Searching: Where it all begins...

Transcription

1 Searching: Where it all begins... CPSC 433 Christian Jacob Dept. of Computer Science Dept. of Biochemistry & Molecular Biology University of Calgary

2 Problem Solving by Searching 1. Introductory Concepts and Examples 2. Basic Search Techniques 3. Heuristic Search 4. Problem Decomposition and AND/OR Graphs 5. Searching in Games

3 Problem Solving by Searching 1. Introductory Concepts and Examples 2. Basic Search Techniques 3. Heuristic Search 4. Problem Decomposition and AND/OR Graphs 5. Searching in Games

4 Problem Solving by Searching 1. Introductory Concepts and Examples a. Problem space b. State space graphs

5 Problem Spaces A problem space is a complete set of possible states, generated by exploring a! possible steps, which may or may not lead from a given start state to a goal state.

6 Blocks Rearrangement Problem [Bratko, 2001]

7 State Space Graph

8 Components of a State Space Graph Start: description with which to label the start node Operators: functions that transform from one state to another, within the constraints of the search problem Goal condition: state description(s) that correspond(s) to goal state(s)

9 State Space Graph Start: Goal:

10 State Space Graph Start: Goal:

11 State Space Graph Start: Goal:

12 State Space Graph Start: Goal:

13 Eight Puzzle Problem Start Goal Start 1: 4 steps Start 2: 5 steps Start 3: 18 steps

14 Eight Puzzle Problem Start Goal

15 Rubik s Cube

16 Chess [Newborn 1997]

17 What it boils down to: Searching in Graphs

18 Problem Solving by Searching 1. Introductory Concepts and Examples 2. Basic Search Techniques 3. Heuristic Search 4. Problem Decomposition and AND/OR Graphs 5. Searching in Games

19 Basic Search Techniques a. Depth-first or backtracking search b. Iterative deepening search c. Breadth-first search

20 Depth-First Search

21 Depth First Search: Eight Puzzle Terminal depth = 5

22 Depth-First Search in Cyclic Graphs Add cycle detection!

23 Depth-First Search Evaluation Good: Since we don t expand all nodes at a level, space complexity is modest. For branching factor b and depth m, we require b" number of nodes to be stored in memory. However, the worst case is still O(b^m). That is when we are forced to search through the whole tree.

24 Depth-First Search Evaluation Bad: If you have deep search trees (or infinite!), DFS may end up running off to infinity and not be able to recover. DFS is neither optimal nor complete. - Complete: If there is a solution, it will find it. - Optimal: The best solution is found.

25 Depth-Limited Search DLS is a modified DFS to avoid its pitfalls: Impose a limit, l, on the maximum depth level. Little changes from DFS, but not optimal. DLS is complete if the limit l we impose is greater than or equal to the depth of our solution space. Time: O(b^l), Space: bl.

26 Basic Search Techniques a. Depth-first or backtracking search b. Iterative deepening search c. Breadth-first search

27 Iterative Deepening Search: A Variant of DLS Depth bound

28 Basic Search Techniques a. Depth-first or backtracking search b. Iterative deepening search c. Breadth-first search

29 Breadth-First Search

30 Eight Puzzle: BFS

31 BFS: Complexity BFS expands from the root, where it expands a fixed number of nodes, say b. On level d = 2 we expand b^2 nodes. On level d = 3 we expand b^3 nodes. Therefore: O(b^d) time complexity All leaf nodes need to be stored in memory. Hence, space complexity = time complexity.

32 Problem Solving by Searching 1. Introductory Concepts and Examples 2. Basic Search Techniques 3. Heuristic Search 4. Problem Decomposition and AND/OR Graphs 5. Searching in Games

33 Heuristic Search a. Greedy Search b. Best-First Heuristic Search A*

34 Greedy Search on a Graph Start 75 A 118 C B D E 80 G F H I Goal State h(n) A 366 B 374 C 329 D 244 E 253 F 178 G 193 H 98 I 0 h(n) = straight-line distance heuristic

35 Greedy Search 75 B State h(n) A 366 B 374 C 329 D 244 E 253 F 178 G 193 H 98 I 0 A E C B h= A 118 C 140 E 80 G 99 F D 97 H 101 I

36 Greedy Search B State h(n) A 366 B 374 C 329 D 244 E 253 F 178 G 193 H 98 I 0 Total distance = = 431 h=253 A E C B A F G 75 h = 366 h = 178 h = 193 A 118 C D E 80 G F H I A E F h = 253 h = 178

37 Greedy Search 75 A 118 C B D State E 80 h(n) A 366 B 374 C 329 D 244 E 253 F 178 G 193 H 98 I 0 G F Total distance = =431 H I A h = 253 E h =366 h=178 h=193 h = 253 E F h = 0 A E F I I A C G h = 253 h = 178 h = 0 B

38 Greedy Search: Optimal? State h(n) B A 366 B 374 C 329 D 244 E 253 F 178 G 193 H 98 I 0 Cost(A E F I )= 431 Cost(A E G H I) = A 118 C 140 E 80 G 99 F D 97 H 101 I

39 Greedy Search: Complete? Straight-line distance A B h(n) 6 5 A Greedy search is incomplete. C D 7 0 B Starting node Worst-case time complexity: O(b^m) Target node D C

40 Heuristic Search a. Greedy Search b. Best-First Heuristic Search A*

41 A* Best-First Heuristic Search f(n) = g(n) + h(n) g(n): cost so far up to node & h: heuristic estimator from node n to the goal '

42 A* Example start g h

43 A* Example start Activate-deactivate Process 1 Process 2

44 A* Example start Activate-deactivate Process 1 Process 2

45 A* Example start Activate-deactivate Process 1 Process 2

46 A* Example start Activate-deactivate Process 1 Process 2

47 A* Example start Activate-deactivate Process 1 Process 2

48 A* Example start Activate-deactivate Process 1 Process 2

49 A* Example start Activate-deactivate Process 1 Process 2

50 A* Example start Activate-deactivate Process 1 Process 2

51 Problem Solving by Searching 1. Introductory Concepts and Examples 2. Basic Search Techniques 3. Heuristic Search 4. Problem Decomposition and AND/OR Graphs 5. Searching in Games

52 AND/OR Graphs OR AND AND

53 AND/OR Solution Tree(s) Solution Tree T: OR AND The original problem P is the root node of T. If P is an OR node then exactly one of its successors (in the AND/OR graph), together with its own solution tree, is in T. If P is an AND node, then all of its successors (in the AND/OR graph), together with their solution trees, are in T.

54 Solution Tree: Example Solution Tree 1

55 Solution Tree: Example Solution Tree 2

56 Route Problem

57 Route Problem

58 Route Problem

59 Route Problem

60 Problem Solving by Searching 1. Introductory Concepts and Examples 2. Basic Search Techniques 3. Heuristic Search 4. Problem Decomposition and AND/OR Graphs 5. Searching in Games

61 Chess Us to move Opponent to move Us to move

62 Two-Person Game AND/OR Graph OR AND

63 Searching in Games a. Minimax Strategy b. Pruning c. Chance

64 Minimax Strategy Our possible configurations Opponent s possible configurations Evaluations of configurations

65 Minimax Strategy Our possible configurations Opponent s possible configurations

66 Minimax Strategy Our possible configurations Opponent s possible configurations

67 Chess Search Tree Structure

68 Searching in Games a. Minimax Strategy b. Pruning c. Chance

69 Speedup through Pruning

70 Speedup through Pruning

71 Speedup through Pruning 3... A2 is worth at most 2 to MAX

72 Speedup through Pruning A2 is worth at most 2 to MAX

73 Alpha-Beta Pruning A sophisticated pruning algorithm Introduced and explained in this week s lab!

74 Searching in Games a. Minimax Strategy b. Pruning c. Chance

75 Give Chance a Chance

76 Weighted Minimax Max Chance Weights Min Chance Max = 6 * *

77 References Bratko, I. (2001). PROLOG Programming for Artificial Inte!igence. New York, Addison-Wesley. Kurzweil, R. (1990). The Age of Inte!igent Machines. Cambridge, MA, MIT Press. Newborn, M. (1997). Kasparov versus Deep Blue. Berlin, Springer-Verlag. Nilsson, N. (1998). Artificial Inte!igence A New Synthesis. San Francisco, CA, Morgan Kaufmann. Russel, S., and Norvig, P. (1995). Artificial Inte!igence A Modern Approach. Upper Saddle River, NJ, Prentice Hall.