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CS 480 Lec 3 Sept 11, 09 Goals: Chapter 3 (uninformed search) project # 1 and # 2 Chapter 4 (heuristic search)

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Search strategies Uninformed Blind search Breadth-first depth-first Iterative deepening Bidirectional Branch and Bound Memoized search Informed Heuristic search Greedy search, hill climbing, Heuristics Important concepts: Completeness Time complexity Space complexity Quality of solution

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Search strategies A search strategy is defined by picking the order of node expansion Strategies are evaluated along the following dimensions: completeness: does it always find a solution if one exists? time complexity: number of nodes generated space complexity: maximum number of nodes in memory optimality: does it always find a least-cost solution? Time and space complexity are measured in terms of b: maximum branching factor of the search tree d: depth of the least-cost solution m: maximum depth of the state space (may be ∞)

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Uninformed search strategies Uninformed search strategies use only the information available in the problem definition Breadth-first search Depth-first search Depth-limited search Iterative deepening search Memoized search Branch and bound search

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Some useful definitions

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Fig 3.5(a) The finite state graph for a flip flop and (b) its transition matrix. Luger: Artificial Intelligence, 6th edition. © Pearson Education Limited, 2009 6

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Fig 3.6(a) The finite state graph and (b) the transition matrix for string recognition example 7

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Search formalism

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Breadth-first search Expand shallowest unexpanded node Implementation: fringe is a FIFO queue, i.e., new successors go at end

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Breadth-first search Expand shallowest unexpanded node Implementation: fringe is a FIFO queue, i.e., new successors go at end

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Breadth-first search Expand shallowest unexpanded node Implementation: fringe is a FIFO queue, i.e., new successors go at end

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Breadth-first search Expand shallowest unexpanded node Implementation: fringe is a FIFO queue, i.e., new successors go at end

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Properties of breadth-first search Complete? Yes (if b is finite) b = branching factor Time? 1+b+b 2 +b 3 +… +b d + b(b d -1) = O(b d+1 ) Space? O(b d+1 ) (keeps every node in memory) Optimal? Yes (if cost = 1 per step) i.e, it always finds the solution at the lowest level of the search tree since the search proceeds level by level. Space is the bigger problem (than time)

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BFS tree for sliding puzzle problem

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BFS algorithm

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Depth-first search Expand deepest unexpanded node Implementation: fringe = LIFO queue, i.e., put successors at front

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Depth-first search Expand deepest unexpanded node Implementation: fringe = LIFO queue, i.e., put successors at front

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Depth-first search Expand deepest unexpanded node Implementation: fringe = LIFO queue, i.e., put successors at front

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Depth-first search Expand deepest unexpanded node Implementation: fringe = LIFO queue, i.e., put successors at front

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Depth-first search Expand deepest unexpanded node Implementation: fringe = LIFO queue, i.e., put successors at front

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Depth-first search Expand deepest unexpanded node Implementation: fringe = LIFO queue, i.e., put successors at front

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Depth-first search Expand deepest unexpanded node Implementation: fringe = LIFO queue, i.e., put successors at front

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Depth-first search Expand deepest unexpanded node Implementation: fringe = LIFO queue, i.e., put successors at front

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Depth-first search Expand deepest unexpanded node Implementation: fringe = LIFO queue, i.e., put successors at front

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Depth-first search Expand deepest unexpanded node Implementation: fringe = LIFO queue, i.e., put successors at front

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Depth-first search Expand deepest unexpanded node Implementation: fringe = LIFO queue, i.e., put successors at front

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Depth-first search Expand deepest unexpanded node Implementation: fringe = LIFO queue, i.e., put successors at front

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Depth-first search

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Properties of depth-first search Complete? No: fails in infinite-depth spaces, spaces with loops Modify to avoid repeated states along path complete in finite spaces Time? O(b m ): terrible if m is much larger than d but if solutions are dense, may be much faster than breadth-first Space? O(bm), i.e., linear space! Optimal? No Why not?

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Depth-limited search = depth-first search with depth limit l, i.e., nodes at depth l have no successors Recursive implementation:

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Iterative deepening search Iterative deepening search is not an appropriate strategy for Latin square problem. DFS will do just as well without the overhead What about sliding piece puzzle?

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Iterative deepening search l =0

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Iterative deepening search l =1

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Iterative deepening search l =2

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Iterative deepening search l =3

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Iterative deepening search Number of nodes generated in a depth-limited search to depth d with branching factor b: N DLS = b 0 + b 1 + b 2 + … + b d-2 + b d-1 + b d Number of nodes generated in an iterative deepening search to depth d with branching factor b: N IDS = (d+1)b 0 + d b 1 + (d-1)b 2 + … + 3b d-2 +2b d-1 + 1b d For b = 10, d = 5, N DLS = 1 + 10 + 100 + 1,000 + 10,000 + 100,000 = 111,111 N IDS = 6 + 50 + 400 + 3,000 + 20,000 + 100,000 = 123,456 Overhead = (123,456 - 111,111)/111,111 = 11%

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Properties of iterative deepening search Complete? Yes Time? (d+1)b 0 + d b 1 + (d-1)b 2 + … + b d = O(b d ) Space? O(bd) Optimal? Yes, if step cost = 1

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Bidirectional search

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Properties of bidirectional search Complete? Yes Time? O(b d/2 ) Space? O(b d/2 ) Optimal? Yes.

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Summary of algorithms

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Graph search

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Summary Problem formulation usually requires abstracting away real-world details to define a state space that can feasibly be explored. Variety of uninformed search strategies. Iterative deepening search uses only linear space and not much more time than other uninformed algorithms.

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Missing link puzzle

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