1. Problem Solving by Searching
CSC361
AI CSC361: Problem Solving & Search

2. Search Strategies: Classification
Search Strategies can be
Uninformed Search Strategy
Informed Search Strategy
Uninformed search strategies use only the information available in the problem definition.
Informed search strategies use other domain specific information.
AI CSC361: Problem Solving & Search

3. Informed Search Strategies
Informed search strategies employ problem specific knowledge to make a choice of which node to select next.  This knowledge is formulated in the form of a function f(x).
Best-first search
Uniform Cost Search
Greedy best-first search
A* search
AO*
AI CSC361: Problem Solving & Search

4. AI CSC361: Problem Solving & Search
Best First Search
Idea: Select the best node to goal check and expand.
Which is the best node?  An evaluation function f(n) tell us which node is the best. The value returned by f(n) gives an estimate of the ‘desirability’ of node n. Or its closeness to the goal state.
So the nodes in the Open List are ordered according to the values returned by f(n) and first node selected.
AI CSC361: Problem Solving & Search

5. AI CSC361: Problem Solving & Search
Best First Search: UCS
Example of Best First Search:
Uniform Cost Search, Greedy Search, A*.
UCS: its evaluation function is the total path cost incurred up to a node n.
f(n) = g(n) = cost from start state to n.
Drawback: g(n) does not direct search towards the goal, because it considers the cost incurred in the past not the cost that will be incurred in the future.
AI CSC361: Problem Solving & Search

6. AI CSC361: Problem Solving & Search

7. Best First Search: Greedy Search
Evaluation function f(n) = h(n) (a heuristic function)
h(n) gives estimate of cost from n to goal
e.g., hSLD(n) = straight-line distance from n to Bucharest
Greedy search expands the node that appears to be closest to the goal
AI CSC361: Problem Solving & Search

8. AI CSC361: Problem Solving & Search
ARAD
SIBIU
TIMSOURA
ZIRIND
253
329
374
ARAD
366
AI CSC361: Problem Solving & Search

9. AI CSC361: Problem Solving & Search

10. AI CSC361: Problem Solving & Search
Greedy Search
Complete? No – can get stuck in loops, e.g., Iasi  Neamt  Iasi  Neamt 
Time? O(bm), but a good heuristic can give dramatic improvement
Space? O(bm) -- keeps all nodes in memory
Optimal? No
AI CSC361: Problem Solving & Search

11. AI CSC361: Problem Solving & Search
A* Search
Idea: Considers cost to reach node n from start state and cost to reach goal from n.
Evaluation function f(n) = g(n) + h(n)
g(n) = cost so far to reach n
h(n) = estimated cost from n to goal
f(n) = estimated total cost of path through n to goal
AI CSC361: Problem Solving & Search

12. AI CSC361: Problem Solving & Search2 3 4
AI CSC361: Problem Solving & Search

13. AI CSC361: Problem Solving & Search5 6
AI CSC361: Problem Solving & Search

14. AI CSC361: Problem Solving & Search
A* Search
Complete? Yes
Time? O(bd) .. Exponential
Space? Keeps all nodes in memory
Optimal? Yes  provided that h(n) is admissible and consistent.
AI CSC361: Problem Solving & Search

15. AI CSC361: Problem Solving & Search
Admissible Heuristic
A heuristic h(n) is admissible if for every node n,
h(n) ≤ h*(n), where h*(n) is the true cost to reach the goal state from n.
An admissible heuristic never overestimates the cost to reach the goal, i.e., it is optimistic
Example: hSLD(n) (never overestimates the actual road distance)
Theorem: If h(n) is admissible, A* using TREE-SEARCH is optimal
AI CSC361: Problem Solving & Search

16. AI CSC361: Problem Solving & Search
Admissible Heuristic
For the 8-puzzle:
h1(n) = No. of misplaced tiles  h1(S)= 8
h2(n) = sum of distances of tiles from their goal position  h2(S) = = 18
AI CSC361: Problem Solving & Search

17. AI CSC361: Problem Solving & Search
Consistent Heuristic
A heuristic is consistent if for every node n, every successor n' of n generated by any action a,
h(n) ≤ c(n,a,n') + h(n')
If h is consistent, we have
f(n') = g(n') + h(n')
= g(n) + c(n,a,n') + h(n')
≥ g(n) + h(n)
= f(n) f(n) is non-decreasing along any path
Theorem: If h(n) is consistent, A* using GRAPH-SEARCH is optimal
AI CSC361: Problem Solving & Search

18. AI CSC361: Problem Solving & Search
Dominance
If h2(n) ≥ h1(n) for all n (both admissible) then h2 dominates h1
h2 is better for search
Typical search costs (average number of nodes expanded):
d=12 IDS = 3,644,035 nodes A*(h1) = 227 nodes A*(h2) = 73 nodes
d=24 IDS = too many nodes A*(h1) = 39,135 nodes A*(h2) = 1,641 nodes
AI CSC361: Problem Solving & Search

19. AI CSC361: Problem Solving & Search
AND-OR Search Graphs
OR Graphs: The state-space graphs dealt with previously were OR graphs; from any state one-of the applicable actions is chosen to reach the goal.
STATE 1
OR Node
AI CSC361: Problem Solving & Search

20. AI CSC361: Problem Solving & Search
AND-OR Search Graphs
AND-OR Graphs: In AND-OR graphs, from a state not one but all the paths may have to be pursued to reach the goal state.
STATE 1
AND Node
AI CSC361: Problem Solving & Search

21. AI CSC361: Problem Solving & Search
AND-OR Search Graphs
How do AND-OR graphs arise?
AND-OR graphs arise when it is advantageous to decompose a state into its components which are independent of each other.
Example: Integration (See the handout)
Example: Rewriting problem – found in syntax analysis.
AI CSC361: Problem Solving & Search

22. AI CSC361: Problem Solving & Search
AND-OR Search Graphs
Example: Rewriting Problem:
Start State: (C, B, Z)
Operators/actions: R1: C  (D, L)
R2: C (B, M)
R3: B (M, M)
R4: Z  (B, B, M)
Goal State: any state consisting only of M’s
AI CSC361: Problem Solving & Search

23. AI CSC361: Problem Solving & Search
AO*
How are AND-OR graphs searched with a best-first strategy?
A* algorithm is not suitable
Should we choose C to expand next?
No, because total cost along C
would be =9;
Along B, 5+1=6
B is the best choice. A B C D
h = 5
h = 3
h = 4 1
AI CSC361: Problem Solving & Search

24. AI CSC361: Problem Solving & Search
AO* Algorithm
General search for AND-OR graphs:
Initialize: {D1,..,Dn}  decompose initial state into independent components Di, if possible.
Loop until {Di} is Empty
Select: Select current path D* from {Di} according to some strategy and a node in D*.
Goal Check: Do all leaves in the current path D* satisfy goal_test()? If Yes, return solution.
AI CSC361: Problem Solving & Search

25. AI CSC361: Problem Solving & Search
AO* Algorithm
Expand: Expand node selected from D*. Decompose each state generated and add the components to {Di}.
End.
AI CSC361: Problem Solving & Search

26. Local Search Algorithms
AI CSC361: Problem Solving & Search

27. Local Search Algorithms
What type of problems can these algorithms be used to solve?
Many problems (e.g. 8-queens, VLSI layout, ..) have the property that the state description of a goal itself contains all the information needed for a solution – the path by which the solution is reached is does not matter.
These algorithms keep a single state and try to improve it.
AI CSC361: Problem Solving & Search

28. Local Search Algorithms
Put n queens on an n × n board with no two queens on the same row, column, or diagonal
AI CSC361: Problem Solving & Search

29. Local Search Algorithms
Generate-and-Test (the basic algorithm)
Hill-Climbing
Simple-Hill-Climbing
Steepest-Ascent-Hill-Climbing
Simulated Annealing
Genetic Algorthms
AI CSC361: Problem Solving & Search

30. Generate-and-Test Algorithm
Generate a possible solution state. (Randomly or by improving the current state)
Test if it is a solution state by comparing with the goal state.
If the solution state has been found quit else go to step 1.
AI CSC361: Problem Solving & Search

31. Hill-Climbing Algorithm
"Like climbing Everest in thick fog with amnesia“
AI CSC361: Problem Solving & Search

32. Hill-Climbing Algorithm
Problem: Algorithm depends on initial state, can get stuck in local maxima.
AI CSC361: Problem Solving & Search

33. Hill-Climbing: 8-Queens
h (n) = number of pairs of queens that are attacking each other, either directly or indirectly
h (n) = 17 for this state
AI CSC361: Problem Solving & Search

34. Hill-Climbing: 8-Queens
h (n) = 1 for this state
AI CSC361: Problem Solving & Search

35. AI CSC361: Problem Solving & Search
Hill-Climbing
Problems with Hill-Climbing:
Local Maxima
Plateaus
Ridges
AI CSC361: Problem Solving & Search

36. AI CSC361: Problem Solving & Search
Summary
Informed search algorithms
Local search algorithms
AI CSC361: Problem Solving & Search