1. Russell and Norvig: Chapter 4 CSMSC 421 – Fall 2005
Heuristic Search
Russell and Norvig: Chapter 4
CSMSC 421 – Fall 2005

2. Modified Search Algorithm
INSERT(initial-node,FRINGE)
Repeat:
If FRINGE is empty then return failure
n  REMOVE(FRINGE)
s  STATE(n)
If GOAL?(s) then return path or goal state
For every state s’ in SUCCESSORS(s)
Create a node n’
INSERT(n’,FRINGE)

3. Search Algorithms Blind search – BFS, DFS, uniform cost
no notion concept of the “right direction”
can only recognize goal once it’s achieved
Heuristic search – we have rough idea of how good various states are, and use this knowledge to guide our search

4. Best-first search Idea: use an evaluation function f(n) for each node
Estimate of desirability
Expand most desirable unexpanded node
Implementation: fringe is queue sorted by descreasing order of desirability
Greedy search
A* search

5. Heuristic Webster's Revised Unabridged Dictionary (1913) (web1913)
Heuristic \Heu*ris"tic\, a. [Greek. to discover.] Serving to discover or find out.
The Free On-line Dictionary of Computing (15Feb98)
heuristic 1. <programming> A rule of thumb, simplification or educated guess that reduces or limits the search for solutions in domains that are difficult and poorly understood. Unlike algorithms, heuristics do not guarantee feasible solutions and are often used with no theoretical guarantee. 2. <algorithm> approximation algorithm.
From WordNet (r) 1.6
heuristic adj 1: (computer science) relating to or using a heuristic rule 2: of or relating to a general formulation that serves to guide investigation [ant: algorithmic] n : a commonsense rule (or set of rules) intended to increase the probability of solving some problem [syn: heuristic rule, heuristic program]

6. Informed Search
Add domain-specific information to select the best path along which to continue searching
Define a heuristic function, h(n), that estimates the “goodness” of a node n.
Specifically, h(n) = estimated cost (or distance) of minimal cost path from n to a goal state.
The heuristic function is an estimate, based on domain-specific information that is computable from the current state description, of how close we are to a goal

7. Greedy Search
f(N) = h(N)  greedy best-first

8. Robot Navigation

9. Robot Navigation
f(N) = h(N), with h(N) = Manhattan distance to the goal 2 1 5 8 7 3 4 6

10. Robot Navigation
f(N) = h(N), with h(N) = Manhattan distance to the goal 8 7 6 5 4 3 2 3 4 5 6 7 5 4 3 5 6 3 2 1 1 2 4
What happened??? 7 7 6 5 8 7 6 5 4 3 2 3 4 5 6

11. Greedy Search f(N) = h(N)  greedy best-first Is it complete?
If we eliminate endless loops, yes
Is it optimal?

12. More informed search
We kept looking at nodes closer and closer to the goal, but were accumulating costs as we got further from the initial state
Our goal is not to minimize the distance from the current head of our path to the goal, we want to minimize the overall length of the path to the goal!
Let g(N) be the cost of the best path found so far between the initial node and N
f(N) = g(N) + h(N)

13. Robot Navigation
f(N) = g(N)+h(N), with h(N) = Manhattan distance to goal
7+2
8+3 2 1 5 8 7 3 4 6
8+3
7+4
7+4
6+5
5+6
6+3
4+7
5+6
3+8
4+7
3+8
2+9
2+9
3+10
7+2
6+1
2+9
3+8
6+1
8+1
7+0
2+9
1+10
1+10
0+11
7+0
7+2
6+1
8+1
7+2
6+3
6+3
5+4
5+4
4+5
4+5
3+6
3+6
2+7
2+7
3+8

14. Can we Prove Anything?
If the state space is finite and we avoid repeated states, the search is complete, but in general is not optimal
If the state space is finite and we do not avoid repeated states, the search is in general not complete
If the state space is infinite, the search is in general not complete

15. Admissible heuristic
Let h*(N) be the true cost of the optimal path from N to a goal node
Heuristic h(N) is admissible if:  h(N)  h*(N)
An admissible heuristic is always optimistic

16. A* Search Evaluation function: f(N) = g(N) + h(N) where:
g(N) is the cost of the best path found so far to N
h(N) is an admissible heuristic
Then, best-first search with this evaluation function is called A* search
Important AI algorithm developed by Fikes and Nilsson in early 70s. Originally used in Shakey robot.

17. Completeness & Optimality of A*
Claim 1: If there is a path from the initial to a goal node, A* (with no removal of repeated states) terminates by finding the best path, hence is:
complete
optimal
requirements:
0 <   c(N,N’)
Each node has a finite number of successors

18. Completeness of A*
Theorem: If there is a finite path from the initial state to a goal node, A* will find it.

19. Proof of Completeness Let g be the cost of a best path to a goal node
No path in search tree can get longer than g/, before the goal node is expanded

20. Optimality of A*
Theorem: If h(n) is admissable, then A* is optimal.

21. Proof of Optimality f(G1) = g(G1) f(N) = g(N) + h(N)  g(N) + h*(N)
Cost of best path
to a goal thru N
f(G1) = g(G1) G1 N G2
f(N) = g(N) + h(N)  g(N) + h*(N)
f(G1)  g(N) + h(N)  g(N) + h*(N)

22. Example of Evaluation Function
f(N) = (sum of distances of each tile to its goal)
+ 3 x (sum of score functions for each tile)
where score function for a non-central tile is 2 if it is
not followed by the correct tile in clockwise order
and 0 otherwise 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8
f(N) =
3x( )
= 49
goal N

23. Heuristic Function
Function h(N) that estimates the cost of the cheapest 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
h(N) = number of misplaced tiles = 6
goal N

24. Heuristic Function
Function h(N) that estimate the cost of the cheapest 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
h(N) = sum of the distances of
every tile to its goal position =
= 13
goal N

25. 8-Puzzle f(N) = h(N) = number of misplaced tiles 3 4 3 4 5 3 3 4 2 4 44 4 2 1

26. 8-Puzzle f(N) = g(N) + h(N) with h(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

27. 8-Puzzle
f(N) = h(N) =  distances of tiles to goal 6 4 5 3 2 5 4 2 1

28. 8-Puzzle 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 N
goal
h1(N) = number of misplaced tiles = 6 is admissible
h2(N) = sum of distances of each tile to goal = is admissible
h3(N) = (sum of distances of each tile to goal) x (sum of score functions for each tile) = is not admissible

29. 8-Puzzle f(N) = g(N) + h(N) with h(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. Robot navigation
f(N) = g(N) + h(N), with h(N) = straight-line distance from N to goal
Cost of one horizontal/vertical step = 1
Cost of one diagonal step = 2

31. About Repeated States N N1 S S1 N2 g(N1) < g(N) h(N) < h(N1)
f(N)=g(N)+h(N)
f(N1)=g(N1)+h(N1) N2
f(N2)=g(N2)+h(N)
g(N1) < g(N)
h(N) < h(N1)
f(N) < f(N1)
g(N2) < g(N)

32. Consistent Heuristic
The admissible heuristic h is consistent (or satisfies the monotone restriction) if for every node N and every successor N’ of N: h(N)  c(N,N’) + h(N’) (triangular inequality) N N’
h(N)
h(N’)
c(N,N’)

33. 8-Puzzle 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 N
goal
h1(N) = number of misplaced tiles
h2(N) = sum of distances of each tile to goal
are both consistent

34. Robot navigation Cost of one horizontal/vertical step = 1
Cost of one diagonal step = 2
h(N) = straight-line distance to the goal is consistent

35. Claims
If h is consistent, then the function f along any path is non-decreasing: f(N) = g(N) + h(N) f(N’) = g(N) +c(N,N’) + h(N’) N N’
h(N)
h(N’)
c(N,N’)

36. Claims
If h is consistent, then the function f along any path is non-decreasing: f(N) = g(N) + h(N) f(N’) = g(N) +c(N,N’) + h(N’) h(N)  c(N,N’) + h(N’) f(N)  f(N’) N N’
h(N)
h(N’)
c(N,N’)

37. Claims
If h is consistent, then the function f along any path is non-decreasing: f(N) = g(N) + h(N) f(N’) = g(N) +c(N,N’) + h(N’) h(N)  c(N,N’) + h(N’) f(N)  f(N’)
If h is consistent, then whenever A* expands a node it has already found an optimal path to the state associated with this node N N’
h(N)
h(N’)
c(N,N’)

38. Avoiding Repeated States in A*
If the heuristic h is consistent, then:
Let CLOSED be the list of states associated with expanded nodes
When a new node N is generated:
If its state is in CLOSED, then discard N
If it has the same state as another node in the fringe, then discard the node with the largest f

39. Complexity of Consistent A*
Let s be the size of the state space
Let r be the maximal number of states that can be attained in one step from any state
Assume that the time needed to test if a state is in CLOSED is O(1)
The time complexity of A* is O(s r log s)

40. Heuristic Accuracy
h(N) = 0 for all nodes is admissible and consistent. Hence, breadth-first and uniform-cost are particular A* !!!
Let h1 and h2 be two admissible and consistent heuristics such that for all nodes N: h1(N)  h2(N).
Then, every node expanded by A* using h2 is also expanded by A* using h1.
h2 is more informed than h1

41. Iterative Deepening A* (IDA*)
Use f(N) = g(N) + h(N) with admissible and consistent h
Each iteration is depth-first with cutoff on the value of f of expanded nodes

42. 8-Puzzle f(N) = g(N) + h(N) with h(N) = number of misplaced tiles4 6
Cutoff=4

43. 8-Puzzle f(N) = g(N) + h(N) with h(N) = number of misplaced tiles4 6 4 6
Cutoff=4

44. 8-Puzzle f(N) = g(N) + h(N) with h(N) = number of misplaced tiles4 6 4 5 6
Cutoff=4

45. 8-Puzzle f(N) = g(N) + h(N) with h(N) = number of misplaced tiles5 4 6 4 5 6
Cutoff=4

46. 8-Puzzle f(N) = g(N) + h(N) with h(N) = number of misplaced tiles6 5 4 6 4 5 6
Cutoff=4

47. 8-Puzzle f(N) = g(N) + h(N) with h(N) = number of misplaced tiles4 6
Cutoff=5

48. 8-Puzzle f(N) = g(N) + h(N) with h(N) = number of misplaced tiles4 6 4 6
Cutoff=5

49. 8-Puzzle f(N) = g(N) + h(N) with h(N) = number of misplaced tiles4 6 4 5 6
Cutoff=5

50. 8-Puzzle f(N) = g(N) + h(N) with h(N) = number of misplaced tiles4 6 4 5 6
Cutoff=5 7

51. 8-Puzzle f(N) = g(N) + h(N) with h(N) = number of misplaced tiles4 5 4 6 5 6 7
Cutoff=5

52. 8-Puzzle f(N) = g(N) + h(N) with h(N) = number of misplaced tiles4 5 5 6 4 5 6 7
Cutoff=5

53. 8-Puzzle f(N) = g(N) + h(N) with h(N) = number of misplaced tiles4 5 5 6 4 5 6 7
Cutoff=5

54. About Heuristics
Heuristics are intended to orient the search along promising paths
The time spent computing heuristics must be recovered by a better search
After all, a heuristic function could consist of solving the problem; then it would perfectly guide the search
Deciding which node to expand is sometimes called meta-reasoning
Heuristics may not always look like numbers and may involve large amount of knowledge

55. What’s the Issue? Search is an iterative local procedure
Good heuristics should provide some global look-ahead (at low computational cost)

56. Another approach… for optimization problems Local Search Algorithms
rather than constructing an optimal solution from scratch, start with a suboptimal solution and iteratively improve it
Local Search Algorithms
Hill-climbing or Gradient descent
Potential Fields
Simulated Annealing
Genetic Algorithms, others…

57. Hill-climbing search
If there exists a successor s for the current state n such that
h(s) < h(n)
h(s) <= h(t) for all the successors t of n,
then move from n to s. Otherwise, halt at n.
Looks one step ahead to determine if any successor is better than the current state; if there is, move to the best successor.
Similar to Greedy search in that it uses h, but does not allow backtracking or jumping to an alternative path since it doesn’t “remember” where it has been.
Not complete since the search will terminate at "local minima," "plateaus," and "ridges."

58. Hill climbing on a surface of states
Height Defined by Evaluation Function

59. Hill climbing
Steepest descent (~ greedy best-first with no search)  may get stuck into local minimum

60. Local-minimum problem
Robot Navigation
Local-minimum problem
f(N) = h(N) = straight distance to the goal

61. Examples of problems with HC
applet

62. Drawbacks of hill climbing
Problems:
Local Maxima: peaks that aren’t the highest point in the space
Plateaus: the space has a broad flat region that gives the search algorithm no direction (random walk)
Ridges: flat like a plateau, but with dropoffs to the sides; steps to the North, East, South and West may go down, but a step to the NW may go up.
Remedy:
Introduce randomness
Random restart.
Some problem spaces are great for hill climbing and others are terrible.

63. What’s the Issue? Search is an iterative local procedure
Good heuristics should provide some global look-ahead (at low computational cost)

64. Hill climbing example 2 8 3 1 6 4 7 5 1 3 8 4 7 6 5 2 start h = -4
goal
h = 0 -2 -5 -5 2 8 3 1 4 7 6 5 1 3 8 4 7 6 5 2
h = -3
h = -1 -3 -4 2 3 1 8 4 7 6 5 3 1 8 4 7 6 5 2
h = -2
h = -3 -4
f(n) = -(number of tiles out of place)

65. Example of a local maximum1 2 5 7 4 8 6 3 -4
start
goal 1 2 5 7 4 8 6 3 1 2 5 7 4 8 6 3 1 2 5 7 4 8 6 3 -4 -3 1 2 5 7 4 8 6 3 -4

66. Potential Fields Idea: modify the heuristic function
Goal is gravity well, drawing the robot toward it
Obstacles have repelling fields, pushing the robot away from them
This causes robot to “slide” around obstacles
Potential field defined as sum of attractor field which get higher as you get closer to the goal and the indivual obstacle repelling field (often fixed radius that increases exponentially closer to the obstacle)

67. Does it always work? No. But, it often works very well in practice
Advantage #1: can search a very large search space without maintaining fringe of possiblities
Scales well to high dimensions, where no other methods work
Example: motion planning
Advantage #2: local method. Can be done online

68. Example: RoboSoccer All robots have same field: attracted to the ball
Repulsive potential to other players
Kicking field: attractive potential to the ball and local repulsive potential if clase to the ball, but not facing the direction of the opponent’s goal. Result is tangent, player goes around the ball.
Single team: kicking field + repulsive field to avoid hitting other players + player position fields (paraboilic if outside your area of the field, 0 inside). Player nearest to the ball has the largest attractive coefficient, avoids all players crowding the ball.
Two teams: identical potential fields.

69. Simulated annealing
Simulated annealing (SA) exploits an analogy between the way in which a metal cools and freezes into a minimum-energy crystalline structure (the annealing process) and the search for a minimum [or maximum] in a more general system.
SA can avoid becoming trapped at local minima.
SA uses a random search that accepts changes that increase objective function f, as well as some that decrease it.
SA uses a control parameter T, which by analogy with the original application is known as the system “temperature.”
T starts out high and gradually decreases toward 0.

70. Simulated annealing (cont.)
A “bad” move from A to B is accepted with a probability
(f(B)-f(A)/T) e
The higher the temperature, the more likely it is that a bad move can be made.
As T tends to zero, this probability tends to zero, and SA becomes more like hill climbing
If T is lowered slowly enough, SA is complete and admissible.

71. The simulated annealing algorithm

72. Simulated Annealing applet
Successful application: circuit routing, traveling sales person (TSP)

73. Summary: Local Search Algorithms
Steepest descent (~ greedy best-first with no search)  may get stuck into local minimum
Better Heuristics: Potential Fields
Simulated annealing
Genetic algorithms

74. When to Use Search Techniques?
The search space is small, and
There is no other available techniques, or
It is not worth the effort to develop a more efficient technique
The search space is large, and
There is no other available techniques, and
There exist “good” heuristics

75. Summary Heuristic function Best-first search
Admissible heuristic and A*
A* is complete and optimal
Consistent heuristic and repeated states
Heuristic accuracy
IDA*