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Touring problems Start from Arad, visit each city at least once. What is the state-space formulation? Start from Arad, visit each city exactly once. What.

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Presentation on theme: "Touring problems Start from Arad, visit each city at least once. What is the state-space formulation? Start from Arad, visit each city exactly once. What."— Presentation transcript:

1 Touring problems Start from Arad, visit each city at least once. What is the state-space formulation? Start from Arad, visit each city exactly once. What is the state-space formulation? Two techniques for modeling constraints Invalidate states Precondition actions 1

2 2 Uninformed search strategies Uninformed search strategies use only the information available in the problem definition

3 3 Tree search algorithms Basic idea: offline, simulated exploration of state space by generating successors of already-explored states (a.k.a.~expanding states)

4 4 Tree search example

5 5

6 6

7 7 Implementation: states vs. nodes A state is a (representation of) a physical configuration A node is a data structure constituting part of a search tree includes state, parent node, action, path cost g(x), depth The Expand function creates new nodes, filling in the various fields and using the SuccessorFn of the problem to create the corresponding states. Two different nodes may refer to the same state (but on different paths)!

8 8 Implementation: general tree search

9 9 Breadth-first search Expand shallowest unexpanded node Implementation: fringe is a FIFO queue, i.e., new successors go at end

10 10 Breadth-first search Expand shallowest unexpanded node Implementation: fringe is a FIFO queue, i.e., new successors go at end

11 11 Breadth-first search Expand shallowest unexpanded node Implementation: fringe is a FIFO queue, i.e., new successors go at end

12 12 Breadth-first search Expand shallowest unexpanded node Implementation: fringe is a FIFO queue, i.e., new successors go at end

13 13 Repeated states Failure to detect repeated states can turn a linear problem into an exponential one!

14 14 Depth-first search Expand deepest unexpanded node Implementation: fringe = LIFO queue, i.e., put successors at front

15 14 Jan 2004CS Blind Search15 Depth-first search Expand deepest unexpanded node Implementation: fringe = LIFO queue, i.e., put successors at front

16 14 Jan 2004CS Blind Search16 Depth-first search Expand deepest unexpanded node Implementation: fringe = LIFO queue, i.e., put successors at front

17 14 Jan 2004CS Blind Search17 Depth-first search Expand deepest unexpanded node Implementation: fringe = LIFO queue, i.e., put successors at front

18 18 Depth-first search Expand deepest unexpanded node Implementation: fringe = LIFO queue, i.e., put successors at front

19 19 Depth-first search Expand deepest unexpanded node Implementation: fringe = LIFO queue, i.e., put successors at front

20 20 Depth-first search Expand deepest unexpanded node Implementation: fringe = LIFO queue, i.e., put successors at front

21 21 Depth-first search Expand deepest unexpanded node Implementation: fringe = LIFO queue, i.e., put successors at front

22 22 Depth-first search Expand deepest unexpanded node Implementation: fringe = LIFO queue, i.e., put successors at front

23 23 Depth-first search Expand deepest unexpanded node Implementation: fringe = LIFO queue, i.e., put successors at front

24 24 Depth-first search Expand deepest unexpanded node Implementation: fringe = LIFO queue, i.e., put successors at front

25 25 Depth-first search Expand deepest unexpanded node Implementation: fringe = LIFO queue, i.e., put successors at front

26 Which one is better? BFS or DFS? 14 Jan 2004CS Blind Search26

27 27 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 (assumed to be finite) m: maximum depth of the state space (may be ∞) Typical scenario: d is finite, but m is not

28 28 Properties of breadth-first search Complete? Yes (if b is finite) 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) Space is the bigger problem (more than time)

29 29 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

30 30 Uniform-cost search Expand least-cost unexpanded node Implementation: fringe = queue ordered by path cost Equivalent to breadth-first if step costs all equal Complete? Yes, if step cost ≥ ε Time? # of nodes with g ≤ cost of optimal solution, O(b ceiling(C*/ ε) ) where C * is the cost of the optimal solution Space? # of nodes with g ≤ cost of optimal solution, O(b ceiling(C*/ ε) ) Optimal? Yes – nodes expanded in increasing order of g(n)

31 To summarize … BFS: memory very high, complexity lower, optimal DFS: memory very low, complexity higher (maybe lower if solutions are dense), not optimal Is there an uninformed search that combines the advantages of these methods? 14 Jan 2004CS Blind Search31

32 32 Depth-limited search = depth-first search with depth limit L, i.e., nodes at depth L have no successors How to determine a good depth? Recursive implementation:

33 33 Iterative deepening search

34 34 Iterative deepening search l =0

35 35 Iterative deepening search l =1

36 36 Iterative deepening search l =2

37 37 Iterative deepening search l =3

38 38 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 = , , ,000 = 111,111 N IDS = , , ,000 = 123,456 Overhead = (123, ,111)/111,111 = 11%

39 39 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

40 40 Summary of algorithms Demo of search algorithms

41 41 Bi-directional search

42 42 Bi-directional search Pros and cons Pros O(b d ) to O(b d/2 ) in terms of time b = 10, d= 6, reduce from 11,111,100 to 22,200 Cons memory requirement, O(b d/2 ) difficult to come up with predecessor function implicit goal test

43 43

44 14 Jan 2004CS Blind Search44 Graph search

45 14 Jan 2004CS Blind Search45 Graph search Will a node be generated twice? Yes Will a node be expanded twice? No

46 14 Jan 2004CS Blind Search46 Graph search

47 Optimality of graph search Uniform search BFS DFS 14 Jan 2004CS Blind Search47


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