KI-Programmierung. Basic Search Algorithms
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1 KI-Programmierung Basic Search Algorithms Bernhard Beckert UNIVERSITÄT KOBLENZ-LANDAU Winter Term 2007/2008 B. Beckert: KI-Programmierung p.1
2 Example: Travelling in Romania Scenario On holiday in Romania; currently in Arad Flight leaves tomorrow from Bucharest B. Beckert: KI-Programmierung p.2
3 Example: Travelling in Romania Scenario On holiday in Romania; currently in Arad Flight leaves tomorrow from Bucharest Goal Be in Bucharest B. Beckert: KI-Programmierung p.2
4 Example: Travelling in Romania Scenario On holiday in Romania; currently in Arad Flight leaves tomorrow from Bucharest Goal Be in Bucharest Formulate problem States: various cities Actions: drive between cities B. Beckert: KI-Programmierung p.2
5 Example: Travelling in Romania Scenario On holiday in Romania; currently in Arad Flight leaves tomorrow from Bucharest Goal Be in Bucharest Formulate problem States: various cities Actions: drive between cities Solution Appropriate sequence of cities e.g.: Arad, Sibiu, Fagaras, Bucharest B. Beckert: KI-Programmierung p.2
6 Example: Travelling in Romania B. Beckert: KI-Programmierung p.3
7 Problem formulation Defined by the following four items 1. Initial state Example: Arad 2. Successor function S Example: S(Arad) = { gozerind, Zerind, gosibiu, Sibiu,... } 3. Goal test Example: x = Bucharest (explicit test) 4. Path cost (optional) Example: sum of distances, number of operators executed, etc. B. Beckert: KI-Programmierung p.4
8 Problem formulation Solution A sequence of operators leading from the initial state to a goal state B. Beckert: KI-Programmierung p.5
9 Selecting a state space Abstraction Real world is absurdly complex State space must be abstracted for problem solving (Abstract) state Set of real states (Abstract) operator Complex combination of real actions Example: Arad Zerind represents complex set of possible routes (Abstract) solution Set of real paths that are solutions in the real world B. Beckert: KI-Programmierung p.6
10 Tree search algorithms Simulated exploration of state space in a search tree by generating successors of already-explored states function TREE-SEARCH( problem, strategy) returns a solution or failure initialize the search tree using the initial state of problem loop do if there are no candidates for expansion then return failure choose a leaf node for expansion according to strategy if the node contains a goal state then return the corresponding solution else expand the node and add the resulting nodes to the search tree end B. Beckert: KI-Programmierung p.7
11 Tree search: Example B. Beckert: KI-Programmierung p.8
12 Tree search: Example B. Beckert: KI-Programmierung p.8
13 Tree search: Example B. Beckert: KI-Programmierung p.8
14 Implementation of search algorithms function TREE-SEARCH( problem, fringe) returns a solution or failure fringe INSERT(MAKE-NODE(INITIAL-STATE[problem]),fringe) loop do if fringe is empty then return failure node REMOVE-FIRST(fringe) if GOAL-TEST[problem] applied to STATE(node) succeeds then return node else fringe INSERT-ALL(EXPAND(node, problem), fringe) end fringe State Expand queue of nodes not yet considered gives the state that is represented by node creates new nodes by applying possible actions to node B. Beckert: KI-Programmierung p.9
15 Search strategies Strategy Defines the order of node expansion Important properties of strategies completeness time complexity space complexity optimality does it always find a solution if one exists? number of nodes generated/expanded maximum number of nodes in memory does it always find a least-cost solution? Time and space complexity measured in terms of b d maximum branching factor of the search tree depth of a solution with minimal distance to root m maximum depth of the state space (may be ) B. Beckert: KI-Programmierung p.10
16 Uninformed search strategies Uninformed search Use only the information available in the problem definition Frequently used strategies Breadth-first search Uniform-cost search Depth-first search Depth-limited search Iterative deepening search B. Beckert: KI-Programmierung p.11
17 Breadth-first search Idea Expand shallowest unexpanded node Implementation fringe is a FIFO queue, i.e. successors go in at the end of the queue B. Beckert: KI-Programmierung p.12
18 Breadth-first search Idea Expand shallowest unexpanded node Implementation fringe is a FIFO queue, i.e. successors go in at the end of the queue B. Beckert: KI-Programmierung p.12
19 Breadth-first search Idea Expand shallowest unexpanded node Implementation fringe is a FIFO queue, i.e. successors go in at the end of the queue B. Beckert: KI-Programmierung p.12
20 Breadth-first search Idea Expand shallowest unexpanded node Implementation fringe is a FIFO queue, i.e. successors go in at the end of the queue B. Beckert: KI-Programmierung p.12
21 Breadth-first search: Example Romania B. Beckert: KI-Programmierung p.13
22 Breadth-first search: Example Romania B. Beckert: KI-Programmierung p.13
23 Breadth-first search: Example Romania B. Beckert: KI-Programmierung p.13
24 Breadth-first search: Example Romania B. Beckert: KI-Programmierung p.13
25 Breadth-first search: Properties Complete Time Space Optimal B. Beckert: KI-Programmierung p.14
26 Breadth-first search: Properties Complete Yes (if b is finite) Time Space Optimal B. Beckert: KI-Programmierung p.14
27 Breadth-first search: Properties Complete Yes (if b is finite) Time 1+b+b 2 + b b d + b(b d 1) O(b d+1 ) i.e. exponential in d Space Optimal B. Beckert: KI-Programmierung p.14
28 Breadth-first search: Properties Complete Yes (if b is finite) Time 1+b+b 2 + b b d + b(b d 1) O(b d+1 ) Space O(b d+1 ) i.e. exponential in d keeps every node in memory Optimal B. Beckert: KI-Programmierung p.14
29 Breadth-first search: Properties Complete Yes (if b is finite) Time 1+b+b 2 + b b d + b(b d 1) O(b d+1 ) Space O(b d+1 ) i.e. exponential in d keeps every node in memory Optimal Yes (if cost = 1 per step), not optimal in general B. Beckert: KI-Programmierung p.14
30 Breadth-first search: Properties Complete Yes (if b is finite) Time 1+b+b 2 + b b d + b(b d 1) O(b d+1 ) Space O(b d+1 ) i.e. exponential in d keeps every node in memory Optimal Yes (if cost = 1 per step), not optimal in general Disadvantage Space is the big problem (can easily generate nodes at 5MB/sec so 24hrs = 430GB) B. Beckert: KI-Programmierung p.14
31 Romania with step costs in km B. Beckert: KI-Programmierung p.15
32 Uniform-cost search Idea Expand least-cost unexpanded node (costs added up over paths from root to leafs) Implementation fringe is queue ordered by increasing path cost Note Equivalent to breadth-first search if all step costs are equal B. Beckert: KI-Programmierung p.16
33 Uniform-cost search B. Beckert: KI-Programmierung p.17
34 Uniform-cost search B. Beckert: KI-Programmierung p.17
35 Uniform-cost search B. Beckert: KI-Programmierung p.17
36 Uniform-cost search B. Beckert: KI-Programmierung p.17
37 Uniform-cost search: Properties Complete Time Space Optimal B. Beckert: KI-Programmierung p.18
38 Uniform-cost search: Properties Complete Yes (if step costs positive) Time Space Optimal B. Beckert: KI-Programmierung p.18
39 Uniform-cost search: Properties Complete Yes (if step costs positive) Time # of nodes with past-cost less than that of optimal solution Space Optimal B. Beckert: KI-Programmierung p.18
40 Uniform-cost search: Properties Complete Yes (if step costs positive) Time Space # of nodes with past-cost less than that of optimal solution # of nodes with past-cost less than that of optimal solution Optimal B. Beckert: KI-Programmierung p.18
41 Uniform-cost search: Properties Complete Yes (if step costs positive) Time Space Optimal # of nodes with past-cost less than that of optimal solution # of nodes with past-cost less than that of optimal solution Yes B. Beckert: KI-Programmierung p.18
42 Depth-first search Idea Expand deepest unexpanded node Implementation fringe is a LIFO queue (a stack), i.e. successors go in at front of queue Note Depth-first search can perform infinite cyclic excursions Need a finite, non-cyclic search space (or repeated-state checking) B. Beckert: KI-Programmierung p.19
43 Depth-first search B. Beckert: KI-Programmierung p.20
44 Depth-first search B. Beckert: KI-Programmierung p.20
45 Depth-first search B. Beckert: KI-Programmierung p.20
46 Depth-first search B. Beckert: KI-Programmierung p.20
47 Depth-first search B. Beckert: KI-Programmierung p.20
48 Depth-first search B. Beckert: KI-Programmierung p.20
49 Depth-first search B. Beckert: KI-Programmierung p.20
50 Depth-first search B. Beckert: KI-Programmierung p.20
51 Depth-first search B. Beckert: KI-Programmierung p.20
52 Depth-first search B. Beckert: KI-Programmierung p.20
53 Depth-first search B. Beckert: KI-Programmierung p.20
54 Depth-first search B. Beckert: KI-Programmierung p.20
55 Depth-first search: Example Romania B. Beckert: KI-Programmierung p.21
56 Depth-first search: Example Romania B. Beckert: KI-Programmierung p.21
57 Depth-first search: Example Romania B. Beckert: KI-Programmierung p.21
58 Depth-first search: Example Romania B. Beckert: KI-Programmierung p.21
59 Depth-first search: Properties Complete Time Space Optimal B. Beckert: KI-Programmierung p.22
60 Depth-first search: Properties Complete Yes: if state space finite No: if state contains infinite paths or loops Time Space Optimal B. Beckert: KI-Programmierung p.22
61 Depth-first search: Properties Complete Yes: if state space finite No: if state contains infinite paths or loops Time O(b m ) Space Optimal B. Beckert: KI-Programmierung p.22
62 Depth-first search: Properties Complete Yes: if state space finite No: if state contains infinite paths or loops Time O(b m ) Space O(bm) (i.e. linear space) Optimal B. Beckert: KI-Programmierung p.22
63 Depth-first search: Properties Complete Yes: if state space finite No: if state contains infinite paths or loops Time O(b m ) Space Optimal O(bm) (i.e. linear space) No B. Beckert: KI-Programmierung p.22
64 Depth-first search: Properties Complete Yes: if state space finite No: if state contains infinite paths or loops Time O(b m ) Space Optimal O(bm) (i.e. linear space) No Disadvantage Time terrible if m much larger than d Advantage Time may be much less than breadth-first search if solutions are dense B. Beckert: KI-Programmierung p.22
65 Iterative deepening search Depth-limited search Depth-first search with depth limit B. Beckert: KI-Programmierung p.23
66 Iterative deepening search Depth-limited search Depth-first search with depth limit Iterative deepening search Depth-limit search with ever increasing limits function ITERATIVE-DEEPENING-SEARCH( problem) returns a solution or failure inputs: problem /* a problem */ for depth 0 to do result DEPTH-LIMITED-SEARCH( problem, depth) if result cutoff then return result end B. Beckert: KI-Programmierung p.23
67 Iterative deepening search with depth limit 0 B. Beckert: KI-Programmierung p.24
68 Iterative deepening search with depth limit 1 B. Beckert: KI-Programmierung p.25
69 Iterative deepening search with depth limit 2 B. Beckert: KI-Programmierung p.26
70 Iterative deepening search with depth limit 3 B. Beckert: KI-Programmierung p.27
71 Iterative deepening search: Example Romania with l = 0 B. Beckert: KI-Programmierung p.28
72 Iterative deepening search: Example Romania with l = 1 B. Beckert: KI-Programmierung p.29
73 Iterative deepening search: Example Romania with l = 1 B. Beckert: KI-Programmierung p.29
74 Iterative deepening search: Example Romania with l = 2 B. Beckert: KI-Programmierung p.30
75 Iterative deepening search: Example Romania with l = 2 B. Beckert: KI-Programmierung p.30
76 Iterative deepening search: Example Romania with l = 2 B. Beckert: KI-Programmierung p.30
77 Iterative deepening search: Example Romania with l = 2 B. Beckert: KI-Programmierung p.30
78 Iterative deepening search: Example Romania with l = 2 B. Beckert: KI-Programmierung p.30
79 Iterative deepening search: Properties Complete Time Space Optimal B. Beckert: KI-Programmierung p.31
80 Iterative deepening search: Properties Complete Yes Time Space Optimal B. Beckert: KI-Programmierung p.31
81 Iterative deepening search: Properties Complete Yes Time (d + 1)b 0 + db 1 +(d 1)b b d O(b d+1 ) Space Optimal B. Beckert: KI-Programmierung p.31
82 Iterative deepening search: Properties Complete Yes Time (d + 1)b 0 + db 1 +(d 1)b b d O(b d+1 ) Space O(bd) Optimal B. Beckert: KI-Programmierung p.31
83 Iterative deepening search: Properties Complete Yes Time (d + 1)b 0 + db 1 +(d 1)b b d O(b d+1 ) Space O(bd) Optimal Yes (if step cost = 1) B. Beckert: KI-Programmierung p.31
84 Iterative deepening search: Properties Complete Yes Time (d + 1)b 0 + db 1 +(d 1)b b d O(b d+1 ) Space O(bd) Optimal Yes (if step cost = 1) (Depth-First) Iterative-Deepening Search often used in practice for search spaces of large, infinite, or unknown depth. B. Beckert: KI-Programmierung p.31
85 Comparison Criterion Breadthfirst Uniformcost Depthfirst Iterative deepening Complete? Yes Yes No Yes Time b d+1 b d b m b d Space b d+1 b d bm bd Optimal? Yes Yes No Yes B. Beckert: KI-Programmierung p.32
86 Comparison Breadth-first search Iterative deepening search B. Beckert: KI-Programmierung p.33
87 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 B. Beckert: KI-Programmierung p.34
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