1. Supplemental slides for CSE 327 Prof. Jeff Heflin
Ch. 3 – Search
Supplemental slides for CSE 327
Prof. Jeff Heflin

2. 8-Puzzle Transition Model7 2 4 5 6 8 3 1
state s
actions a
blank-right
blank-left
blank-up
blank-down 7 2 4 5 6 8 3 1 7 2 4 5 6 8 3 1 7 4 5 2 6 8 3 1 7 2 4 5 3 6 8 1
RESULT(s,a)

3. 8-puzzle Search Tree initial state 7 2 4 5 8 6 3 1 7 2 4 8 6 5 3 1 7 2

4. Tree Search Algorithm
function Tree-Search(problem) returns a solution, or failure
initialize the frontier using the initial state of problem loop do if the frontier is empty then return failure choose a leaf node and remove it from the frontier if the node contains a goal state then return the corresponding solution expand the chosen node, adding the resulting nodes to the frontier
From Figure 3.7, p. 77

5. Repeated States A B C A C A B G B C A B G initial state goal state
State space
Search tree A B A C G B C
initial state
goal state A C A B G B C A B G

6. Graph Search Algorithm
function Graph-Search(problem) returns a solution, or failure
initialize the frontier using the initial state of problem initialize the explored set to be empty loop do if the frontier is empty then return failure choose a leaf node and remove it from the frontier if the node contains a goal state then return the corresponding solution add the node to the explored set expand the chosen node, adding the resulting nodes to the frontier, but only if not in the frontier or explored set
From Figure 3.7, p. 77

7. Depth-first Search A not generated on frontier B C in memory deleted D1 A
not generated
on frontier 2 7 B C
in memory
deleted 3 6 8 9 D E F G
blue = in memory, green = on frontier, red = out of memory, clear = not generated 4 5 H I

8. Breadth-first Search A not generated on frontier B C in memory deleted1 A
not generated
on frontier 2 3 B C
in memory
deleted 4 5 6 7 D E F G 8 9 H I

9. Uniform Cost Search I B G not generated on frontier C G in memory
State space
Search tree 1 10 I G I
g(n)=0 4 B 5 2 3 C B
g(n)=4 G
g(n)=10
not generated 4
on frontier 3 C G
g(n)=7
g(n)=9
in memory
deleted

10. Uninformed Search Summary
depth-first
breadth-first
uniform cost
queuing
add to front (LIFO)
add to back (FIFO)
by path cost
complete? no
yes
yes, if all step costs are greater than 0
optimal?
yes, if identical step costs
time
expensive
space
modest

11. A* Example n’s state h(n) A g B 5 C 7 D 8 E 1 F G D 9 3 B 4 2 5 G A ED 9 3 B 4 2 5 G A E 5 6 2 F C
Have already expanded the same state. No need to calculate f(n) 1 A
g(n)=0, h(n)=6, f(n)=0+6=6 3 B 4 D E
Lower cost node is on frontier 2
f=2+5=7
f=3+8=11 f= 5
+1=6 E 5* G B C
f=12+0=12 f=
9+5=14
f=10+7=17

12. Heuristics in A* Use A* to solve the 8-puzzle:
initial state:
goal state: 1 2 3 4 6 8 7 5 1 2 3 4 5 6 7 8
path cost is the total number of moves made
Consider these heuristics
H1: The number of tiles out of place
H2: Sum of distances of tiles from goal positions
Ignore moves that return you to the previous state

13. A* on 8-puzzle Heuristic = H1 1 f=0+3=3 2 f=1+3=4 3 f=1+3=4 f=2+3=56 8 7 5
Heuristic = H1 2 1 2 3 4 6 8 7 5
f=1+3=4 3 1 2 3 4 6 7 5 8
f=1+3=4 1 2 3 4 8 7 6 5
f=2+3=5 1 2 3 4 6 8 7 5
f=2+4=6 4 1 2 3 4 6 7 5 8
f=2+2=4 1 2 4 6 3 7 5 8
f=2+4=6
Whether or not this node is expanded depends on how you break ties. It could be expanded at any time or not at all. 1 3 4 2 6 7 5 8
f=3+3=6 1 2 3 4 6 7 5 8
f=3+3=6 5 1 2 3 4 5 6 7 8
f=3+1=4
The notation f=x+y=z is used to show the f(n) calculation for each node, where f=f(n)=z, x=g(n), y=h(n). 1 2 3 4 5 6 7 8
f=4+2=6 6 1 2 3 4 5 6 7 8
f=4+0=4

14. A* on 8-puzzle Heuristic = H2 1 f=0+4=4 f=1+5=6 2 f=1+3=4 3 f=2+2=47 5
Heuristic = H2 1 2 3 4 6 8 7 5
f=1+5=6 2 1 2 3 4 6 7 5 8
f=1+3=4 3 1 2 3 4 6 7 5 8
f=2+2=4 1 2 4 6 3 7 5 8
f=2+4=6 1 3 4 2 6 7 5 8
f=3+3=6 1 2 3 4 6 7 5 8
f=3+3=6 4 1 2 3 4 5 6 7 8
f=3+1=4 1 2 3 4 5 6 7 8
f=4+2=6 5 1 2 3 4 5 6 7 8
f=4+0=4

15. Summary of Search Algorithms
type
ordering
optimal?
complete?
efficient?
depth-first
uninformed
LIFO no
if lucky
breadth-first
FIFO
yesa
yes
uniform cost
g(n)
yesb
greedy
informed
h(n)
usually A*
g(n)+h(n)
yesc
a – if step costs are identical
b – if step costs > 0
c – if heuristic is admissible and consistent