1. CS 312: Algorithm Design & Analysis Lecture #37: A* (cont.); Admissible Heuristics Credit: adapted from slides by Stuart Russell of UC Berkeley. This work is licensed under a Creative Commons Attribution-Share Alike 3.0 Unported License.Creative Commons Attribution-Share Alike 3.0 Unported License

2. Announcements  Project #7: TSP  Whiteboard: due today  Early: next Wednesday (4/8)  Due: next Friday (4/10)  Competition (“The Bake-Off”)  Big 3 problems in the Competition set survey  3 problems in the One-Time Competition set survey  Hall of Fame (Bake-Off results) presented last day of class (4/13)  Optional: Cookie Bake-Off  Bring to final exam  A bit of extra HW credit for winning this bake-off

3. Objectives  Prove the optimality of A*  Understand Admissible Heuristics  Revisit the idea of “Relaxed Problems” more deeply

4. Search algorithms

5. A. Beam Search  Q. How could you change A* or B&B to be more memory efficient?  Various forms of Beam search:  Agenda size  Threshold from best  Sacrifices guarantees of optimality

6. Theorem

7. Proof of Optimality of A *  Suppose some sub-optimal goal state G 2 has been generated and is on the agenda. Let n be an unexpanded state on the agenda such that n is on a shortest (optimal) path to the optimal goal state G. Assume h() is admissible. Focus on G 2 : f(G 2 ) = g(G 2 )since h(G 2 ) = 0 g(G 2 ) > g(G) since G 2 is suboptimal

8. Proof of Optimality of A *  Suppose some sub-optimal goal state G 2 has been generated and is on the agenda. Let n be an unexpanded state on the agenda such that n is on a shortest (optimal) path to the optimal goal state G. Assume h() is admissible. f(G 2 ) = g(G 2 )since h(G 2 ) = 0 g(G 2 ) > g(G) since G 2 is suboptimal Focus on G: f(G) = g(G)since h(G) = 0 f(G 2 ) > f(G)substitution

9. Proof of Optimality of A *  Suppose some sub-optimal goal state G 2 has been generated and is on the agenda. Let n be an unexpanded state on the agenda such that n is on a shortest (optimal) path to the optimal goal state G. Assume h() is admissible. Now focus on n: h(n) ≤ h*(n)since h is admissible g(n) + h(n) ≤ g(n) + h * (n) algebra f(n) = g(n) + h(n)definition f(G) = g(n) + h*(n) by assumption f(n) ≤ f(G)substitution Hence f(G 2 ) > f(n), and A * will never select G 2 for expansion. f(G 2 ) = g(G 2 )since h(G 2 ) = 0 g(G 2 ) > g(G) since G 2 is suboptimal f(G) = g(G)since h(G) = 0 f(G 2 ) > f(G)substitution

10. Optimality of A *  A * expands states in order of increasing f value  Gradually adds "f-contours" of states  Like Uniform cost search, but more directed!  Contour i has all states with f=f i, where f i < f i+1

11. Consistent heuristics  A heuristic is consistent if for every state 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

12. Solving the 8-puzzle with A*  states?  actions?  goal test?  path cost g(n)?

13. Example: The 8-puzzle  states?  actions?  goal test?  path cost g(n)?

14. Example: The 8-puzzle  states?  actions?  goal test?  path cost g(n)?

15. Example: The 8-puzzle  states? locations of tiles  actions? move blank left, right, up, down  goal test? = goal state (given)  path cost g(n)? 1 per move

16. Example: The 8-puzzle  What does the state space look like? How deep?  Note: optimal solution of n-Puzzle family is NP- hard  Not known to be in NP

17. Admissible heuristics?

19. Relaxed problems  A problem with fewer restrictions is called a relaxed problem  Theorem: The cost of an optimal solution to a relaxed problem is an admissible heuristic for the original problem  E.g., our bound function for TSP

20. Relaxed problems  If the rules of the 8-puzzle are relaxed  so that a tile can move anywhere  then h 1 (n) gives the shortest solution  If the rules are relaxed  so that a tile can move to any adjacent square  then h 2 (n) gives the shortest solution

21. Admissible heuristics E.g., for the 8-puzzle:  h 1 (n) = number of misplaced tiles  h 2 (n) = total Manhattan distance (i.e., no. of squares from desired location of each tile)  h 1 (S) = ?  h 2 (S) = ?

22. Admissible heuristics E.g., for the 8-puzzle:  h 1 (n) = number of misplaced tiles  h 2 (n) = total Manhattan distance (i.e., no. of squares from desired location of each tile)  h 1 (S) = ? 8  h 2 (S) = ? 3+1+2+2+2+3+3+2 = 18

23. Dominance; i.e. Tighter Bounds  If h 2 (n) ≥ h 1 (n) for all n (both admissible)  then h 2 dominates h 1  h 2 is better for search  Typical search costs (average number of states expanded) for d-puzzle:  d=11 (3x4) Iterative Deepening Search (IDS) = 3,644,035 states A * (h 1 ) = 227 states A * (h 2 ) = 73 states  d=24 (5 x 5) IDS = too many states A * (h 1 ) = 39,135 states A * (h 2 ) = 1,641 states

24. Compare and Contrast  A* vs. Branch & Bound  Reasoning about an admissible heuristic in A* is like reasoning about optimistic bounds in B&B  A* doesn’t include a BSSF  Thus, B&B is potentially more memory efficient

25. Assignment  HW #28