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Notes Dijstra s Algorithm Corrected syllabus

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Tree Search Implementation Strategies Require data structure to model search tree Tree Node : State ( e. g. Sibiu ) Parent ( e. g. Arad ) Action ( e. g. GoTo ( Sibiu )) Path cost or depth ( e. g. 140) Children ( e. g. Faragas, Oradea ) ( optional, helpful in debugging )

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Queue Methods : Empty ( queue ) Returns true if there are no more elements Pop ( queue ) Remove and return the first element Insert ( queue, element ) Inserts element into the queue InsertFIFO ( queue, element ) – inserts at the end InsertLIFO ( queue, element ) – inserts at the front InsertPriority ( queue, element, value ) – inserts sorted by value

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INFORMED SEARCH

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Search goal start Uninformed searchInformed search

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Informed Search What if we had an evaluation function h ( n ) that gave us an estimate of the cost associated with getting from n to the goal h ( n ) is called a heuristic

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Romania with step costs in km h(n)

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Greedy best - first search Evaluation function f ( n ) = h ( n ) ( heuristic ) e. g., f ( n ) = h SLD ( n ) = straight - line distance from n to Bucharest Greedy best - first search expands the node that is estimated to be closest to goal

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Romania with step costs in km nh(n)f(n)

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Best - First Algorithm

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Performance of greedy best - first search Complete ? Optimal ?

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Failure case for best - first search

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Performance of greedy best - first search Complete ? No – can get stuck in loops, e. g., Iasi Neamt Iasi Neamt Optimal ? No

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Complexity of greedy best first search Time ? O ( b m ), but a good heuristic can give dramatic improvement Space ? O ( b m ) -- keeps all nodes in memory

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What can we do better ?

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A * search Ideas : Avoid expanding paths that are already expensive Consider Cost to get here ( known ) – g ( n ) Cost to get to goal ( estimate from the heuristic ) – h ( n )

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A * Evaluation functions 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 start goal n g(n) h(n) f(n)

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ng(n)h(n)f(n)

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A * Heuristics A heuristic h ( n ) is admissible if for every node n, h ( n ) h * ( n ), where h * ( n ) is the true 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 )

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What happens if heuristic is not admissible ? Will still find solution ( complete ) But might not find best solution ( not optimal )

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Properties of A * ( w / admissible heuristic ) Complete ? Yes ( unless there are infinitely many nodes with f f ( G ) ) Optimal ? Yes Time ? Exponential, approximately O ( b d ) in the worst case Space ? O ( b m ) Keeps all nodes in memory

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The heuristic h ( x ) guides the performance of A * Let d ( x ) be the actual distance between S and G h ( x ) = 0 : A * is equivalent to Uniform - Cost Search h ( x ) <= d ( x ) : guarantee to compute the shortest path ; the lower the value h ( x ), the more node A * expands h ( x ) = d ( x ) : follow the best path ; never expand anything else ; difficult to compute h ( x ) in this way ! h ( x ) > d ( x ) : not guarantee to compute a best path ; but very fast h ( x ) >> g ( x ) : h ( n ) dominates -> A * becomes the best first search

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Admissible heuristics

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E. g., for the 8- puzzle :

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Admissible heuristics E. g., for the 8- puzzle : h 1 ( n ) = number of misplaced tiles h 2 ( n ) = summed Manhattan distance for all tiles ( i. e., no. of squares from desired location of each tile ) h 1 ( S ) = ? h 2 ( S ) = ?

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Admissible 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 ) h 1 ( S ) = ? 8 h 2 ( S ) = ? = 18 Which is better ?

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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 What does better mean ? All searches we ve discussed are exponential in time

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Comparison of algorithms ( number of nodes expanded ) DIterative deepeningA*(teleporting tiles)A* (manhattan distance) * *

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Visually

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Where do heuristics come from ? From people Knowledge of the problem From computers By considering a simpler version of the problem Called a relaxation

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Relaxed problems 8- puzzle If the rules of the 8- puzzle are relaxed so that a tile can move anywhere, then h 1 ( n ) gives the shortest solution If the rules are relaxed so that a tile can move to any adjacent square, then h 2 ( n ) gives the shortest solution Consider the example of straight line distance ( Romania navigation ) Is that a relaxation ?

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Iterative - Deepening A * ( IDA *) Further reduce memory requirements of A * Regular Iterative - Deepening : regulated by depth IDA *: regulated by f ( n )= g ( n )+ h ( n )

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Questions ?

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