1. Principles of Computing – UFCFA3-30-1
Week-14
Instructor : Mazhar H Malik
Global College of Engineering and Technology

2. Time and Space Complexity AND
Shortest Path Finding Algorithms

3. Classes of Algorithms Simple recursive algorithms.
Backtracking algorithms.
Divide and conquer algorithms.
Dynamic programming algorithms.
Greedy algorithms.
Evolutionary algorithms.
Randomized algorithms.

4. Problem Formulation - Romania
Start
Finish

5. Problem Formulation - Romania
Initial State  Arad
Goal State  Bucharest
Operator
o driving between cities
o state space consists of all 20 cities in the graph
Goal Test
o is the current state Bucharest?
o a solution is a path from the initial to the goal state
Path cost is a function of time/distance/risk/petrol/…

6. Uninformed Search Strategies
Uninformed strategies
use only the information
available in the problem definition, Also known as
blind searching
Breadth-first search
Uniform-cost search
Depth-first search
Depth-limited search
Iterative deepening search

7. Comparing Uninformed Search Strategies
Completeness
Will a solution always be found if one exists?
Time
How long does it take to find the solution?
Often represented as the number of nodes searched
Space
How much memory is needed to perform the search?
Often represented as the maximum number of nodes stored at once
Optimal
Will the optimal (least cost) solution be found?

8. 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
AI: Chapter 3: Solving Problems by Searching 7

9. Breadth-First Search Recall from Data Structures the basic
algorithm for a breadth-first search on a
graph or tree
Expand the shallowest unexpanded node
Place all new successors at the end of a FIFO
queue

10. Breadth-First Search

11. Breadth-First Search

12. Breadth-First Search

13. Breadth-First Search

14. Properties of Breadth-First Search
Complete
Yes if b (max branching factor) is finite
Time
– 1 + b + b2 + … + bd = O(bd)
exponential in d
Space
O(bd)
Keeps every node in memory
This is the big problem; an agent that generates nodes at 10 MB/sec will produce 860 MB in 24 hours
Optimal
Yes (if cost is 1 per step); not optimal in general

15. Properties of Breadth-First Search

16. Breadth First Search outcome
The memory
requirements are a breadth-first search
bigger than is
problem for
execution time
Exponential-complexity search problems cannot be solved by uniformed methods for any but the smallest instances

17. Depth-First Search
Recall from Data Structures the basic algorithm for a depth-first search on a graph or tree
Expand the deepest unexpanded node
Unexplored successors are placed on a stack
until fully explored

18. Depth-First Search

19. Depth-First Search

20. Depth-First Search

21. Depth-First Search

22. Depth-First Search

23. Depth-First Search

24. Depth-First Search

25. Depth-First Search

26. Depth-First Search

27. Depth-First Search

28. Depth-First Search

29. Depth-First Search

30. Depth-First Search Complete Time Space Optimal
No: fails in infinite-depth spaces, spaces with loops
Modify to avoid repeated spaces along path
Yes: in finite spaces
Time
O(bm)
Not great if m is much larger than d
But if the solutions are dense, this may be faster than breadth-first
search
Space
O(bm)…linear space
Optimal No

31. Depth-Limited Search
A variation of depth-first search that uses a depth limit
Alleviates the problem of unbounded trees
Search to a predetermined depth l (“ell”)
Nodes at depth l have no successors
Same as depth-first search if l = ∞
Can terminate for failure and cutoff

32. Depth-Limited Search

33. Depth-Limited Search Complete Time Space Optimal Yes if l = d O(bl)
No if l = d

34. Summery

35. NP Complete Problem
The list below contains some well-known problems that are NP-complete when expressed as decision problems.
Boolean satisfiability problem (SAT)
Knapsack problem.
Hamiltonian path problem.
Travelling salesman problem (decision version)
Subgraph isomorphism problem.
Subset sum problem.
Clique problem.
Vertex cover problem.