1. Search Strategies
CMPT 420 / CMPG 720

3. Uninformed Search Strategies
Uninformed search strategies use only the information available in the problem definition
Depth-first search
Iterative deepening search

4. Breadth-first search
Expand shallowest unexpanded node

5. Breadth-first search
Expand shallowest unexpanded node

6. Breadth-first search
Expand shallowest unexpanded node

7. Breadth-first search
Examine siblings before children

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

11. Example: Romania

12. Graph Representation Adjacency list representation of G = (V, E)
An array of V lists, one for each vertex in V
Each list Adj[u] contains all the vertices v such that there is an edge between u and v
Adj[u] contains the vertices adjacent to u (in arbitrary order)
Can be used for both directed and undirected graphs 1 2 5 / 1 2 5 4 3 2 1 5 3 4 / 3 2 4 4 2 5 3 / 5
Undirected graph 4 1 2

13. Graph Representation Adjacency matrix representation of G = (V, E)
Assume vertices are numbered 1, 2, … V 
The representation consists of a matrix A V x V :
aij = if (i, j)  E
0 otherwise 1 2 3 4 5
For undirected graphs matrix A is symmetric:
aij = aji
A = AT 1 1 1 2 5 4 3 2 1 3 1 4 1
Undirected graph 5 1

14. Breadth-first search
Is BFS guaranteed to find a solution if one exists?

15. Breadth-first search
Is BFS guaranteed to find a solution if one exists?
Problem?

16. Breadth-first search Guaranteed to find a solution if one exists
Problem: it might take a very long time to find a solution!
note how much of tree is expanded that doesn't contribute towards goal

17. Properties of breadth-first search
Complete?
Yes
Optimal?
Time?
Space?

18. Properties of breadth-first search
Complete?
Yes
Optimal?
Yes (if cost = 1 per step)
Time?
Space?

19. Properties of breadth-first search
Complete?
Yes
Optimal?
Yes (if cost = 1 per step)
Time?
1+b+b2+b3+… +bd = O(bd)
exponential in d
Space?

20. Properties of breadth-first search
Complete?
Yes
Optimal?
Yes (if cost = 1 per step)
Time?
1+b+b2+b3+… +bd = O(bd)
exponential in d
Space?
O(bd) (O(bd-1) nodes in the explored set and O(bd) nodes in the frontier)

21. Uniform-cost search Expand the node with the lowest path cost g(n)
Implementation:

22. Uniform-cost search Expand the node with the lowest path cost g(n)
Implementation:
Frontier = priority queue ordered by path cost g(n)

23. Example: Romania

24. Properties of uniform-cost search
Equivalent to breadth-first if step costs all equal
Complete?
Yes
Optimal?
Yes – nodes expanded in increasing order of path cost
Time?
O(bd)
Space?

25. Depth-first search
Expand deepest unexpanded node

26. Depth-first search
Expand deepest unexpanded node

27. Depth-first search
Examine children before siblings

28. Depth-first search
Examine children before siblings

29. Depth-first search
Examine children before siblings

30. Depth-first search
Examine children before siblings

31. Depth-first search
Examine children before siblings

32. Depth-first search
Examine children before siblings

33. Depth-first search
Examine children before siblings

34. Depth-first search
Examine children before siblings

35. Depth-first search
Examine children before siblings

36. Depth-first search
Implementation:
Frontier = LIFO queue

37. Depth-first search Example

38. More efficient (in some cases) than breadth-first search
Problem: what if some branch is very long: can spend a lot of time searching useless cases

39. Properties of depth-first search
Complete?
Yes (if every branch has finite number of states)
Optimal? No

40. Properties of depth-first search
Complete?
Yes (if every branch has finite number of states)
Optimal? No
Time?
O(bm): terrible if m is much larger than d
but if solutions are dense, may be much faster than breadth-first

41. Comparing Uninformed Search Strategies
Time and space complexity are measured in
b – maximum branching factor of the search tree
m – maximum depth of the state space
d – depth of the least cost solution

42. Properties of depth-first search
Complete?
Yes (if every branch has finite number of states)
Optimal? No
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!

43. Depth-limited search Depth-first search with depth limit l,
i.e., nodes at depth l have no successors
20 cities in map of Romania
l =19
Any city can be reached from any other city in at most 9 steps.
Diameter gives a better depth limit.

44. Properties of Depth-limited search
Complete?
Yes (if the goal is within the limit)
Optimal? No
Time?
O(bl)
Space?

45. Properties of Depth-limited search
Depth-first search can be viewed as a special case with l = infinity
Main advantage that the search process can't descend forever (in a branch without solution).
However, it will not find a solution if all solutions require a depth greater than the limit.
(This is likely when l is unknown.)

46. How to overcome this and maintain the advantages of depth limited search?

47. Iterative deepening search
Gradually increases the limit – 0, 1, 2, …
If don't find solution, increase depth and try again
Combines the benefits of depth-first and breadth-first search.

48. Iterative deepening search l =0

49. Iterative deepening search l =1

50. Iterative deepening search l =2

51. Iterative deepening search l =3

52. Properties of iterative deepening search
Complete?
Yes
Optimal?
Yes, if step cost = 1
Space?
O(bd), linear space
Time?
d b1 + (d-1)b2 + … + (1)bd = O(bd)

53. Time consuming??
Can show this doesn't really increase the number of examined nodes by much over depth-first search:
For large graphs, "most" of the nodes are at the outermost edge

54. If branching factor is b, then there are b2 nodes in the 2nd layer, b3 in the 3rd, bn in the nth
Thus number of nodes at depth d is
1 + b + b2 + b3   bd
If d =4, b =10, get
 + 103 + 104 = 11,111 nodes
For iterative deepening, need to consider some nodes more than once:
b(d) + b2(d - 1) + b3(d - 2)  bd
d = 4, b = 10 gives
1*(4 + 1) + 10 *  *  *  = 12,345 visitations: a 10% increase
in general, if branching factor is 10, visit nodes 11% more often by iterative deepening

55. Properties of iterative deepening search
Combines the benefits of
depth-first search
Memory space O(bd)
linear!
breadth-first search
Always complete and optimal

56. Bi-Directional Search
Start the process simultaneously from the initial start and backwards from the goal state
whether the frontiers of the two intersect
To reduce the complexity
bd/2 + bd/2 < bd