4b-search-4p - Best-First Search Idea use a function f for...

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Artificial Intelligence Informed Search and Exploration Readings: Chapter 4 of Russell & Norvig. Best-First Search Idea: use a function f for each node n to estimate of “desirability” Strategy: Alwasy expand unexpanded node n with least f ( n ) Implementation: fringe is a priority queue sorted in decreasing order of f ( n ) Special cases: greedy search: f ( n ) = h ( n ) A search: f ( n ) = g ( n ) + h ( n ) where g ( n ) is the real cost from the initial node to n and h ( n ) is an estimate of the cost from n to a goal. Properties of Heuristic Function Admissible: h ( n ) h ( n ) for any node, where h ( n ) is the true cost from n to the nearest goal. Consistent: h ( n ) + c ( n, a, n ) h ( n ) , where n = R ESULT ( A , N ) ( n ) . Optimality/Completeness of A* Search If the problem is solvable, A* always finds an optimal solution when the standard assumptions are satisfied, the heuristic function is admissible. A* is optimally efficient for any heuristic function h : No other optimal strategy expands fewer nodes than A*.
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Complexity of A* Search Worst-case time complexity: still exponential ( O ( b d ) ) unless the error in h is bounded by the logarithm of the actual path cost. That is, unless | h ( n ) - h ( n ) | ≤ O (log h ( n )) where h ( n ) = actual cost from n to goal. Worst-Case Space Complexity: O ( b m ) as in greedy best-first. A* generally runs out of memory before running out of time. Improvements: IDA*, SMA*. How to Obtain Admissible Heuristics Admissible heuristics can be derived from the exact solution cost of a relaxed version of the problem Example: the 8-puzzle h 1 ( n ) = number of misplaced tiles How: If the rules of the 8-puzzle are relaxed so that a tile can move anywhere , then h 1 ( n ) gives the shortest solution h 2 ( n ) = total Manhattan distance (i.e., number of squares from desired location of each tile) How: If the rules are relaxed so that a tile can move to any adjacent square , then h 2 ( n ) gives the shortest solution. Key point: the optimal cost of a relaxed problem is no greater than the optimal cost of the real problem.
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