1. Solving Problems by Searching
Chapter 3

2. Outline Problem-solving agents Problem types Problem formulation
Basic search algorithms
Greedy best-first search
A* search
Heuristic Search

3. 8-puzzle problem 5 7 4 6 1 3 8 2 state description
3-by-3 array: each cell contains one of 1-8 or blank symbol
two state transition descriptions
84 moves: one of 1-8 numbers moves up, down, right, or left
4 moves: one black symbol moves up, down, right, or left
The number of nodes in the state-space graph:
9! ( = 362,880 )

4. Formulating the State Space
For huge search space we need,
Careful formulation
Implicit representation of large search graphs
Efficient search method

5. 8-queens problem
Place 8-queens in the position such that no queen can attack the others

6. Implicit State-Space Graphs
Basic components to an implicit representation of a state-space graph
1. Description of start node
2. Actions: Functions of state transformation
3. Goal condition: true-false valued function
Types of search process
Uninformed search: no problem specific information
Heuristic search: existence of problem-specific information

7. 3. Breadth-First Search Procedure
1. Apply all possible operators (successor function) to the start node.
2. Apply all possible operators to all the direct successors of the start node.
3. Apply all possible operators to their successors till goal node found.
 Expanding : applying successor function to a node

9. Breadth-First Search Advantage Disadvantage
Finds the path of minimal length to the goal.
Disadvantage
Requires the generation and storage of a tree whose size is exponential the the depth of the shallowest goal node
Uniform-cost search [Dijkstra 1959]
Expansion by equal cost rather than equal depth

10. Problem-solving agents

11. Example: Travel in Romania

12. Example: Travel in Romania
On holiday in Romania; currently in Arad.
Flight leaves tomorrow from Bucharest
Formulate goal:
be in Bucharest
Formulate problem:
states: various cities
actions: drive between cities
Find solution:
sequence of cities, e.g., Arad, Sibiu, Fagaras, Bucharest

13. Problem types Deterministic, fully observable  single-state problem
Agent knows exactly which state it will be in; solution is a sequence
Non-observable  sensorless problem (conformant problem)
Agent may have no idea where it is; solution is a sequence
Nondeterministic and/or partially observable  contingency problem
percepts provide new information about current state
often interleave, search, execution
Unknown state space  exploration problem

14. Single-state problem formulation
A problem is defined by four items:
initial state e.g., "at Arad“
actions or successor function S(x) = set of action–state pairs
e.g., S(Arad) = {<Arad  Zerind, Zerind>, … }
goal test, can be
explicit, e.g., x = "at Bucharest"
implicit, e.g., Checkmate(x)
path cost (additive)
e.g., sum of distances, number of actions executed, etc.
c(x,a,y) is the step cost, assumed to be ≥ 0
A solution is a sequence of actions leading from the initial state to a goal state

15. Selecting a state space
Real world is very complex generally
 state space must be abstracted for problem solving
(Abstract) state = set of real states
(Abstract) action = complex combination of real actions
e.g., "Arad  Zerind" represents a complex set of possible routes, detours, rest stops, etc.
For guaranteed realizability, any real state "in Arad“ must get to some real state "in Zerind“
(Abstract) solution =
set of real paths that are solutions in the real world
Each abstract action should be "easier" than the original problem

16. Tree search algorithms
Basic idea:
offline, simulated exploration of state space by generating successors of already-explored states (a.k.a.~expanding states)

17. Tree search example

18. Tree search example

19. Tree search example

20. Implementation: general tree search

21. Implementation: states vs. nodes
A state is a (representation of) a physical configuration
A node is a data structure constituting part of a search tree includes state, parent node, action, path cost g(x), depth
The Expand function creates new nodes, filling in the various fields and using the SuccessorFn of the problem to create the corresponding states.

22. Search strategies
A search strategy is defined by picking the order of node expansion
Strategies are evaluated along the following dimensions:
completeness: does it always find a solution if one exists?
time complexity: number of nodes generated
space complexity: maximum number of nodes in memory
optimality: does it always find a least-cost solution?
Time and space complexity are measured in terms of
b: maximum branching factor of the search tree
d: depth of the least-cost solution
m: maximum depth of the state space (may be ∞)

23. Uninformed search strategies
Uninformed search strategies use only the information available in the problem definition
Breadth-first search
Uniform-cost search
Depth-first search
Depth-limited search
Iterative deepening search

24. Breadth-first search Implementation:
fringe is a FIFO queue, i.e., new successors go at end

25. Breadth-first search Implementation:
fringe is a FIFO queue, i.e., new successors go at end

26. Breadth-first search Implementation:
fringe is a FIFO queue, i.e., new successors go at end

27. Chapter 3 Uninformed Search
Problem-solving agents
Problem types
Problem formulation
Example problems
Basic search algorithms

28. Breadth-first search Implementation:
fringe is a FIFO queue, i.e., new successors go at end

29. Properties of breadth-first search
Complete? Yes (if b is finite)
Time? 1+b+b2+b3+… +bd + b(bd-1) = O(bd+1)
Space? O(bd+1) (keeps every node in memory)
Optimal? Yes (if cost = 1 per step)
Space is the bigger problem (more than time)

30. Uniform-cost search Implementation:
fringe = queue ordered by path cost
Time? # of nodes with g ≤ cost of optimal solution, O(bceiling(C*/ ε)) where C* is the cost of the optimal solution
Optimal? Yes – nodes expanded in increasing order of g(n)

31. Depth-first search Implementation:
fringe = LIFO queue, i.e., put successors at front

32. Depth-first search Implementation:
fringe = LIFO queue, i.e., put successors at front

33. Depth-first search Implementation:
fringe = LIFO queue, i.e., put successors at front

34. Depth-first search Implementation:
fringe = LIFO queue, i.e., put successors at front

35. Depth-first search Implementation:
fringe = LIFO queue, i.e., put successors at front

36. Depth-first search Implementation:
fringe = LIFO queue, i.e., put successors at front

37. Depth-first search Implementation:
fringe = LIFO queue, i.e., put successors at front

38. Depth-first search Implementation:
fringe = LIFO queue, i.e., put successors at front

39. Depth-first search Implementation:
fringe = LIFO queue, i.e., put successors at front

40. Depth-first search Implementation:
fringe = LIFO queue, i.e., put successors at front

41. Depth-first search Implementation:
fringe = LIFO queue, i.e., put successors at front

42. Depth-first search Implementation:
fringe = LIFO queue, i.e., put successors at front

43. Properties of depth-first search
Complete? No: fails in infinite-depth spaces, spaces with loops
Modify to avoid repeated states along path
 complete in finite spaces
Time? O(bm): terrible if m is much larger than d
but if solutions are dense, may be much faster than breadth-first
Space? O(bm), i.e., linear space!
Optimal? No

44. Depth-limited search = depth-first search with depth limit l,
Recursive implementation:

45. Iterative deepening search

46. 5. Iterative Deepening
Advantage
Linear memory requirements of depth-first search
Guarantee for goal node of minimal depth
Procedure
Successive depth-first searches are conducted – each with depth bounds increasing by 1

47. Iterative deepening search
Number of nodes generated in a depth-limited search to depth d with branching factor b:
NDLS = b0 + b1 + b2 + … + bd-2 + bd-1 + bd
Number of nodes generated in an iterative deepening search to depth d with branching factor b:
NIDS = (d+1)b0 + d b^1 + (d-1)b^2 + … + 3bd-2 +2bd-1 + 1bd
For b = 10, d = 5,
NDLS = , , ,000 = 111,111
NIDS = , , ,000 = 123,456
Overhead = (123, ,111)/111,111 = 11%

48. Iterative deepening search
Complete? Yes
Time? (d+1)b0 + d b1 + (d-1)b2 + … + bd = O(bd)
Space? O(bd)
Optimal? Yes, if step cost = 1

49. Summary of algorithms

50. Repeated states
Failure to detect repeated states can turn a linear problem into an exponential one!

51. Graph search

52. Outline Problem-solving agents Problem types Problem formulation
Basic search algorithms
Informed Search
Greedy best-first search
A* search
Heuristic Search

53. Best-first search Idea: use an evaluation function f(n) for each node
estimate of "desirability“
Expand most desirable unexpanded node
Implementation:
Special cases:
greedy best-first search
A* search

54. Greedy Best-first Search
Evaluation function
f(n) = h(n) (heuristic)
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

55. Greedy best-first search
Complete? No – can get stuck in loops, e.g., Iasi  Neamt  Iasi  Neamt 
Time? O(bm), but a good heuristic can give dramatic improvement
Space? O(bm) -- keeps all nodes in memory
Optimal? No

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

57. A* search example

58. 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, i.e., it is optimistic
Example: hSLD(n) (never overestimates the actual road distance)
Theorem: If h(n) is admissible, A* using TREE-SEARCH is optimal

59. Optimality of A* (proof)
Suppose some suboptimal goal G2 has been generated and is in the fringe. Let n be an unexpanded node in the fringe such that n is on a shortest path to an optimal goal G.
f(G2) = g(G2) since h(G2) = 0
g(G2) > g(G) since G2 is suboptimal
f(G) = g(G) since h(G) = 0
f(G2) > f(G) from above

60. Optimality of A* (proof)
Suppose some suboptimal goal G2 has been generated and is in the fringe. Let n be an unexpanded node in the fringe such that n is on a shortest path to an optimal goal G.
f(G2) > f(G) from above
h(n) ≤ h^*(n) since h is admissible
g(n) + h(n) ≤ g(n) + h*(n)
f(n) ≤ f(G)
Hence f(G2) > f(n), and A* will never select G2 for expansion

61. Consistent Heuristics
A heuristic is consistent if for every node n, every successor n' of n generated by any action a, then
h(n) ≤ c(n,a,n') + h(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)
i.e., f(n) is non-decreasing along any path.
Theorem: If h(n) is consistent, A* using GRAPH-SEARCH is optimal

62. Optimality of A* 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

63. Properties of A*
Complete? Yes (unless there are infinitely many nodes with f ≤ f(G) )
Time? Exponential
Space? Keeps all nodes in memory
Optimal? Yes

64. Heuristic Function Admissible heuristics
E.g., for the 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) = ?

65. Admissible heuristics
E.g., for the 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) = ? 8
h2(S) = ? = 18

66. Dominance If h2(n) ≥ h1(n) for all n (both admissible)
then h2 dominates h1
===》 h2 is better for search
Typical search costs (average number of nodes expanded):
d=12 IDS = 3,644,035 nodes (ITERATIVE-DEEPENING-SEARCH) A*(h1) = 227 nodes A*(h2) = 73 nodes
d=24 IDS = too many nodes A*(h1) = 39,135 nodes A*(h2) = 1,641 nodes

67. Relaxed Problems
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

68. Summary
Problem formulation usually requires abstracting away real-world details to define a state space that can feasibly be explored
Variety of uninformed search strategies
Iterative deepening sarch uses only linear space and not much more time than other uninformed algorithms