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An Introduction to Artificial Intelligence Lecture 4a: Informed Search and Exploration Ramin Halavati In which we see how information about the state space can prevent algorithms from blundering about the dark.

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Outline Best-first search Greedy best-first search A * search Heuristics Local search algorithms Hill-climbing search Simulated annealing search Local beam search Genetic algorithms

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UNINFORMED? Uninformed: Path CostGoal Test –To search the states graph/tree using Path Cost and Goal Test.

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INFORMED? Informed: –More data about states such as distance to goal. –Best First Search Almost Best First Search Heuristic –h(n): estimated cost of the cheapest path from n to goal. h(goal) = 0. –Not necessarily guaranteed, but seems fine.

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Greedy Best First Search Compute estimated distances to goal. Expand the node which gains the least estimate.

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Greedy Best First Search Example Heuristic: Straight Line Distance (H SLD )

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Greedy Best First Search Example

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Properties of Greedy Best First Search Complete? –No, can get stuck in loop. Time? –O(b m ), but a good heuristic can give dramatic improvement Space? –O(b m ), keeps all nodes in memory Optimal? –No, it depends

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A * search Idea: avoid expanding paths that are already expensive Evaluation function f(n) = g(n) + h(n) –g(n) = cost so far to reach n –h(n) = estimated cost from n to goal –f(n) = estimated total cost of path through n to goal

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A * search example

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A* vs Greedy

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Admissible Heuristics h(n) is admissible: –if for every node n, h(n) ≤ h * (n), h * (n): the true cost from n to goal. –Never Overestimates. –It’s Optimistic. –Example: h SLD (n) (never overestimates the actual road distance)

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A* is Optimal Theorem: If h(n) is admissible, A * using TREE-SEARCH is optimal –TREE-SEARCH: To re-compute the cost of each node, each time you reach it. –GRAPH-SEARCH: To store the costs of all nodes, the first time you reach em.

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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. 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 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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Consistent Heuristics h(n) is consistent if: –for every node n, –every successor n' of n generated by any action a, –h(n) ≤ c(n,a,n') + h(n') Consistency: –Monotonicity –Triangular Inequality. –Usually at no extra cost! Theorem: If h(n) is consistent, A* using GRAPH-SEARCH is optimal

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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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Properties of A* Complete? Yes (unless there are infinitely many nodes with f ≤ f(G) ) Time? Exponential Space? Keeps all nodes in memory (b d ) Optimal? Yes A* prunes all nodes with f(n)>f(Goal). A* is Optimally Efficient.

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How to Design Heuristics? E.g., for the 8-puzzle: h 1 (n) = number of misplaced tiles h 2 (n) = total Manhattan distance (i.e., no. of squares from desired location of each tile)

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Admissible heuristics h 1 (n) = Number of misplaced tiles h 2 (n) = Total Manhattan distance

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Effective Branching Factor If A* finds the answer –by expanding N nodes, –using heuristic h(n), –At depth d, –b* is effective branching factor if: 1+b*+(b*) 2 +…+(b*) d = N+1

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Dominance If h 2 (n) ≥ h 1 (n) for all n (both admissible) then h 2 dominates h 1. h 2 is better for search. h 2 is more realistic. h (n)=max(h 1 (n), h 2 (n),…,h m (n)) Heuristic must be efficient.

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How to Generate Heuristics? Formal Methods –Relaxed Problems –Pattern Data Bases Disjoint Pattern Sets –Learning ABSOLVER, 1993 –A new, better heuristic for 8 puzzle. –First heuristic for Rubik’s cube.

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“ Relaxed Problem ” Heuristic A problem with fewer restrictions on the actions. The cost of an optimal solution to a relaxed problem is an admissible heuristic for the original problem. 8-puzzle: –Main Rule: A tile can be moved from square A to B if A is horizontally or vertically adjacent to B and B is empty. –Relaxed Rules: A tile can move from square A to square B if A is adjacent to B. (h 2 ) A tile can move from square A to square B if B is blank. A tile can move from square A to square B. (h 1 )

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“ Sub Problem ” Heuristic The cost to solve a subproblem. It IS admissible.

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“ Pattern Database ” Heuristics To store the exact solution cost to some sub-problems.

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“ Disjoint Pattern ” Databases Disjoint Pattern Databases. –To add the result of several Pattern- Database heuristics. Speed Up: 10 3 times for 15-Puzzle and 10 6 times for 24-Puzzle. Separablity: Rubik’s cube vs. 8-Puzzle.

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Learning Heuristics from Experience Machine Learning Techniques. Feature Selection –Linear Combinations

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BACK TO MAIN SEARCH METHOD What’s wrong with A*? It’s both Optimal and Optimally Efficient. –MEMORY

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Memory Bounded Heuristic Search Iterative Deepening A* (IDA*) –Similar to Iterative Deepening Depth First Search –Bounded by f-cost. –Memory: b*d

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Recursive Best First Search Main Idea: –To search a level with limited f-cost, based on other open nodes with continuous update.

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Recursive Best First Search

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Recursive Best First Search, Sample

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Complete? Yes, given enough space. Space? b * d Optimal? Yes, if admissible. Time? Hard to analyze. It depends…

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Memory, more memory … A*:b d IDA*, RBFS:b*d What about exactly 10 MB?

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Memory-Bounded A* MA* Simplified Memory Bounded A* (SMA*) –To store as many nodes as possible (the A* trend). –When memory is full, remove the worst current node and update its parent.

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SMA* Example

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SMA* Code

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To be continued …

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