Uninformed Search

1. Uninformed Search ECE457 Applied Artificial Intelligence Spring 2008 Lecture #2

2. Outline • Problem-solving by searching • Uninformed search techniques • Russell & Norvig, chapter 3 ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 2

3. Problem-solving by searching • An agent needs to perform actions to get from its current state to a goal. • This process is called searching. • Central in many AI systems • Theorem proving, VLSI layout, game playing, navigation, scheduling, etc. ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 3

4. Requirements for searching • Define the problem • Represent the search space by states • Define the actions the agent can perform and their cost • Define a goal • What is the agent searching for? • Define the solution • The goal itself? • The path (i.e. sequence of actions) to get to the goal? ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 4

5. Assumptions • Goal-based agent • Environment • Fully observable • Deterministic • Sequential • Static • Discrete • Single agent ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 5

6. Formulating problems • A well-defined problem has: • An initial state • A set of actions • A goal test • A concept of cost ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 6

7. Well-Defined Problem Example • Initial state • Action • Move blank left, right, up or down, provided it does not get out of the game • Goal test • Are the tiles in the “goal state” order? • Cost • Each move costs 1 • Path cost is the sum of moves ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 7

8. Well-Defined Problem Example • Travelling salesman problem • Find the shortest round trip to visit each city exactly once • Initial state • Any city • Set of actions • Move to an unvisited city • Goal test • Is the agent in the initial city after having visited every city? • Concept of cost • Action cost: distance between cities • Path cost: total distance travelled ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 8

9. Example: 8-puzzle left down left down left down right up ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 9

10. Parent Root Child Branching factor (b) Node (state) Expanding a node Edge (action) Maximum depth (m) Fringe Leaf Search Tree ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 10

11. Properties of Search Algos. • Completeness • Is the algorithm guaranteed to find a goal node, if one exists? • Optimality • Is the algorithm guaranteed to find the best goal node, i.e. the one with the cheapest path cost? • Time complexity • How many nodes are generated? • Space complexity • What’s the maximum number of nodes stored in memory? ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 11

12. Types of Search • Uninformed Search • Only has the information provided by the problem formulation (initial state, set of actions, goal test, cost) • Informed Search • Has additional information that allows it to judge the promise of an action, i.e. the estimated cost from a state to a goal ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 12

13. Breath-First Search ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 13

14. Breath-First Search • Complete, if b is finite • Optimal, if path cost is equal to depth • Guaranteed to return the shallowest goal (depth d) • Time complexity = O(bd+1) • Space complexity = O(bd+1) ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 14

15. Breath-First Search • Upper-bound case: goal is last node of depth d • Number of generated nodes:b+b²+b³+…+bd+(bd+1-b) = O(bd+1) • Space & time complexity: all generated nodes ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 15

16. Uniform-Cost Search • Expansion of Breath-First Search • Explore the cheapest node first (in terms of path cost) • Condition: No zero-cost or negative-cost edges. • Minimum cost is є • Breath-First Search is Uniform-Cost Search with constant-cost edges ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 16

17. Uniform-Cost Search • Complete given a finite tree • Optimal • Time complexity = O(bC*/є+1) ≥O(bd+1) • Space complexity = O(bC*/є+1) ≥O(bd+1) ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 17

18. Uniform-Cost Search • Upper-bound case: goal has path cost C*, all other actions have minimum cost of є • Depth explored before taking action C*: C*/є • Depth of fringe nodes: C*/є + 1 • Space & time complexity: all generated nodes: O(bC*/є+1) є C* є є є є є є є є є є є є є є ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 18

19. Depth-First Search ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 19

20. Depth-First Search • Complete, if m is finite • Not optimal • Time complexity = O(bm) • Space complexity = bm+1 = O(bm) • Can be reduced to O(m) with recursive algorithm ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 20

21. Depth-First Search • Upper-bound case for space: goal is last node of first branch • After that, we start deleting nodes • Number of generated nodes: b nodes at each of m levels • Space complexity: all generated nodes = O(bm) ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 21

22. Depth-First Search • Upper-bound case for time: goal is last node of last branch • Number of nodes generated:b nodes for each node of m levels (entire tree) • Time complexity: all generated nodes O(bm) ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 22

23. Depth-Limited Search • Depth-First Search with depth limit l • Avoids problems of Depth-First Search when trees are unbounded • Depth-First Search is Depth-Limited Search with l =  ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 23

24. Depth-Limited Search • Complete, if l > d • Not optimal • Time complexity = O(bl) • Space complexity = O(bl) ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 24

25. Depth-Limited Search • Upper-bound case for space: goal is last node of first branch • After that, we start deleting nodes • Number of generated nodes: b nodes at each of l levels • Space complexity: all generated nodes = O(bl) ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 25

26. Depth-Limited Search • Upper-bound case for time: goal is last node of last branch • Number of nodes generated:b nodes for each node of l levels (entire tree to depth l) • Time complexity: all generated nodes O(bl) ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 26

27. Iterative Deepening Search • Depth-First Search with increasing depth limit l • Repeat depth-limited search over and over, with l = l + 1 • Avoids problems of Depth-First Search when trees are unbounded • Avoids problem of Depth-Limited Search when goal depth d > l ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 27

28. Iterative Deepening Search • Complete , if b is finite • Optimal, if path cost is equal to depth • Guaranteed to return the shallowest goal • Time complexity = O(bd) • Space complexity = O(bd) • Nodes on levels above d are generated multiple times ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 28

29. Iterative Deepening Search • Upper-bound case for space: goal is last node of first branch • After that, we start deleting nodes • Number of generated nodes: b nodes at each of d levels • Space complexity: all generated nodes = O(bd) ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 29

30. Iterative Deepening Search • Upper-bound case for time: goal is last node of last branch • Number of nodes generated:b nodes for each node of d levels (entire tree to depth d) • Time complexity: all generated nodes O(bd) ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 30

31. Depth Searches ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 31

32. Summary of Searches 1: Assuming b finite (common in trees) 2: Assuming equal action costs 3: Assuming all costs  є 4: Unless m finite (uncommon in trees) 5: Unless l precisely selected ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 32

33. Summary / Example • Going from Arad to Bucharest ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 33

34. Summary / Example • Initial state • Being in Arad • Action • Move to a neighbouring city, if a road exists. • Goal test • Are we in Bucharest? • Cost • Move cost = distance between cities • Path cost = distance travelled since Arad ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 34

35. Summary / Example • Breath-First Search ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 35

36. Summary / Example Arad 0 Zerind 75 Sibiu 140 Timisoara 118 Oradea 146 Lugoj 229 Fagaras 239 Rimnicu 220 Craiova 366 Pietsti 317 Mehadia 299 Bucharest 418 Dobreta 374 Arad 0 Arad 0 Zerind 75 Sibiu 140 Timisoara 118 Arad 0 Zerind 75 Sibiu 140 Timisoara 118 Oradea 146 Arad 0 Zerind 75 Sibiu 140 Timisoara 118 Oradea 146 Lugoj 229 Arad 0 Zerind 75 Sibiu 140 Timisoara 118 Oradea 146 Lugoj 229 Fagaras 239 Rimnicu 220 Arad 0 Zerind 75 Sibiu 140 Timisoara 118 Oradea 146 Lugoj 229 Fagaras 239 Rimnicu 220 Craiova 366 Pietsti 317 Mehadia 299 Bucharest 418 Dobreta 374 Arad 0 Zerind 75 Sibiu 140 Timisoara 118 Oradea 146 Lugoj 229 Fagaras 239 Rimnicu 220 Craiova 366 Pietsti 317 Mehadia 299 Arad 0 Zerind 75 Sibiu 140 Timisoara 118 Oradea 146 Lugoj 229 Fagaras 239 Rimnicu 220 Craiova 366 Pietsti 317 Mehadia 299 Bucharest 450 Arad 0 Zerind 75 Sibiu 140 Timisoara 118 Oradea 146 Lugoj 229 Fagaras 239 Rimnicu 220 Craiova 366 Pietsti 317 Mehadia 299 Bucharest 450 Dobreta 374 Arad 0 Zerind 75 Sibiu 140 Timisoara 118 Oradea 146 Lugoj 229 Fagaras 239 Rimnicu 220 Craiova 366 Pietsti 317 Mehadia 299 Bucharest 418 Dobreta 374 Arad 0 Zerind 75 Sibiu 140 Timisoara 118 Oradea 146 Lugoj 229 Fagaras 239 Rimnicu 220 Craiova 366 Pietsti 317 Mehadia 299 Bucharest 418 Dobreta 374 Arad 0 Zerind 75 Sibiu 140 Timisoara 118 Oradea 146 Lugoj 229 Fagaras 239 Rimnicu 220 Arad 0 Zerind 75 Sibiu 140 Timisoara 118 Oradea 146 Lugoj 229 Fagaras 239 Rimnicu 220 Craiova 366 Pietsti 317 • Uniform-Cost Search ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 36

37. Summary / Example • Depth-First Search ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 37

38. Summary / Example • Depth-Limited Search, l = 4 ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 38

39. Summary / Example • Iterative Deepening Search ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 39

40. Repeated States Example: 8-puzzle left down left down left down right up ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 40

41. Repeated States • Unavoidable in problems where • Actions are reversible • Multiple paths to the same state are possible • Can greatly increase the number of nodes in a tree • Or even make a finite tree infinite! ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 41

42. D A B C D E D B A C C C C D D B D E E D D E E E E E E E E E E E E E E D Repeated States • Each state generates a single child twice • 26 different states • 225 leaves (i.e. state Z) • Over 67Mnodes in the tree ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 42

43. Repeated States • Maintain a closed list of visited states • Closed list (for expanded nodes) vs. open list (for fringe nodes) • Detect and discard repeated states upon generation • Increases space complexity ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 43