# Lecture 4 - Reading Material Sections 3.3 3.5 Optimal...

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Reading Material Sections 3.3 – 3.5 “Optimal Rectangle Packing: New Results” By R. Korf (optional) “Optimal Rectangle Packing: A Meta-CSP Approach” (optional) Sections 4.1 – 4.2

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Best-first search Idea: use an evaluation function f(n) for each node estimate of the desirability - Expand most desirable unexpanded node Implementation : Order the nodes in fringe in decreasing order of desirability Special cases: greedy best-first search A * search
Romania with step costs in km

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Greedy best-first search Evaluation function f(n) = h(n) ( h euristic) = estimate of cost from n to goal • e.g., h SLD (n) = straight-line distance from n to Bucharest Greedy best-first search expands the node that appears to be closest to goal
Greedy best-first search example

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Greedy best-first search example
Greedy best-first search example

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Greedy best-first search example
Properties of greedy best-first search Complete? Yes (with assumptions) 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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A * search Idea: avoid expanding paths that are already expensive Evaluation function f(n) = g(n) + h(n) g(n) = actual 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
Romania with step costs in km

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A * search example
A * search example

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A * search example
A * search example

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A * search example
A * search example

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Admissible heuristics A heuristic h(n) is admissible if for every node n , h(n) ≤ h * (n), where h * (n) is the true optimal 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)
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

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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
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Lecture 4 - Reading Material Sections 3.3 3.5 Optimal...

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