Tree/Graph Search. q IDDFS-? B C. Search CSL452 - ARTIFICIAL INTELLIGENCE 2

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1 Informed Search

2 Tree/Graph Search q IDDFS-? A B C D E F G Search CSL452 - ARTIFICIAL INTELLIGENCE 2

3 Informed (Heuristic) Search q Heuristic problem specific knowledge o finds solutions more efficiently q New terms o g(n) cost from the initial state to the state at node n o h(n) estimated cost from the state at node n to a goal state o f(n) evaluation function, cost estimate from the initial state to a goal state passing to through a state at node n q Search o Greedy Best First Search o A* Search Informed Search CSL452 - ARTIFICIAL INTELLIGENCE 3

4 Greedy Best First Search (GBFS) q Expand the most desirable node o Desirability is measured through the evaluation function f(n) and here f(n)= h(n) q Implementation: priority queue based on h(n) Informed Search CSL452 - ARTIFICIAL INTELLIGENCE 4

5 GBFS - Analysis q Completeness: No, Can get stuck in loops o can be made complete with repeated state checking q Optimality: No q Time Complexity: O(b m ) q Space Complexity: O(b m ) o keeps all nodes in memory Informed Search CSL452 - ARTIFICIAL INTELLIGENCE 5

6 A* search q Idea: avoid expanding paths that are already expensive. q Implementation: priority queue based on the evaluation function f(n) o f(n)=g(n)+h(n) Ø g(n) cost so far to reach the node n Ø h(n) estimated cost to goal from n Ø f(n) estimated total cost of path through n to goal o Identical to uniform cost search(ucs) except that we use g+h instead of only g. Informed Search CSL452 - ARTIFICIAL INTELLIGENCE 6

7 A* - Example Informed Search CSL452 - ARTIFICIAL INTELLIGENCE 7

8 A* search - Analysis q Completeness: Yes, unless there are infinitely many nodes with f f(g) q Optimality:??? Informed Search CSL452 - ARTIFICIAL INTELLIGENCE 8

9 Admissible Heuristic q A* uses an admissible heuristic h(n) h(n) apple h (n), 8n o h*(n) is the true cost to the goal node from node n o h(n) 0, for all n Search noes Heuristic value h* h_1 h_2 h_3 h_4 h_5 Informed Search CSL452 - ARTIFICIAL INTELLIGENCE 9

10 Example Heuristic Functions q 8-puzzle problem q Examples: o Number of misplaced tiles o Total Manhattan distance Informed Search CSL452 - ARTIFICIAL INTELLIGENCE 10

11 Example Heuristic Functions q Romania Tourist Problem Arad Bucharest Craiova Drobeta Eforie Fagaras Giurgiu Hirsova Iasi Lugoj Mehadia Neamt Oradea Pitesti Rimnicu Vilcea Sibiu Timisoara Urziceni Vaslui Zerind Figure 3.22 Values of h SLD straight-line distances to Bucharest. q Examples: o straight line distance- never overestimates the actual road distance. Informed Search CSL452 - ARTIFICIAL INTELLIGENCE 11

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13 Visualizing A* Search q A* expands nodes of increasing! value Uniform cost search Informed Search CSL452 - ARTIFICIAL INTELLIGENCE 13

14 A* Search Optimality q Proof by Contradiction o Let N be the goal node reached by A*, and suppose there is another goal node N. Then g(n) g(n ) o Assume to the contrary g(n )<g(n) o When we picked N for expansion either N or an ancestor of N, say N, must have been on the queue. Since we picked N for expansion o f(n) F(N ) implies g(n)+h(n) g(n )+h(n ) o g(n) g(n )+h(n ) (h(n)=0 - for a goal node) o g(n )=g(n )+effort(n,n ), h(n ) effort(n,n ) o g(n ) g(n )+h(n ) - contradiction 19/01/16 CSL452 - ARTIFICIAL INTELLIGENCE 14

15 A* search - Analysis q Completeness: Yes, unless there are infinitely many nodes with!!(") q Optimality: Yes q Time Complexity: exponential in [relative error in h * length of the solution] q Space Complexity: Keeps all nodes in memory Informed Search CSL452 - ARTIFICIAL INTELLIGENCE 15

16 Admissibility, Monotonicity, Pathmax Correction q Is orange h admissible? q Is green h admissible? q Does f(c) make sense? o f(b) = = 8.9 o f(c) = = 0.2 q Path cost estimate reduces o not good, though optimal solution is reachable q Apply Pathmax correction o f(n) = max(f(parent(n)),g(n)+h(n)) o makes f monotonic along the path B C D A G 9 19/01/16 CSL452 - ARTIFICIAL INTELLIGENCE 16

17 Monotonic Heuristic q Consistent Heuristic q A heuristic is monotonic if h(n) c(n,a,n )+h(n ) q If h is monotonic o f(n ) = g(n )+h(n ) = g(n)+c(n,a,n )+h(n ) o Therefore, f(n ) g(n)+h(n) = f(n) q f is monotonic along any path. Triangle Inequality 19/01/16 CSL452 - ARTIFICIAL INTELLIGENCE 17

18 Heuristic Functions q h1 number of misplaced tiles - 8 q h2- Manhattan distance =18 q True cost - 26 Informed Search CSL452 - ARTIFICIAL INTELLIGENCE 18

19 Accuracy of the Heuristic q Effective branching factor b* q If the total number of nodes generated by A* for a particular problem is N, and the solution depth is d, then b* is the branching factor that a uniform tree of depth d would have in order to contain N+1 nodes. Search Cost (nodes generated) Effective Branching Factor d IDS A (h 1 ) A (h 2 ) IDS A (h 1 ) A (h 2 ) Figure 3.29 Comparison of the search costs and effective branching factors for the ITERATIVE-DEEPENING-SEARCH and A algorithms with h 1, h 2. Data are averaged over 100 instances of the 8-puzzle for each of various solution lengths d. 19/01/16 CSL452 - ARTIFICIAL INTELLIGENCE 19

20 Accuracy of the Heuristic q Is h2 always better than h1? q Yes h2 dominates h1 q domination efficiency (of search) Search Cost (nodes generated) Effective Branching Factor d IDS A (h 1 ) A (h 2 ) IDS A (h 1 ) A (h 2 ) Figure 3.29 Comparison of the search costs and effective branching factors for the ITERATIVE-DEEPENING-SEARCH and A algorithms with h 1, h 2. Data are averaged over 100 instances of the 8-puzzle for each of various solution lengths d. 19/01/16 CSL452 - ARTIFICIAL INTELLIGENCE 20