Admissible heuristics A heuristic h(n) is admissible if for every node n, h(n) ≤ h*(n), where h*(n) is the true cost to reach the goal state from n. An admissible heuristic never overestimates the cost to reach the goal, i.e., it is optimistic Example: h SLD(n) (never overestimates the actual road distance) This … Consistent Heuristics I Suppose two nodes u and v are connected by an edge. A heuristic function h is consistent or monotone if it satisﬁes the following: h(u) ≤e(u,v)+h(v) where e(u,v) is the edge distance from u to v. Admissible vs Consistent Heuristics. It has so long been thought that HS yields minimal cost solution graphs only if the heuristic satisfies the so-called ‘consistency condition’. I All consistent heuristics are admissible. I’m currently taking an AI class this semester. I find the topic extremely interesting and fun to learn, but that isn’t to say that there aren’t topics that confuse me. 1 Consistent (monotonic) heuristic Definition: A consistent heuristic is one for which, for every pair of nodes Admissible Heuristic: A heuristic function h(n) is said to be admissible on (G,Γ) iﬀ h(n) ≤ h∗(n) for every n ∈ G Consistent Heuristic: A heuristic function h(n) is said to be consistent (or monotone) on G iﬀ for any pair of nodes, n0 and n, the triangle inequality holds: h(n0) ≤ k(n0,n)+h(n) Heuristic Accuracy • Let h 1 and h 2 be two consistent heuristics such that for all nodes N: h 1 (N) h 2 (N) • h 2 is said to be more accurate (or more informed) than h 1 h 1 (N) = number of misplaced tiles h 2 (N) = sum of distances of every tile to its goal position h 2 is more accurate than … (1 mark for clarity of description, mentioning some of the other info here, or giving an example) (3 marks) iv.Manhattan distance, Euclidean Distance, Tiles-out-of-place are three examples. For your example, there is no additional information available regarding the two heuristics. Admissible Heuristic Let h*(N) be the cost of the optimal path from N to a goal node The heuristic function h(N) is admissible 16 if: 0 ≤h(N) ≤h*(N) An admissible heuristic function is always optimistic ! A* is optimal if heuristic is admissible. For example, we know that the eucledian distance is admissible for searching the shortest path (in terms of actual distance, not path cost). It is shown here that the requirement that the heuristic be consistent can be relaxed to the one that the heuristic be merely admissible. (Proof left to the reader.) UCS is a special case (h = 0) Graph search: A* optimal if heuristic is consistent UCS optimal (h = 0 is consistent) Consistency implies admissibility In general, most natural admissible heuristics tend to be consistent, especially if from relaxed problems For the best paths, and an “admissible” heuristic, set D to the lowest cost between adjacent squares. 3.Use a heuristic that’s not only admissible, but also consistent. 2 3 Admissible Heuristics • A heuristic h(n) is admissible if for every node n, h(n) ≤ h*(n) where h*(n) is the true cost to reach the goal state from n. • An admissible heuristic never overestimates the cost to reach the goal Admissible Heuristics • Is the Straight Line Distance heuristic h SLD If the heuristic is admissible and consistent A* nds a solution with the fewest number of exapansions. Note also that any consistent heuristic is admissible (but not always vice-versa). In the absence of obstacles, and on terrain that has the minimum movement cost D, moving one step closer to the goal should increase g by D and decrease h by D. A prime example is the difference between admissible and consistent heuristics.
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