next node to expand based on the least cost, it does not see if the expanded frontier leads to a goal state. Hence, it may sometimes lead to an infinite loop or a longer path (because it may lead to a dead end).