This document summarizes material from a course on advanced topics in artificial intelligence search. It covers uninformed search techniques like depth-first search and breadth-first search as well as informed search techniques like A* search that use heuristics. The document explains that heuristics estimate how close a state is to the goal and discusses properties like admissibility that are required for A* to find optimal solutions. Examples of heuristic functions for pathfinding problems are provided.

1. CS 767: Advanced Topics in AI
Search
Instructor: Dr. Muhammad Aqib
University Institute of Information Technology
PMAS-Arid Agriculture University Rawalpindi
(slides courtesy Dan Klein, Pieter Abbeel, University of California, Berkeley)

2. Informed Search
o Uninformed Search
o DFS
o BFS
o UCS
▪ Informed Search
▪ Heuristics
▪ Greedy Search
▪ A* Search
▪ Graph Search
2

3. Search Heuristics
▪ A heuristic is:
▪ A function that estimates how close a state is to a goal
▪ Designed for a particular search problem
▪ Pathing?
▪ Examples: Manhattan distance, Euclidean distance for
pathing
10
5
11.
2
3

4. Example: Heuristic Function
h(x)
4

5. Greedy Search
5

6. Greedy Search
o Expand the node that seems closest…
o Is it optimal?
o No. Resulting path to Bucharest is not the shortest!
6

7. Greedy Search
o Strategy: expand a node that you think is
closest to a goal state
o Heuristic: estimate of distance to nearest
goal for each state
o A common case:
o Best-first takes you straight to the (wrong)
goal
o Worst-case: like a badly-guided DFS
…
b
…
b
[Demo: contours greedy empty (L3D1)]
[Demo: contours greedy pacman small maze (L3D4)]
7

8. A* Search
8

9. A* Search
UCS Greedy
A*
9

10. Combining UCS and Greedy
o Uniform-cost orders by path cost, or backward cost g(n)
o Greedy orders by goal proximity, or forward cost h(n)
o A* Search orders by the sum: f(n) = g(n) + h(n)
S a d
b
G
h=5
h=6
h=2
1
8
1
1
2
h=6
h=0
c
h=7
3
e h=1
1
Example: Teg
S
a
b
c
e
d
d
G
G
g = 0
h=6
g = 1
h=5
g = 2
h=6
g = 3
h=7
g = 4
h=2
g = 6
h=0
g = 9
h=1
g = 10
h=2
g = 12
h=0
10

11. When should A* terminate?
o Should we stop when we enqueue a goal?
o No: only stop when we dequeue a goal
S
B
A
G
2
3
2
2
h = 1
h = 2
h = 0
h = 3
S 0 3 3
g h +
S->A 2 2 4
S->B 2 1 3
S->B->G 5 0 5
S->A->G 4 0 4
11

12. Is A* Optimal?
o What went wrong?
o Actual bad goal cost < estimated good goal cost
o We need estimates to be less than actual costs!
A
G
S
1 3
h = 6
h = 0
5
h = 7
g h +
S 0 7 7
S->A 1 6 7
S->G 5 0 5
12

13. Admissible Heuristics

14. Idea: Admissibility
Inadmissible (pessimistic) heuristics
break optimality by trapping
good plans on the fringe
Admissible (optimistic) heuristics
slow down bad plans but
never outweigh true costs

15. Admissible Heuristics
o A heuristic h is admissible (optimistic) if:
where is the true cost to a nearest goal
o Examples:
o Coming up with admissible heuristics is most of what’s
involved in using A* in practice.
15 11.5
0.0

16. Optimality of A* Tree Search

17. Optimality of A* Tree Search
Assume:
o A is an optimal goal node
o B is a suboptimal goal node
o h is admissible
Claim:
o A will exit the fringe before B
…

18. Optimality of A* Tree Search: Blocking
Proof:
o Imagine B is on the fringe
o Some ancestor n of A is on the
fringe, too (maybe A!)
o Claim: n will be expanded before B
1. f(n) is less or equal to f(A)
Definition of f-cost
Admissibility of h
…
h = 0 at a goal

19. Optimality of A* Tree Search: Blocking
Proof:
o Imagine B is on the fringe
o Some ancestor n of A is on the
fringe, too (maybe A!)
o Claim: n will be expanded before B
1. f(n) is less or equal to f(A)
2. f(A) is less than f(B)
B is suboptimal
h = 0 at a goal
…

20. Optimality of A* Tree Search: Blocking
Proof:
o Imagine B is on the fringe
o Some ancestor n of A is on the
fringe, too (maybe A!)
o Claim: n will be expanded before B
1. f(n) is less or equal to f(A)
2. f(A) is less than f(B)
3. n expands before B
o All ancestors of A expand before B
o A expands before B
o A* search is optimal
…

21. Properties of A*
…
b
…
b
Uniform-Cost A*

22. UCS vs A* Contours
o Uniform-cost expands equally in
all “directions”
o A* expands mainly toward the
goal, but does hedge its bets to
ensure optimality
Start Goal
Start Goal
[Demo: contours UCS / greedy / A* empty (L3D1)]
[Demo: contours A* pacman small maze (L3D5)]

23. Graph Search

24. Tree Search: Extra Work!
o Failure to detect repeated states can cause exponentially more
work.
Search Tree
State Graph

25. Graph Search
o In BFS, for example, we shouldn’t bother expanding the circled nodes
(why?)
S
a
b
d p
a
c
e
p
h
f
r
q
q c G
a
q
e
p
h
f
r
q
q c G
a

26. Graph Search
o Idea: never expand a state twice
o How to implement:
o Tree search + set of expanded states (“closed set”)
o Expand the search tree node-by-node, but…
o Before expanding a node, check to make sure its state has
never been expanded before
o If not new, skip it, if new add to closed set
o Important: store the closed set as a set, not a list
o Can graph search wreck completeness? Why/why not?
o How about optimality?

27. A* Graph Search Gone Wrong?
S
A
B
C
G
1
1
1
2
3
h=2
h=1
h=4
h=1
h=0
S
(0+2)
A
(1+4)
B
(1+1)
C
(2+1)
G
(5+0)
C
(3+1)
G
(6+0)
State space
graph
Search
tree
Closed
Set:
S B C A

28. Consistency of Heuristics
o Main idea: estimated heuristic costs ≤ actual costs
o Admissibility: heuristic cost ≤ actual cost to goal
h(A) ≤ actual cost from A to G
o Consistency: heuristic “arc” cost ≤ actual cost for each
arc
h(A) – h(C) ≤ cost(A to C)
o Consequences of consistency:
o The f value along a path never decreases
h(A) ≤ cost(A to C) + h(C)
o A* graph search is optimal
3
A
C
G
h=4 h=1
1
h=2

29. Optimality of A* Search
o With a admissible heuristic, Tree A* is optimal.
o With a consistent heuristic, Graph A* is optimal.
o See slides, also video lecture from past years for details.
o With h=0, the same proof shows that UCS is optimal.

30. A*: Summary

31. A*: Summary
o A* uses both backward costs and (estimates of) forward
costs
o A* is optimal with admissible / consistent heuristics
o Heuristic design is key: often use relaxed problems

32. Tree Search Pseudo-Code

33. Graph Search Pseudo-Code

34. The One Queue
o All these search algorithms are
the same except for fringe
strategies
o Conceptually, all fringes are priority
queues (i.e. collections of nodes
with attached priorities)
o Practically, for DFS and BFS, you
can avoid the log(n) overhead from
an actual priority queue, by using
stacks and queues
o Can even code one implementation
that takes a variable queuing object

35. Search and Models
o Search operates over
models of the world
o The agent doesn’t
actually try all the plans
out in the real world!
o Planning is all “in
simulation”
o Your search is only as
good as your models…

36. Search Gone Wrong?
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