1. Informed Search

2. Informed Search Strategies
uses problem-specific knowledge beyond the definition of the problem itself
Greedy best-first search
A* search

3. Romania with step costs in km

4. Informed Search Idea: use an evaluation function f(n) for each node
estimate of "desirability"
Expand most desirable unexpanded node
Implementation:
Order the nodes in frontier in decreasing order of desirability
Special cases:
greedy best-first search
A* search

5. Greedy best-first search
Evaluation function f(n) = h(n) (heuristic)
= estimate of cost from n to goal
e.g., hSLD(n) = straight-line distance from n to Bucharest
Greedy best-first search expands the node that appears to be closest to goal

6. Romania with step costs in km

7. Greedy best-first search example

8. Greedy best-first search example

9. Greedy best-first search example

10. Greedy best-first search example

11. Properties of greedy best-first search
Complete?
Yes
Optimal? No
Time?
O(bm), but a good heuristic can give dramatic improvement
Space?
O(bm)

12. A* search Best-known form of informed search.
Idea: avoid expanding paths that are already expensive
Evaluation function f(n) = g(n) + h(n)
f(n) = estimated total cost of path through n to goal
g(n) = cost so far to reach n
h(n) = estimated cost from n to goal

13. A* search
Conditions for optimality:
Admissibility
Consistency

14. Admissible heuristics
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
Example: hSLD(n) (never overestimates the actual road distance)

15. Consistency (Monotonicity)
A heuristic h(n) is consistent if, for every node parent and every successor child of parent,
h(parent) <= c(parent->child) + h(child)
triangle inequality

16. Romania with step costs in km

17. A* search example

18. A* search example

19. A* search example

20. A* search example

21. A* search example

22. A* search example

23. A* Search Optimal Time Complete Yes
The better the heuristic, the better the time
Best case h is perfect, O(d)
Worst case h = 0, O(bd) same as Uniform Cost Search
Space
O(bd)

24. Heuristic Functions
To use A* a heuristic function must be used that never overestimates the number of steps to the goal

25. Heuristic Function
Function h(N) that estimates the cost of the path from node N to goal node.
Example: 8-puzzle 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8
h1(N) = number of misplaced tiles
= 6 N
goal

26. Manhattan distances

27. Heuristic Function
Function h(N) that estimate the cost of the path from node N to goal node.
Example: 8-puzzle
h2(N) = sum of the distances of
every tile to its goal position =
= 13 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 N
goal

28. 8-Puzzle f(N) = h1(N) = number of misplaced tiles 3 4 3 4 5 3 3 4 2 44 4 2 1

29. 8-Puzzle f(N) = g(N) + h1 (N) with h1(N) = number of misplaced tiles
3+3
3+4
1+5
1+3
2+3
2+4
5+2
5+0
0+4
3+4
3+2
4+1

30. 8-Puzzle
f(N) = h2(N) =  distances of tiles to goal 6 4 5 3 2 5 4 2 1

31. 8-Puzzle EXERCISE: f(N) = g(N) + h2(N)
with h2(N) =  distances of tiles to goal
0+4

32. Heuristic quality and dominance
1200 random problems with solution lengths from 2 to 24.
h2 dominates h1 and is better for search

33. Relaxed problems 1 2 3 4 5 6 7 8
A problem with fewer restrictions on the actions is called a relaxed problem.
The cost of an optimal solution to a relaxed problem is an admissible heuristic for the original problem
If the rules of the 8-puzzle are relaxed so that a tile can move anywhere, then h1(n) gives the shortest solution
If the rules are relaxed so that a tile can move to any adjacent square, then h2(n) gives the shortest solution