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Blind Search Russell and Norvig: Chapter 3, Sections 3.4 – 3.6 Slides adapted from: robotics.stanford.edu/~latombe/cs121/2003/home.htm by Prof. Jean-Claude Latombe

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Blind Search Depth first search Breadth first search Iterative deepening No matter where the goal is, these algorithms will do the same thing.

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Depth-First 1 fringe

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Depth-First 1 2 fringe

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Depth-First 1 2 3 fringe

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Breadth First 1 fringe

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Breadth First 1 2 fringe

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Breadth First 1 2 fringe 3

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Breadth First 1 2 fringe 3 4

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Generic Search Algorithm Path search(start, operators, is_goal) { fringe = makeList(start); while (state=fringe.popFirst()) { if (is_goal(state)) return pathTo(state); S = successors(state, operators); fringe = insert (S, fringe); } return NULL; } Depth-first: insert=prepend; Breadth-first: insert=append

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Performance Measures of Search Algorithms Completeness Is the algorithm guaranteed to find a solution when there is one? Optimality Is this solution optimal? Time complexity How long does it take? Space complexity How much memory does it require?

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Important Parameters Maximum number of successors of any state branching factor b of the search tree Minimal length of a path in the state space between the initial and a goal state depth d of the shallowest goal node in the search tree

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Evaluation of Breadth-first Search b: branching factor d: depth of shallowest goal node Complete Optimal if step cost is 1 Number of nodes generated: 1 + b + b 2 + … + b d = (b d+1 -1)/(b-1) = O(b d ) Time and space complexity is O(b d )

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Big O Notation g(n) is in O(f(n)) if there exist two positive constants a and N such that: for all n > N, g(n) a f(n)

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Time and Memory Requirements d#NodesTimeMemory 2111.01 msec11 Kbytes 411,1111 msec1 Mbyte 6~10 6 1 sec100 Mb 8~10 8 100 sec10 Gbytes 10~10 10 2.8 hours1 Tbyte 12~10 12 11.6 days100 Tbytes 14~10 14 3.2 years10,000 Tb Assumptions: b = 10; 1,000,000 nodes/sec; 100bytes/node

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Evaluation of Depth-first Search b: branching factor d: depth of shallowest goal node m: maximal depth of a leaf node Complete only for finite search tree Not optimal Number of nodes generated: 1 + b + b 2 + … + b m = O(b m ) Time complexity is O(b m ) Space complexity is O(bm) or O(m)

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Depth-Limited Strategy Depth-first with depth cutoff k (maximal depth below which nodes are not expanded) Three possible outcomes: Solution Failure (no solution) Cutoff (no solution within cutoff)

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Iterative Deepening Strategy Repeat for k = 0, 1, 2, …: Perform depth-first with depth cutoff k Complete Optimal if step cost =1 Space complexity is: O(bd) or O(d) Time complexity is: (d+1)(1) + db + (d-1)b 2 + … + (1) b d = O(b d ) Same as BFS! WHY???

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Calculation db + (d-1)b 2 + … + (1) b d = b d + 2b d-1 + 3b d-2 +… + db = b d (1 + 2b -1 + 3b -2 + … + db -d ) b d ( i=1,…, ib (1-i) ) = b d ( b/(b-1) ) 2

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Comparison of Strategies Breadth-first is complete and optimal, but has high space complexity Bad when branching factor is high Depth-first is space efficient, but neither complete nor optimal Bad when search depth is infinite Iterative deepening is asymptotically optimal

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Uniform-Cost Strategy Each step has some cost > 0. The cost of the path to each fringe node N is g(N) = costs of all steps. The goal is to generate a solution path of minimal cost. The queue FRINGE is sorted in increasing cost. S 0 1 A 5 B 15 C SG A B C 5 1 10 5 5 G 11 G 10

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Repeated States 8-queens No assembly planning Few 123 45 678 8-puzzle and robot navigation Many search tree is finitesearch tree is infinite

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Avoiding Repeated States Requires comparing state descriptions Breadth-first strategy: Keep track of all generated states If the state of a new node already exists, then discard the node

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Avoiding Repeated States Depth-first strategy: Solution 1: Keep track of all states associated with nodes in current path If the state of a new node already exists, then discard the node Avoids loops Solution 2: Keep track of all states generated so far If the state of a new node has already been generated, then discard the node Space complexity of breadth-first

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Summary Search strategies: breadth-first, depth- first, and variants Evaluation of strategies: completeness, optimality, time and space complexity Avoiding repeated states

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