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1 Heuristic Search Chapter 4

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2 Outline Heuristic function Greedy Best-first search Admissible heuristic and A* Properties of A* Algorithm IDA*

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3 Heuristic Search Heuristic: A rule of thumb generally based on expert experience, common sense to guide problem-solving process In search, use a heuristic function that estimates how far we are from a goal. How do we use heuristics?

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4 Romania with step costs in km

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5 Greedy best-first search example

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9 Robot Navigation

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10 Robot Navigation 0211 587 7 3 4 7 6 7 632 8 6 45 233 36524435 546 5 6 4 5 f(N) = h(N), with h(N) = Manhattan distance to the goal

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11 Properties of greedy best-first search Complete? No – can get stuck in loops; Yes, if we can avoid repeated states Time? O(b m ), but a good heuristic can give dramatic improvement Space? O(b m ) -- keeps all nodes in memory Optimal? No

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12 A* Search A* search combines Uniform-cost and Greedy Best-first Search Evaluation function: f(N) = g(N) + h(N) where: g(N) is the cost of the best path found so far to N h(N) is an admissible heuristic f(N) is the estimated cost of cheapest solution through N 0 < c(N,N’) (no negative cost steps). Order the nodes in the fringe in increasing values of f(N)

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13 A * search example

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14 A * search example

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15 A * search example

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16 A * search example

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17 A * search example

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18 A * search example

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19 Admissible heuristic Let h*(N) be the cost of the optimal path from N to a goal node Heuristic h(N) is admissible if: 0 h(N) h*(N) An admissible heuristic is always optimistic

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20 8-Puzzle 123 456 78 12 3 4 5 67 8 Ngoal h 1 (N) = number of misplaced tiles = 6 is admissible h 2 (N) = sum of distances of each tile to goal = 13 is admissible What heuristics are overestimates?

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21 Completeness & Optimality of A* Claim: If there is a path from the initial to a goal node, A* using TREE-SEARCH terminates by finding the best path, hence is: complete optimal

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22 Optimality of A * (proof) Suppose some suboptimal goal G 2 has been generated and is in the fringe. Let n be an unexpanded node in the fringe such that n is on a shortest path to an optimal goal G. We can show f(n) < f(G 2 ), so A* would not have selected G 2.

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23 Optimality of A * (proof) Cont’d: f(G 2 ) = g(G 2 )since h(G 2 ) = 0 g(G 2 ) > g(G) since G 2 is suboptimal f(G) = g(G) since h(G) = 0 f(G 2 ) > f(G) from above

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24 Optimality of A * (proof) Cont’d: f(G 2 )> f(G) from above h(n)≤ h*(n) since h is admissible g(n) + h(n)≤ g(n) + h * (n) f(n) ≤ f(G) Hence f(G 2 ) > f(n), and A * will never select G 2 for expansion

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25 Exampel: Graph Search returns a suboptimal solution h=7 21 5 4 h=6 h=1h=0 S A B G h is admissible

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26 Exampel: Graph Search returns a suboptimal solution h=7 21 5 4 h=6 h=1h=0 S A B G h is admissible

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27 Exampel: Graph Search returns a suboptimal solution h=7 21 5 4 h=6 h=1h=0 S A B G h is admissible

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28 Exampel: Graph Search returns a suboptimal solution h=7 21 5 4 h=6 h=1h=0 S A B G h is admissible

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29 Exampel: Graph Search returns a suboptimal solution h=7 21 5 4 h=6 h=1h=0 S A B G h is admissible

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30 Consistent Heuristic The admissible heuristic h is consistent (or satisfies the monotone restriction) if for every node N and every successor N’ of N: h(N) c(N,N’) + h(N’) (triangular inequality) A consisteny heuristic is admissible. N N’ h(N) h(N’) c(N,N’)

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31 Exampel: Graph Search returns a suboptimal solution h=7 21 5 4 h=6 h=1h=0 S A B G h is admissible but not consistent; e.g. h(S)=7 c(S,A) + h(A) = 5 ? No.

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32 8-Puzzle 123 456 78 12 3 4 5 67 8 Ngoal h 1 (N) = number of misplaced tiles h 2 (N) = sum of distances of each tile to goal are both consistent. But do you see why?

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33 Claims If h is consistent, then the function f along any path is non-decreasing: f(N) = g(N) + h(N) f(N’) = g(N) +c(N,N’) + h(N’) h(N) c(N,N’) + h(N’) f(N) f(N’) If h is consistent, then whenever A* expands a node it has already found an optimal path to the state associated with this node N N’ h(N) h(N’) c(N,N’)

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34 Optimality of A * A * expands nodes in order of increasing f value Gradually adds "f-contours" of nodes Contour i has all nodes with f=f i, where f i < f i+1

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35 Avoiding Repeated States in A* If the heuristic h is consistent, then: Let CLOSED be the list of states associated with expanded nodes When a new node N is generated: If its state is in CLOSED, then discard N If it has the same state as another node in the fringe, then discard the node with the largest f

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36 Heuristic Accuracy h(N) = 0 for all nodes is admissible and consistent. Hence, breadth-first and uniform- cost are particular A* !!! Let h 1 and h 2 be two admissible and consistent heuristics such that for all nodes N: h 1 (N) h 2 (N). Then, every node expanded by A* using h 2 is also expanded by A* using h 1. h 2 is more informed than h 1 h2 dominates h1 Which heuristic for 8-puzzle is better?

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37 Complexity of A* Time: exponential Space: can keep all nodes in memory If we want save space, use IDA*

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38 Iterative Deepening A* (IDA*) Use f(N) = g(N) + h(N) with admissible and consistent h Each iteration is depth-first with cutoff on the value of f of expanded nodes

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39 8-Puzzle 4 6 f(N) = g(N) + h(N) with h(N) = number of misplaced tiles Cutoff=4

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40 8-Puzzle 4 4 6 f(N) = g(N) + h(N) with h(N) = number of misplaced tiles Cutoff=4 6

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41 8-Puzzle 4 4 6 f(N) = g(N) + h(N) with h(N) = number of misplaced tiles Cutoff=4 6 5

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42 8-Puzzle 4 4 6 f(N) = g(N) + h(N) with h(N) = number of misplaced tiles Cutoff=4 6 5 5

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43 4 8-Puzzle 4 6 f(N) = g(N) + h(N) with h(N) = number of misplaced tiles Cutoff=4 6 5 56

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44 8-Puzzle 4 6 f(N) = g(N) + h(N) with h(N) = number of misplaced tiles Cutoff=5

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45 8-Puzzle 4 4 6 f(N) = g(N) + h(N) with h(N) = number of misplaced tiles Cutoff=5 6

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46 8-Puzzle 4 4 6 f(N) = g(N) + h(N) with h(N) = number of misplaced tiles Cutoff=5 6 5

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47 8-Puzzle 4 4 6 f(N) = g(N) + h(N) with h(N) = number of misplaced tiles Cutoff=5 6 57

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48 8-Puzzle 4 4 6 f(N) = g(N) + h(N) with h(N) = number of misplaced tiles Cutoff=5 6 5 7 5

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49 8-Puzzle 4 4 6 f(N) = g(N) + h(N) with h(N) = number of misplaced tiles Cutoff=5 6 5 7 55

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50 8-Puzzle 4 4 6 f(N) = g(N) + h(N) with h(N) = number of misplaced tiles Cutoff=5 6 5 7 55

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51 About Heuristics Heuristics are intended to orient the search along promising paths The time spent computing heuristics must be recovered by a better search After all, a heuristic function could consist of solving the problem; then it would perfectly guide the search Deciding which node to expand is sometimes called meta-reasoning Heuristics may not always look like numbers and may involve large amount of knowledge

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52 When to Use Search Techniques? The search space is small, and There are no other available techniques, or It is not worth the effort to develop a more efficient technique The search space is large, and There is no other available techniques, and There exist “good” heuristics

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53 Summary Heuristic function Greedy Best-first search Admissible heuristic and A* A* is complete and optimal Consistent heuristic and repeated states Heuristic accuracy IDA*

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