1. Informed Search and Exploration
Chapter 4
Informed Search and Exploration

2. Outline Best-First Search Greedy Best-First Search A* Search
Heuristics
Variances of A* Search

3. Tree Search (Reviewed, Fig. 3.9)
A search strategy is defined by picking the order of node expansion
function TREE-SEARCH ( problem, strategy ) returns a solution, or failure
initialize the search tree using the initial state of problem
loop do if there are no candidates for expansion then return failure
choose a leaf node for expansion according to strategy
if the node contains a goal state then
return the corresponding solution
else expand the node and add the resulting nodes to the search tree

4. Search Strategies Uninformed Search Strategies
by systematically generating new states and testing against the goal
Informed Search Strategies
by using problem-specific knowledge beyond the definition of the problem to find solutions more efficiently

5. Best-First Search An instance of general Tree-Search or Graph-Search
A node is selected for expansion based on an evaluation function, f(n), with the lowest evaluation
an estimate of desirability  expand most desirable unexpanded node
Implementation
a priority queue that maintains the fringe in ascending order of f-values
Best-first search is venerable but inaccurate

6. Best-First Search (cont.-1)
Heuristic search uses problem-specific knowledge: evaluation function
Choose the seemingly-best node based on the cost of the corresponding solution
Need estimate of the cost to a goal
e.g., depth of the current node
sum of distances so far
Euclidean distance to goal, etc.
Heuristics: rules of thumb
Goal: to find solutions more efficiently

7. Best-First Search (cont.-2)
Heuristic Function
h(n) = estimated cost of the cheapest path from node n to a goal (h(n) = 0, for a goal node)
Special cases
Greedy Best-First Search (or Greedy Search)
minimizing estimated cost from the node to reach a goal
A* Search
minimizing the total estimated solution cost

8. Heuristic
Heuristic is derived from heuriskein in Greek, meaning “to find” or “to discover”
The term heuristic is often used to describe rules of thumb or advices that are generally effective, but not guaranteed to work in every case
In the context of search, a heuristic is a function that takes a state as an argument and returns a number that is an estimate of the merit of the state with respect to the goal

9. Heuristic (cont.)
A heuristic algorithm improves the average-case performance, but does not necessarily improve the worst-case performance
Not all heuristic functions are beneficial
The time spent evaluating the heuristic function in order to select a node for expansion must be recovered by a corresponding reduction in the size of search space explored
Useful heuristics should be computationally inexpensive!

10. Romania with Step Costs in km
Straight-line distances to Bucharest
Arad 366
Bucharest 0
Craiova 160
Drobeta 242
Eforie 161
Fagaras 176
Giurgiu 77
Hirsova 151
Iasi 226
Lugoj 244
Mehadia 241
Neamt 234
Oradea 380
Pitesti 100
Rimnicu Vilcea 193
Sibiu 253
Timisoara 329
Urziceni 80
Vaslui 199
Zerind 374

11. Greedy Best-First Search
Greedy best-first search expands the nodes that appears to be closest to the goal
Evaluation function = Heuristic function
f(n) = h(n) = estimated best cost to goal from n
h(n) = 0 if n is a goal
e.g., hSLD(n) = straight-line distance for route-finding
function GREEDY-SEARCH ( problem ) returns a solution, or failure
return BEST-FIRST-SEARCH ( problem, h )

12. Greedy Best-First Search Example

13. Analysis of Greedy Search
Complete?
Yes (in finite space with repeated-state checking)
No (start down an infinite path and never return to try other possibilities)
(e.g., from Iasi to Fagaras)
Susceptible to false starts Isai  Neamt (dead end)
No repeated states checking Isai  Neamt  Isai  Neamt   (oscillation)

14. Analysis of Greedy Search (cont.)
Optimal? No
(e.g., from Arad to Bucharest)
Arad → Sibiu → Fagaras → Bucharest
(450 = , is not shortest)
Arad → Sibiu → Rim → Pitesti → Bucharest
(418 = )
Time? best: O(d), worst: O(bm)
m: the maximum depth
like depth-first search
a good heuristic can give dramatic improvement
Space? O(bm): keep all nodes in memory

15. A* Search Avoid expanding paths that are already expansive
To minimizing the total estimated solution cost
Evaluation function f(n) = g(n) + h(n)
f(n) = estimated cost of the cheapest solution through n
g(n) = path cost so far to reach n
h(n) = estimated cost of the cheapest path from n to goal
function A*-SEARCH ( problem ) returns a solution, or failure
return BEST-FIRST-SEARCH ( problem, g+h )

16. Romania with Step Costs in km (remind)
Straight-line distances to Bucharest
Arad 366
Bucharest 0
Craiova 160
Drobeta 242
Eforie 161
Fagaras 176
Giurgiu 77
Hirsova 151
Iasi 226
Lugoj 244
Mehadia 241
Neamt 234
Oradea 380
Pitesti 100
Rimnicu Vilcea 193
Sibiu 253
Timisoara 329
Urziceni 80
Vaslui 199
Zerind 374

17. A* Search Example

18. Romania with Step Costs in km (remind)
Straight-line distances to Bucharest
Arad 366
Bucharest 0
Craiova 160
Drobeta 242
Eforie 161
Fagaras 176
Giurgiu 77
Hirsova 151
Iasi 226
Lugoj 244
Mehadia 241
Neamt 234
Oradea 380
Pitesti 100
Rimnicu Vilcea 193
Sibiu 253
Timisoara 329
Urziceni 80
Vaslui 199
Zerind 374
150

19. Admissible Heuristic A* search uses an admissible heuristic h(n)
h(n) never overestimates the cost to the goal from n
n, h(n)  h*(n), where h*(n) is the true cost to reach the goal from n (also, h(n)  0, so h(G) = 0 for any goal G)
e.g., h(n) is not admissible
g(X) + h(X) = 102
g(Y) + h(Y) = 74
Optimal path is not found!
e.g., straight-line distance hSLD(n) never overestimates the actual road distance

20. Admissible Heuristic (cont.)
e.g., 8-puzzle
h1(n) = number of misplaced tiles
h2(n) = total Manhattan distance
i.e., no. of squares from desired location of each tile
h1(S) =
h2(S) =
A* is complete and optimal if h(n) is admissible 8
= 18

21. Optimality of A* (proof)
Suppose some suboptimal goal G2 has been generated and is in the queue
Let n be an unexpanded node on a shortest path to an optimal goal G

22. Optimality of A* (cont.-1)
C*: cost of the optimal solution path
A* may expand some nodes before selecting a goal node
Assume: G is an optimal and G2 is a suboptimal goal
f(G2) = g(G2) + h(G2) = g(G2) > C* (1)
For some n on an optimal path to G, if h is admissible, then
f(n) = g(n) + h(n)  C* (2)
From (1) and (2), we have
f(n)  C* < f(G2)
So, A* will never select G2 for expansion

23. Monotonicity (Consistency) of Heuristic
A heuristic is consistent if
h(n)  c(n, a, n’) + h(n’)
the estimated cost of reaching the goal from n is no greater than the step cost of getting to successor n plus the estimated cost of reaching the goal from 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’)  f(n)
i.e., f(n) is non-decreasing along any path
Theorem: If h(n) is consistent, A* is optimal n n’ G
h(n)
h(n’)
c(n,a,n’)

24. Optimality of A* (cont.-2)
A* expands nodes in order of increasing f value
Gradually adds f-contours of nodes
Contour i has all nodes with f=fi, where fi < fi+1

25. Why Use Estimate of Goal Distance?
Order in which uniform-cost looks at nodes. A and B are same distance from start, so will be looked at before any longer paths. No ”bias” toward goal. A B
goal
start
Order of examination using dist. From start + estimates of dist. to goal. Notes “bias” toward the goal; points away from goal look worse.
Assume states are points the Euclidean plane

26. Analysis of A* Search Complete? Yes
unless there are infinitely many nodes with f  f(G)
Optimal? Yes, if the heuristic is admissible
Time? Exponential in [relative error in h* length of solution]
Space? O(bd), keep all nodes in memory
Optimally Efficient? Yes
i.e., no other optimal algorithms is guaranteed to expand fewer nodes than A*
A* is not practical for many large-scale problems
since A* usually runs out of space long before it runs out of time

27. Heuristic Functions Example Two commonly used candidates for 8-puzzle
Start State
Goal State
Example
for 8-puzzle
branching factor  3
depth = 22
# of states = 322  3.1  1010
9! / 2 = 181,400 (reachable distinct states)
for 15-puzzle
# of states  1013
Two commonly used candidates
h1(n) = number of misplaced tiles = 8
h2(n) = total Manhattan distance
(i.e., no. of squares from desired location to each tile)
= = 18

28. Effect of Heuristic Accuracy on Performance
h2 dominates (is more informed than) h1

29. Effect of Heuristic Accuracy on Performance (cont.)
Effective Branching Factor b* is defined by
N + 1 = 1 + b* + (b*)2 +‧‧‧+(b*)d
N : total number of nodes generated by A*
d : solution depth
b* : branching factor that a uniform tree of depth d would have to
have in order to contain N+1 nodes
e.g., A* finds a solutions at depth 5 using 52 nodes, then b* = 1.92
A well designed heuristic would have a value of b* close to 1, allowing fairly large problems to be solved
A heuristic function h2 is said to be more informed than h1 (or h2 dominates h1) if both are admissible and
n, h2(n)  h1(n)
A* using h2 will never expand more nodes than A* using h1

30. Inventing Admissible Heuristic Functions
Relaxed problems
problems with fewer restrictions on the actions
The cost of an optimal solution to a relaxed problem is an admissible heuristic for the original problem
e.g., 8-puzzle
A tile can move from square A to square B if
A is horizontally or vertically adjacent to B and B is blank. (P if R and S)
3 relaxed problems
A tile can move from square A to square B if A is horizontally or vertically adjacent to B. (P if R) --- derive h2
A tile can move from square A to square B if B is blank. (P if S)
A tile can move from square A to square B. (P) --- derive h1

31. Inventing Admissible Heuristic Functions (cont.-1)
Composite heuristics
Given h1, h2, … , hm; none dominates any others
h(n) = max { h1(n), h2(n), … , hm(n) }
Subproblem
The cost of the optimal solution of the subproblem is a lower bound on the cost of the complete problem
To get tiles 1, 2, 3 and 4 into their correct positions, without worrying about what happens to other tiles.
Start State
Goal State ‧

32. Inventing Admissible Heuristic Functions (cont.-2)
Weighted evaluation functions
fw(n) = (1-w)g(n) + wh(n)
Learn the coefficients for features of a state
h(n) = w1  f1(n), , wk  fk(n)
Search cost
Good heuristics should be efficiently computable

33. Memory-Bounded Heuristic Search
Overcome the space problem of A*, without sacrificing optimality or completeness
IDA* (iterative deepening A*)
a logic extension of iterative deepening search to use heuristic information
the cutoff used is the f-cost (g+h) rather than depth
RBFS (recursive best-first search)
MA* (memory-bounded A*)
SMA* (simplified MA*)
is similar to A*, but restricts the queue size to fit into the available memory

34. Iterative Deepening A*
Iterative deepening is useful for reducing memory requirement
At each iteration, perform DFS with an f-cost limit
IDA* is complete and optimal (with the same caveats as A* search)
Space complexity: O (bf * / )  O(bd)
f* : the optimal solution cost
 : the smallest operator cost
Time complexity: O(bd)
 : the number of different f values
In the worst case, if A* expands N nodes, IDA* will expand 1+2+…+N = O(N2) nodes
IDA* uses too little memory

35. Iterative Deepening A* (cont.-1)
first, each iteration expands all nodes inside the contour for the current f-cost
peeping over to find out the next contour lines
once the search inside a given contour has been complete
a new iteration is started using a new f-cost for the next contour

36. Iterative Deepening A* (cont.-2)
function IDA* ( problem ) returns a solution sequence
inputs: problem, a problem
local variables: f-limit, the current f-COST limit
root, a node
root  MAKE-NODE( INITIAL-STATE[ problem ])
f-limit  f-COST( root )
loop do
solution, f-limit  DFS-CONTOUR( root , f-limit )
if solution is non-null then return solution
if f-limit =  then return failure
end

37. Iterative Deepening A* (cont.-3)
function DFS-CONTOUR ( node, f-limit )
returns a solution sequence and a new f-COST limit
inputs: node, a node
f-limit, the current f-COST limit
local variables: next-f, the f-COST limit for the next contour, initially 
if f-COST[ node ] > f-limit then return null, f-COST[ node ]
if GOAL-TEST[ problem ]( STATE[ node ]) then return node, f-limit
for each node s in SUCCESSORS( node ) do
solution, new-f  DFS-CONTOUR( s, f-limit )
if solution is non-null then return solution, f-limit
next-f  MIN( next-f, new-f )
end
return null, next-f

38. Recursive Best-First Search (RBFS)
Keeps track of the f-value of the best-alternative path available
if current f-values exceeds this alternative f-value, then backtrack to alternative path
upon backtracking change f-value to the best f-value of its children
re-expansion of this result is thus still possible

39. Recursive Best-First Search (cont.-1)
Arad
366
Timisoara
447
Zerind
449
Sibiu
393
646
Fagaras
415
Oradea
671
413 ∞
450
417
Bucharest
418
Craiova
615
Rim Vil
607
Pitesti
526
553
Arad
366
Timisoara
447
Zerind
449
Sibiu
393
646
Fagaras
415
Oradea
671
413 ∞
Craiova
526
417
553
Rim Vil
Pitesti
Arad
366
Timisoara
447
Zerind
449
Sibiu
393
646
Fagaras
415
Oradea
671
413 ∞
591
Bucharest
450
417
Rim Vil

40. Recursive Best-First Search (cont.-2)
function RECURSIVE-BEST-FIRST-SEARCH ( problem ) returns a solution, or failure
RBFS( problem, MAKE-NODE( INITIAL-STATE[ problem ] ),  )
function RBFS ( problem, node, f_limit )
returns a solution, or failure and a new f-COST limit
if GOAL-TEST[ problem ]( STATE[ node ]) then return node
successors  EXPAND( node, problem )
if successors is empty then return failure, 
for each node s in successors do
f [ s ]  max( g( s ) + h ( s ), f [ node ])
repeat
best  the lowest f-value node in successors
if f [ best ]  f_limit then return failure, f [ best ]
alternative  the second lowest f-value among successors
result, f [ best ]  RBFS( problem, best, MIN( f_limit, alternative ))
if result  failure then return result
f_limit : 上頁 tree 的方框
EXPAND() : return 值是子節點，而不是 fringe 40

41. Analysis of RBFS RBFS is a bit more efficient than IDA* Complete? Yes
still excessive node generation (mind changes)
Complete? Yes
Optimal? Yes, if the heuristic is admissible
Time? Exponential
difficult to characterize, depend on accuracy of h(n) and how often best path changes
Space? O(bd)
IDA* and RBFS suffer from too little memory

42. Simplified Memory Bounded A* (SMA*)
SMA* expands the (newest) best leaf and deletes the (oldest) worst leaf
Aim: find the lowest-cost goal node with enough memory
Max Nodes = 3
A – root node
D,F,I,J – goal nodes
Label: current f-Cost

43. Simplified Memory Bounded A* (cont.-1)
3. memory is full update (A) f-Cost for the min chlid
expand G, drop the higher f-Cost leaf (B)
5. drop H and add I G memorize H update (G) f-Cost for the min child update (A) f-Cost
6. I is goal node, but may not be the best solution the path through G is not so great, so B is generated for the second time
7. drop G and add C A memorize G C is non-goal node C mark to infinite
8. drop C and add D
B memorize C D is a goal node, and it is lowest f-Cost node then terminate
‧How about J has a cost of 19 instead of 24 ?

44. Simplified Memory Bounded A* (cont.-2)
Aim: find the lowest-cost goal node with enough memory
Max Nodes = 3
A – root node
D,F,I,J – goal nodes
Label: current f-Cost
0+12=12 A 10 8
10+5=15
8+5=13 B G 10 10 8 16
20+5=25
20+0=20
16+2=18
24+0=24 C D H I 10 10 8 8
30+5=35
30+0=30
24+0=24
24+5=29 E F J K

45. Simplified Memory Bounded A* (cont.-3)15 G 24 B
I is goal node, but may not be the best solution
the path through G is not so great
so B is generate for the second time A
15 (24) C
25 infinite B 15
drop G and add C
A memorize G
C is non-goal node
C mark to infinite A
20 (24) D 20 B
20 (infinite)
drop C and add D
B memorize C
D is a goal node and it is lowest f-cost node then terminate
How about J has a cost of 19 instead of 24 ?? A
15 (15) G
24 (infinite) I 24
drop H and add I
G memorize H
update (G) f-cost for the min child
update (A) f-cost A
13 (15) G 13 H
18 infinite
memory is full A 12 A 12 B 15 A 13 G B 15
memory is full
update (A) f-cost for the min child
expand G, drop the higher f-cost leaf (B)

46. Simplified Memory Bounded A* (cont.-4)
function SMA* ( problem ) returns a solution sequence
inputs: problem, a problem
local variables: Queue, a queue of nodes ordered by f-COST
Queue  MAKE-QUEUE({ MAKE-NODE( INITIAL-STATE[ problem ])})
loop do
if Queue is empty then return failure
n  deepest least f-COST node in Queue
if GOAL-TEST( n ) then return success
s  NEXT-SUCCESSOR( n )
if s is not a goal and is at maximum depth then
f( s )  
else f( s )  MAX( f( n ), g( s ) + h( s ))
if all of n‘s successors have been generated then
update n‘s f-COST and those of its ancestors if necessary
if SUCCESSORS( n ) all in memory then remove n from Queue
if memory is full then
delete shallowest, highest f-COST node in Queue
remove it from its parent’s successor list
insert its parent on Queue if necessary
insert s on Queue
end

47. SMA* Algorithm
SMA* makes use of all available memory M to carry out the search
It avoids repeated states as far as memory allows
Forgotten nodes: shallow nodes with high f-cost will be dropped from the fringe
A forgotten node will only be regenerated if all other child nodes from its ancestor node have been shown to look worse
Memory limitations can make a problem intractable

48. Analysis of SMA* Complete? Yes, if M  d Optimal? Yes, if M  d *
i.e., if there is any reachable solution
i.e., the depth of the shallowest goal node is less than the memory size
Optimal? Yes, if M  d *
i.e., if any optimal solution is reachable; otherwise, it returns the best reachable solution
Optimally efficiently? Yes, if M  bm
Time?
It is often that SMA* is forced to switch back and forth continually between a set of candidate solution paths
Thus, the extra time required for repeated generation of the same node
Memory limitations can make a problem intractable from the point of view of computation time
Space? Limited

49. HW2, 4/11 deadline
Write an A* programs to solve the path finding problem.