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1. Informed Search Algorithms Chapter 4Feb 2 2007

2. Belief States (Chap 3)

3. Graph Search (Chap 3)

4. Graph Search: Analysis • Time and Space complexity: proportional to state space (< O(bd)) • Optimality: may not be optimal if algorithm throws away a cheaper path to a node that is already closed • Uniform-cost and breadth-first graph search are optimal when step cost is constant

5. Chap 4 – Informed Search • Best-first search • Greedy best-first search • Recursive best-first search • A* search • Heuristics • Local search algorithms • Hill-climbing search • Simulated annealing search • Local beam search • Genetic algorithms sections 4.1, 4.2today’s focus

6. Review: Tree Search • A search strategy is defined by picking the order of node expansion

7. Romania with Step Costs in KM additional information that can inform the expansion strategy

8. Best-First Search • Idea: use an evaluation functionf(n) for each node • f(n) is an estimate of "desirability" • Expand most desirable unexpanded node • Implementation: Order the nodes in fringe in decreasing order of desirability (some form of priority queue) • Special cases: • greedy best-first search • A* search

9. Greedy Best-First Search • Evaluation function f(n) = h(n) (heuristic)= estimate of cost from n to goal • example: hSLD(n) = straight-line distance from n to Bucharest • Greedy best-first search expands the node that appears to be closest to goal

10. Greedy Best-First Search

11. Greedy Best-First Search

12. Greedy Best-First Search

13. Greedy Best-First Search

14. Greedy Best-First Search • Complete? No – can get stuck in loops, e.g., Iasi  Neamt  Iasi  Neamt (when trying to get from Iasi to Fagaras) • Time?O(bm), but a good heuristic can give dramatic improvement • Space?O(bm), keeps all nodes in memory • Optimal? No

15. 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

16. A* Search Example

17. A* Search Example

18. A* Search Example

19. A* Search Example

20. A* Search Example

21. A* Search Example

22. A* Exercise How will A* get from Iasi to Fagaras?

23. 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

24. Admissible Heuristics Admissible Heuristics for the 8-puzzle: h1(n) = number of misplaced tiles h1(S) = 8 h2(n) = total Manhattan distance (sum the walking distance from each square to its desired location) h2(S) = 3+1+2+2+2+3+3+2 = 18

25. Optimality of A*(tree) • Suppose some suboptimal goal G2has 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 + h(G2) = g(G2) > C*, since G2 is a goal on a non-optimal path (C* is the optimal cost) f(n) = g(n) + h(n) <= C*, since h is admissible f(n) <= C* < f(G2), so G2 will never be expanded A* will not expand goals on sub-optimal paths

26. Consistent Heuristics • A heuristic is consistent if for every node n, every successor n' of n generated by any action a, 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) = f(n) • i.e., f(n) is non-decreasing along any path. • Theorem: If h(n) is consistent, A* using GRAPH-SEARCH is optimal • consistency is also called monotonicity

27. Optimality of A*(graph) • 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

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

29. Properties of A* • A* is also optimally efficient • no other optimal algorithm is guaranteed to expand few nodes, if they did they might miss the optimal solution • Good news: complete, optimal and optimally efficient • Bad news: time complexity is bad, space complexity is worse

30. Dominance • If h2(n) ≥ h1(n) for all n (both admissible),then h2dominatesh1 h2is better for search • Typical search costs (average number of nodes expanded)for 8-puzzle: d=12 IDS = 3,644,035 nodes A*(h1) = 227 nodes A*(h2) = 73 nodes d=24 IDS = too many nodes A*(h1) = 39,135 nodes A*(h2) = 1,641 nodes

31. Comparing 8-puzzle Heuristics h2 dominates h1 h2 is better heuristic

32. 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 • Reason: the optimal solution to the original problem is, by definition, also a solution to the relaxed problem and must be at least as expensive as the optimal solution to the relaxed problem.

33. Relaxed Problems • The hSLD(n) function for the Romania travel problem is derived from a relaxed problem: • original problem: drive following roads • relaxed problem: fly in straight lines

34. Relaxed Problems • 8-puzzle relaxed problem 1:Any tile can be moved anywhere.This derives h1(n) = number of misplaced tiles • 8-puzzle relaxed problem 2:Any tile can move to any adjacent square.This derives h2(n) = total Manhattan distance

35. Exercise XXXXXXXXXXXXXXXXXXXX..........XXXGXXXX..XX.XXXXXXXX.XXXXXX.....XXXXXX.XXXXXX.XX..XXXXXX.XXXXXX.XX.........XXX....XXXXXXXXXXXXXXXXX...........XXXXXXXXXXXXXXXXX..SXXXXXXXXXXXX....XXX Design a heuristic for the maze problem

36. Overcoming Space Limitations • A* may run into space limitations • Memory-bounded heuristic search attempts to avoid space complexity • IDA* = iterative deepening A* • RBFS = recursive best-first search • MA*, SMA* = memory-bounded A*

37. IDA* • A* and best-first keep all nodes in memory • IDA* (iterative deeping A*) is similar to standard iterative deeping, expect the cutoff used is the f-cost (g+h), rather than the depth. • Cutoff value is set to smallest f-cost of any node that exceeded cutoff in previous iteration

38. RBFS • Recursive best-first search (RBFS) tries to mimic best-first in linear space • Keeps track of the f-value of the best alternative path from any ancestor of the current node • If current node exceed that limit, it unwinds back to the alternate path • It will often have to re-expand discarded subtrees

39. RBFS f-value of best alternate path

40. RBFS

41. RBFS

42. IDA*, RBFS: too little memory • IDA* only remembers current f-cost limit • RBFS cannot use more memory, even if available • Result: Neither can remember repeated states and may explore same state many times

43. Memory-bounded A* • MA* and SMA* (simplified MA*) work like A* until memory is full. • SMA*: if memory is full, drop the worst leaf node and push its f-value back to its parent • Subtrees are regenerated only when all other paths have been shown to be worse than the forgotten path • Trashing behavior can result if memory is small compared to size of candidate subpaths