Uninformed Search strategies. AIMA sections 3.4,3.5

Transcription

1 AIMA sections 3.4,3.5

2 search use only the information available in the problem denition Breadth-rst search Uniform-cost search Depth-rst search Depth-limited search Iterative deepening search

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

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

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

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

7 Properties of breadth-rst search Complete??

8 Properties of breadth-rst search Complete?? Yes (if b is nite) Time??

9 Properties of breadth-rst search Complete?? Yes (if b is nite) Time?? 1 + b + b 2 + b b d + b(b d 1) = O(b d+1 ), i.e., exp. in d Space??

10 Properties of breadth-rst search Complete?? Yes (if b is nite) Time?? 1 + b + b 2 + b b d + b(b d 1) = O(b d+1 ), i.e., exp. in d Space?? O(b d+1 ) (keeps every node in memory) Optimal??

11 Properties of breadth-rst search Complete?? Yes (if b is nite) Time?? 1 + b + b 2 + b b d + b(b d 1) = O(b d+1 ), i.e., exp. in d Space?? O(b d+1 ) (keeps every node in memory) Optimal?? Yes (if cost = 1 per step); not optimal in general

12 Properties of breadth-rst search Complete?? Yes (if b is nite) Time?? 1 + b + b 2 + b b d + b(b d 1) = O(b d+1 ), i.e., exp. in d Space?? O(b d+1 ) (keeps every node in memory) Optimal?? Yes (if cost = 1 per step); not optimal in general Space is the big problem; can easily generate nodes at 100MB/sec so 24hrs = 8640GB.

13 Uniform cost search Expand least-cost unexpanded node (i.e., minimum step cost) Implementation: fringe = queue ordered by path cost, lowest rst Equivalent to breadth-rst if step costs all equal Complete?? Yes, if step cost ɛ Time?? # of nodes with g cost of optimal solution, O(b C /ɛ ) where C is the cost of the optimal solution Space?? # of nodes with g cost of optimal solution, O(b C /ɛ ) Optimal?? Yesnodes expanded in increasing order of g(n)

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

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

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

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

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

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

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

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

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

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

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

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

26 Properties of depth-rst search Complete??

27 Properties of depth-rst search Complete?? No: fails in innite-depth spaces, spaces with loops Modify to avoid repeated states along path complete in nite spaces Time??

28 Properties of depth-rst search Complete?? No: fails in innite-depth spaces, spaces with loops Modify to avoid repeated states along path complete in nite spaces Time?? O(b m ): terrible if m is much larger than d but if solutions are dense, may be much faster than breadth-rst Space??

29 Properties of depth-rst search Complete?? No: fails in innite-depth spaces, spaces with loops Modify to avoid repeated states along path complete in nite spaces Time?? O(b m ): terrible if m is much larger than d but if solutions are dense, may be much faster than breadth-rst Space?? O(bm), i.e., linear space! Optimal??

30 Properties of depth-rst search Complete?? No: fails in innite-depth spaces, spaces with loops Modify to avoid repeated states along path complete in nite spaces Time?? O(b m ): terrible if m is much larger than d but if solutions are dense, may be much faster than breadth-rst Space?? O(bm), i.e., linear space! Optimal?? No!

31 Depth-limited search DFS + depth limit l: nodes at depth l have no successors Recursive implementation: function Depth-Limited-( problem, limit) returns soln/fail/cuto Recursive-DLS(Make-Node(Initial-State[problem]), problem, limit) function Recursive-DLS(node, problem, limit) returns soln/fail/cuto cuto-occurred? false if Goal-Test(problem, State[node]) then return node else if Depth[node] = limit then return cuto else for each successor in Expand(node, problem) do result Recursive-DLS(successor, problem, limit) if result = cuto then cuto-occurred? true else if result failure then return result if cuto-occurred? then return cuto else return failure

32 Iterative deepening search function Iterative-Deepening-( problem) returns a solution inputs: problem, a problem for depth 0 to do result Depth-Limited-( problem, depth) end if result cuto then return result

33 Iterative deepening search

34 Iterative deepening search

35 Iterative deepening search

36 Iterative deepening search

37 Properties of iterative deepening search Complete??

38 Properties of iterative deepening search Complete?? Yes Time??

39 Properties of iterative deepening search Complete?? Yes Time?? (d + 1)b 0 + db 1 + (d 1)b b d = O(b d ) Space??

40 Properties of iterative deepening search Complete?? Yes Time?? (d + 1)b 0 + db 1 + (d 1)b b d = O(b d ) Space?? O(bd) Optimal??

41 Properties of iterative deepening search Complete?? Yes Time?? (d + 1)b 0 + db 1 + (d 1)b b d = O(b d ) Space?? O(bd) Optimal?? Yes, if step cost = 1 Can be modied to explore uniform-cost tree

42 BFS Vs IDS Numerical comparison for b = 10 and d = 5, solution at far right leaf: N(IDS) = , , , 000 = 123, 456 N(BFS) = , , , , 990 = 1, 111, 101 IDS does better because other nodes at depth d are not expanded BFS can be modied to apply goal test when a node is generated

43 Summary of algorithms Criterion BF UC DF DL ID Complete? Yes Yes, No Yes, if l d Yes Time b d+1 b C /ɛ b m b l b d Space b d+1 b C /ɛ bm bl bd Optimal? Yes Yes No Yes, if l d Yes *: complete if branching factor is nite : complete if step cost is ɛ : optimal if step costs are all identical

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

45 Graph search function Graph-( problem, fringe) returns a solution, or failure closed an empty set fringe Insert(Make-Node(Initial-State[problem]), fringe) loop do end if fringe is empty then return failure node Remove-Front(fringe) if Goal-Test(problem, State[node]) then return node if State[node] is not in closed then add State[node] to closed fringe InsertAll(Expand(node, problem), fringe)

46 Summary Variety of uninformed search Iterative deepening search uses only linear space and not much more time than other uninformed algorithms Graph search can be exponentially more ecient than tree search

47 Exercise: Space Dimension BFS vs IDS Assume: i) a well balanced search tree; ii) the goal state is the last one to be expanded in its level (e.g., the rightmost). if the branching factor is 3, the shallowest goal state is at depth 3 (root has depth 0) and we proceed breadth rst how many nodes are generated? if the branching factor is 4, the shallowest goal state is at depth 3 (root has depth 0) we proceed with an iterative deepening approach, how many nodes are generated? what happens if goal test is performed when inserting in the fringe instead of when removing (as it is in tree-search)?

48 Exercise: formalizing and solving problem through search The Wolf Sheep and Cabbage Problem A man owns a wolf, a sheep and a cabbage: He is on a river bank with a boat that can carry him with only one of his goodies at a time. The man wants to reach the other bank with his wolf, sheep and cabbage, but he knows that wolves eat sheeps, and sheeps eat cabbages, so he cannot leave them alone on a bank. Formalize the WSC problem as a search problem Use breadth rst to nd a solution

49 Exercise: formalizing and solving problem through search The Missionaries and Cannibals Three missionaries and three cannibals are on the same river bank, and want to cross it. They have a boat that can carry two people at most. Cannibals should never outnumber missionaries, on any bank, as they could eat them. Formalize the MC problem as a search problem Give a solution

50 Exercise: Optimality for Graph Dierences between dierent search Consider the state space graph in the gure, all moves cost 1. S1 R S2 R S3 G U S D Answer to the following questions: S0 State whether a Graph version of BFS woul always return the optimal solution for this problem, if not provide an execution where this is not the case. State whether a Graph version of IDS woul always return the optimal solution for this problem, if not provide an execution where this is not the case.