Heuristic Functions. Lecture Heuristic Search. Stefan Edelkamp

Transcription

1 Heuristic Functions Lecture Heuristic Search Stefan Edelkamp

2 1 Overview Definition of heuristic estimator functions Admissible heuristics Consistent heuristics Heuristics in Puzzles: (n 2 1)-Puzzle and known extensions to it, Rubik s Cube, Sokoban, Atomix, etc. Heuristics in application areas: Action and Route Planning and Multiple Sequence Alignment Overview 1

3 2 Definition Heuristic A search heuristic is an information to guide the traversal into the direction of the set of goals In a weighted state space problem graph G = (V, E, s, T, w) a heuristic h is a node evaluation function mapping V to IR + 0. We require for t in T : h(t) = 0 all other nodes v V we have h(v) 0 so that the goal check for u simplifies to compare h(u) with 0 Definition Heuristic 2

4 3 Consistent and Admissible Estimates Many heuristics have monotone estimator functions: A heuristic h is consistent, if h(u) h(v) + w(u, v) for all edges e = (u, v) Some heuristics that are not consistent have at least the following property: A heuristic h is admissible if it is a lower bound estimate that underestimates the shortest remaining distance to the goal. Consistent and Admissible Estimates 3

5 Relation Theorem Consistent estimates are admissible. Proof If h is consistent we have h(v) h(u) w(u, v) for all (u, v) E. Let p = (v 0,..., v k ) be any path from u = v 0 to t = v k. Then we have w(p) = k 1 i=0 k 1 i=0 w(v i, v i+1 ) ( h(vi+1 ) h(v i ) ) = h(t) h(u) = h(u). This is especially true if p is an optimal path. Therefore, h(u) δ(u, T ) Consistent and Admissible Estimates 4

6 4 (n 2 1)-Puzzle A well-known estimate for the (n 2 1)-Puzzle is the Manhattan Distance (MD) For each two states S = ((x 1, y 1 ), (x 2, y 2 ),..., (x n 2 1, y n 2 1 )) and S = ((x 1, y 1 ), (x 2, y 2 ),..., (x n 2 1, y n 2 )) it is defined as 1 MD(S, S ) = n 2 1 i=1 ( xi x i + y i y i ). Lemma The Manhattan-Distance for the (n 2 1)-Puzzle is consistent Proof The difference in heuristic values is at most 1, i.e. h(v) h(u) 1, for all u, v S. This means h(v) h(u) 1 and h(u) h(v) 1. Togther with w(u, v) = 1 the latter inequality implies h(v) h(u) + w(u, v) 0. (n 2 1)-Puzzle 5

7 5 General-Sliding-Tile For the GST domain heuristic MD refers to the reference points of the pieces and is much weaker. The idea is to compute the distances dist between all instances k j of a type F j in common, that is dist(f j, F j ) = k j i=1 x i k j x i i=1 + k j i=1 y i k j y i i=1. A lower bound is then given by MD GST (S, S ) = k j=1 dist(f j, F j ). General-Sliding-Tile 6

8 6 Rubik s Cube In Rubik s Cube large pre-computed look-up table stores all solutions length of a simplified version The data is retrieved in the overall search process as heuristic estimate. Rubik s cube relaxations are found by removing color labels from the cubies For solving the cube in 1997 either all corner cubies were blank (leaving all edge cubies colored) or all corner and six edge cubies were colored (leaving six edge cubies blank). The abstraction scheme behind this idea will be studied in upcoming lectures We will also see that this heuristic is consistent Rubik s Cube 7

9 7 Sokoban A lower bound estimate for PUSHES is found using the minimum matching approach: For every goal field the shortest ball path to every ball constitue a graph in which the perfect matching has to be found. If n is the number of balls then this graph has 2n nodes and n 2 /4 edges. Lemma The heuristic is consistent Proof Moving one ball reduces the individual shortest path to each goal by at most one, and any matching will include only one of the updated shortest path distance value Sokoban 8

10 8 Atomix The condition that an atom slides as far as possible is removed: it may stop at any closer position; these moves are called generalized moves Atomix with generalized moves has an undirected search graph and is NP-hard An atom is further allowed to slide through other atoms or share a place with another atom The goal distance in this model can be summed up for all atoms to yield an admissible heuristic for the original problem. Lemma The heuristic is consistent Proof The h-values of child states can differ from that of the parent state by 0, +1 or 1. Atomix 9

11 9 Route Planning In Route Planning may may define the estimate h(u) = t u 2, where 2 denotes the Euclidean distance metric It is a lower bound, since the shortest way to the goal is at least as long as the air distance. Lemma For w(u, v) = u v 2 the heuristic h(u) = t u 2 is consistent Proof h(u) = t u 2 t v 2 + u v 2 = h(v) + w(u, v) by the triangle inequality of the Euclidean plane. Route Planning 10

12 10 Action Planning The relaxation a + of an action a = (pre(a), eff(a) +, eff(a) ) is defined as a + = (pre(a), eff(a) +, ). The relaxation of a planning problem is the one in which all actions are substituted by their relaxed counterparts We have 1. any solution that solves the original plan also solves the relaxed one; 2. all preconditions and goals can be achieved if and only if they can in the relaxed task. The relaxed planning heuristic h + is defined as the length of the shortest plan, that solves the relaxed problem Action Planning 11

13 Consistency The relaxed planning heuristic h + is admissible by Condition 1. Lemma The relaxed planning heuristic is consistent Proof Suppose plan with cost h + (v) has been established. It can be extended to a plan from u by adding the operator that leads from u to v. Therefore, we have h + (u) h + (v) + w(u, v). Solving relaxed plans is still computationally hard. It can, however, efficiently be approximated Action Planning 12