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**Blind Search by Prof. Jean-Claude Latombe**

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 3 2 fringe

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Breadth First 1 3 2 4 fringe

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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 + b2 + … + bd = (bd+1-1)/(b-1) = O(bd) Time and space complexity is O(bd)

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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) af(n)

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**Time and Memory Requirements**

#Nodes Time Memory 2 111 .01 msec 11 Kbytes 4 11,111 1 msec 1 Mbyte 6 ~106 1 sec 100 Mb 8 ~108 100 sec 10 Gbytes 10 ~1010 2.8 hours 1 Tbyte 12 ~1012 11.6 days 100 Tbytes 14 ~1014 3.2 years 10,000 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 + b2 + … + bm = O(bm) Time complexity is O(bm) 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)b2 + … + (1) bd = O(bd) Same as BFS! WHY???

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**Calculation db + (d-1)b2 + … + (1) bd = bd + 2bd-1 + 3bd-2 +… + db**

bd(Si=1,…, ib(1-i)) = bd (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 S G A B C 5 1 15 10 1 A 5 B 15 C G 11 G 10

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**Repeated States No Few Many 1 2 3 4 5 6 7 8 search tree is finite**

8-queens No assembly planning Few 1 2 3 4 5 6 7 8 8-puzzle and robot navigation Many search tree is finite search 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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