Search Strategies

1. Search Strategies CPS4801

2. 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

3. Breadth-first search • Expand shallowest unexpanded node Implementation: • Frontieris a FIFO queue, i.e., new successors go at end

4. Breadth-first search • Expand shallowest unexpanded node Implementation: • Frontieris a FIFO queue, i.e., new successors go at end

5. Breadth-first search • Expand shallowest unexpanded node Implementation: • Frontieris a FIFO queue, i.e., new successors go at end

6. Breadth-first search • Expand shallowest unexpanded node Implementation: • Frontieris a FIFO queue, i.e., new successors go at end

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

8. Properties of breadth-first search • O(bd) is scary. • Assuming b=10 • Spaceis the bigger problem (more than time)

9. Uniform-cost search • Expand least-cost unexpanded node • (Cheapest search)Implementation: • Frontier = queue ordered by path cost

10. Example: Romania

11. Properties of uniform-cost search • Equivalent to breadth-first if step costs all equal • Complete? Yes, if step cost ≥ εTime? # of nodes with g ≤ cost of optimal solution, O(bceiling(C*/ ε)) where C* is the cost of the optimal solution • Space? # of nodes with g≤ cost of optimal solution, O(bceiling(C*/ ε))Optimal? Yes – nodes expanded in increasing order of g(n)

12. Depth-first search • Expand deepest unexpanded node Implementation: • Frontier= LIFO queue, i.e., put successors at front

13. Depth-first search • Expand deepest unexpanded node Implementation: • Frontier= LIFO queue, i.e., put successors at front

14. Depth-first search • Expand deepest unexpanded node Implementation: • Frontier= LIFO queue, i.e., put successors at front

15. Depth-first search • Expand deepest unexpanded node Implementation: • Frontier= LIFO queue, i.e., put successors at front

16. Depth-first search • Expand deepest unexpanded node Implementation: • Frontier= LIFO queue, i.e., put successors at front

17. Depth-first search • Expand deepest unexpanded node Implementation: • Frontier= LIFO queue, i.e., put successors at front

18. Depth-first search • Expand deepest unexpanded node Implementation: • Frontier= LIFO queue, i.e., put successors at front

19. Depth-first search • Expand deepest unexpanded node Implementation: • Frontier= LIFO queue, i.e., put successors at front

20. Depth-first search • Expand deepest unexpanded node Implementation: • Frontier= LIFO queue, i.e., put successors at front

21. Depth-first search • Expand deepest unexpanded node Implementation: • Frontier= LIFO queue, i.e., put successors at front

22. Depth-first search • Expand deepest unexpanded node Implementation: • fringe = LIFO queue, i.e., put successors at front

23. Depth-first search • Expand deepest unexpanded node Implementation: • Frontier= LIFO queue, i.e., put successors at front

24. 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

25. 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.

26. Iterative deepening search • Gradually increases the limit – 0, 1, 2, … • Combines the benefits of depth-first and breadth-first search.

27. Iterative deepening search l =0

28. Iterative deepening search l =1

29. Iterative deepening search l =2

30. Iterative deepening search l =3

31. Properties of iterative deepening search • Complete? Yes • Time?d b1 + (d-1)b2 + … + (1)bd= O(bd) • Space?O(bd) • Optimal?Yes

32. Summary of algorithms

33. Informed Search Strategies • uses problem-specific knowledge beyond the definition of the problem itself

35. Best-first search • Idea: use an evaluation functionf(n) for each node • estimate of "desirability" • Expand most desirable unexpanded node • Implementation: Order the nodes in frontier in decreasing order of desirability • Special cases: • greedy best-first search • A*search

36. Romania with step costs in km

37. Greedy best-first search • Evaluation function f(n) = h(n) (heuristic) • = estimate of cost from n to goal • 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

38. Greedy best-first search example

39. Greedy best-first search example

40. Greedy best-first search example

41. Greedy best-first search example

42. Properties of greedy best-first search • Complete? No – can get stuck in loops, e.g., Iasi Neamt Iasi Neamt • Time?O(bm) • Space?O(bm) -- keeps all nodes in memory • Optimal? No

43. A* search • Idea: avoid expanding paths that are already 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

44. Romania with step costs in km

45. A* search example

46. A* search example

47. A* search example

48. A* search example

49. A* search example

50. A* search example