Advanced Artificial Intelligence (DT4019, HT10)

Transcription

1 Advanced Artificial Intelligence (DT4019, HT10) Problem Solving and Search: Informed Search Strategies (I) Federico Pecora School of Science and Technology Örebro University Federico Pecora Advanced Artificial Intelligence Problem Solving and Search: Informed Search Strategies (I) (Lecture 3) 1 / 36

2 Outline 1 Greedy Best-First Search 2 Consistent Heuristics Admissibility, Consistency and Dominance Federico Pecora Advanced Artificial Intelligence Problem Solving and Search: Informed Search Strategies (I) (Lecture 3) 2 / 36

3 General Implementaiton of Tree Search (I) Function Tree-Search(problem, fringe) returns a solution, or failure fringe insert(make-node(initial-state(problem)), fringe) while true do if fringe = /0 then return failure node remove-front(fringe) if goal-test(problem,state(node)) then return solution(node) fringe insert-all(expand(node, problem), fringe) Federico Pecora Advanced Artificial Intelligence Problem Solving and Search: Informed Search Strategies (I) (Lecture 3) 3 / 36

4 General Implementaiton of Tree Search (II) Function expand(node, problem) returns a set of nodes successors /0 foreach action, result sucessor(problem, state(node)) do s a new node set-parent(s, node) set-action(s, action) set-state(s, result) set-path-cost(s, path-cost(node) + c(state(node), action, result)) set-depth(s, depth(node)+1) successors successors {s} return successors Federico Pecora Advanced Artificial Intelligence Problem Solving and Search: Informed Search Strategies (I) (Lecture 3) 4 / 36

5 Strategies in Tree Search A search strategy is defined by picking the order of node expansion Strategies are evaluated along four dimensions Completeness: does it always find a solution if one exists? Time complexity: number of nodes generated/expanded Space complexity: maximum number of nodes in memory Optmiality: does it always find a least-cost solution? Time and space comlexity are measured in terms of b: maximum branching factor of the search tree d: depth of the least-cost solution m: maximum depth of the state space (may be ) Federico Pecora Advanced Artificial Intelligence Problem Solving and Search: Informed Search Strategies (I) (Lecture 3) 5 / 36

6 Informed Search: Best-First BFS, DFS, DLS, IDS have a fixed strategy for deciding order of node expansion UCS is somewhat more sophisticated, as it takes into account path cost Best-First search: order nodes based on estimate of desirablility define an evaluation function indicating desirability Implementation: fringe is a queue sorted in decreasing order of desirability Federico Pecora Advanced Artificial Intelligence Problem Solving and Search: Informed Search Strategies (I) (Lecture 3) 6 / 36

7 Informed Search: Best-First Measure of desirability : estimated distance to the goal estimate given by a heuristic function h : S N Greedy search order nodes in queue with decreasing h A* search order nodes in queue with combination of h and g i.e., also take into account cost of already explored path Note: UCS orders nodes in queue according to increasing g Federico Pecora Advanced Artificial Intelligence Problem Solving and Search: Informed Search Strategies (I) (Lecture 3) 7 / 36

8 Greedy Best-First Search Outline 1 Greedy Best-First Search 2 Consistent Heuristics Admissibility, Consistency and Dominance Federico Pecora Advanced Artificial Intelligence Problem Solving and Search: Informed Search Strategies (I) (Lecture 3) 8 / 36

9 Greedy Best-First Search Romania with Step Costs in km Federico Pecora Advanced Artificial Intelligence Problem Solving and Search: Informed Search Strategies (I) (Lecture 3) 9 / 36

10 Greedy Best-First Search Greedy Best-First Search Heuristic function: h(n) = estimate of path cost to closest goal Example: h SLD (n) = straight line distance from n to Bucharest Greedy search expands node that appears to be closest to goal Federico Pecora Advanced Artificial Intelligence Problem Solving and Search: Informed Search Strategies (I) (Lecture 3) 10 / 36

11 Greedy Best-First Search Greedy Best-First Search: Example Federico Pecora Advanced Artificial Intelligence Problem Solving and Search: Informed Search Strategies (I) (Lecture 3) 11 / 36

12 Greedy Best-First Search Greedy Best-First Search: Example Federico Pecora Advanced Artificial Intelligence Problem Solving and Search: Informed Search Strategies (I) (Lecture 3) 11 / 36

13 Greedy Best-First Search Greedy Best-First Search: Example Federico Pecora Advanced Artificial Intelligence Problem Solving and Search: Informed Search Strategies (I) (Lecture 3) 11 / 36

14 Greedy Best-First Search Greedy Best-First Search: Example Federico Pecora Advanced Artificial Intelligence Problem Solving and Search: Informed Search Strategies (I) (Lecture 3) 11 / 36

15 Greedy Best-First Search Greedy Best-First Search: Properties Completeness: Time complexity: Space complexity: Optmiality: Note: not optimal because we can be lead down a wrong path by an overly optimistic heuristic need to bias the search in favor of going back to early paths Federico Pecora Advanced Artificial Intelligence Problem Solving and Search: Informed Search Strategies (I) (Lecture 3) 12 / 36

16 Greedy Best-First Search Greedy Best-First Search: Properties Completeness: no can get stuck in loops, e.g., Iasi Neamt Iasi Neamt... but complete in finite space with repeated state checking Time complexity: Space complexity: Optmiality: Note: not optimal because we can be lead down a wrong path by an overly optimistic heuristic need to bias the search in favor of going back to early paths Federico Pecora Advanced Artificial Intelligence Problem Solving and Search: Informed Search Strategies (I) (Lecture 3) 12 / 36

17 Greedy Best-First Search Greedy Best-First Search: Properties Completeness: no can get stuck in loops, e.g., Iasi Neamt Iasi Neamt... but complete in finite space with repeated state checking Time complexity: O(b m ) Space complexity: Optmiality: Note: not optimal because we can be lead down a wrong path by an overly optimistic heuristic need to bias the search in favor of going back to early paths Federico Pecora Advanced Artificial Intelligence Problem Solving and Search: Informed Search Strategies (I) (Lecture 3) 12 / 36

18 Greedy Best-First Search Greedy Best-First Search: Properties Completeness: no can get stuck in loops, e.g., Iasi Neamt Iasi Neamt... but complete in finite space with repeated state checking Time complexity: O(b m ) Space complexity: O(b m ) (keeps all nodes) Optmiality: Note: not optimal because we can be lead down a wrong path by an overly optimistic heuristic need to bias the search in favor of going back to early paths Federico Pecora Advanced Artificial Intelligence Problem Solving and Search: Informed Search Strategies (I) (Lecture 3) 12 / 36

19 Greedy Best-First Search Greedy Best-First Search: Properties Completeness: no can get stuck in loops, e.g., Iasi Neamt Iasi Neamt... but complete in finite space with repeated state checking Time complexity: O(b m ) Space complexity: O(b m ) (keeps all nodes) Optmiality: no Note: not optimal because we can be lead down a wrong path by an overly optimistic heuristic need to bias the search in favor of going back to early paths Federico Pecora Advanced Artificial Intelligence Problem Solving and Search: Informed Search Strategies (I) (Lecture 3) 12 / 36

20 Greedy Best-First Search Greedy Best-First Search: Properties Completeness: no can get stuck in loops, e.g., Iasi Neamt Iasi Neamt... but complete in finite space with repeated state checking Time complexity: O(b m ) Space complexity: O(b m ) (keeps all nodes) Optmiality: no Note: not optimal because we can be lead down a wrong path by an overly optimistic heuristic need to bias the search in favor of going back to early paths Federico Pecora Advanced Artificial Intelligence Problem Solving and Search: Informed Search Strategies (I) (Lecture 3) 12 / 36

21 Outline 1 Greedy Best-First Search 2 Consistent Heuristics Admissibility, Consistency and Dominance Federico Pecora Advanced Artificial Intelligence Problem Solving and Search: Informed Search Strategies (I) (Lecture 3) 13 / 36

22 Orignially appears in [Hart et al., 1968] Idea: avoid expanding paths that are already expensive Heuristic function: f (n) = g(n) + h(n) g(n) = path cost to node n h(n) = estimated cost from n to goal f (n) = estimated total cost of path through n to goal A search uses an admissible heuristic h(n) h (n) where h (n) is the true cost from n to goal h(n) 0 n, h(goal) = 0 Example: h SLD (n) = never over-estimates the actual road distance Federico Pecora Advanced Artificial Intelligence Problem Solving and Search: Informed Search Strategies (I) (Lecture 3) 14 / 36

23 : Example Federico Pecora Advanced Artificial Intelligence Problem Solving and Search: Informed Search Strategies (I) (Lecture 3) 15 / 36

24 : Example Federico Pecora Advanced Artificial Intelligence Problem Solving and Search: Informed Search Strategies (I) (Lecture 3) 15 / 36

25 : Example Federico Pecora Advanced Artificial Intelligence Problem Solving and Search: Informed Search Strategies (I) (Lecture 3) 15 / 36

26 : Example Federico Pecora Advanced Artificial Intelligence Problem Solving and Search: Informed Search Strategies (I) (Lecture 3) 15 / 36

27 : Example Federico Pecora Advanced Artificial Intelligence Problem Solving and Search: Informed Search Strategies (I) (Lecture 3) 15 / 36

28 : Example Federico Pecora Advanced Artificial Intelligence Problem Solving and Search: Informed Search Strategies (I) (Lecture 3) 15 / 36

29 : Properties Completeness: Time complexity: Space complexity: Optmiality: Theorem: A search is optimal Note: efficiency of A depends on heuristic function Federico Pecora Advanced Artificial Intelligence Problem Solving and Search: Informed Search Strategies (I) (Lecture 3) 16 / 36

30 : Properties Completeness: yes unless there are an infinite number of nodes with f (n) f (goal) Time complexity: Space complexity: Optmiality: Theorem: A search is optimal Note: efficiency of A depends on heuristic function Federico Pecora Advanced Artificial Intelligence Problem Solving and Search: Informed Search Strategies (I) (Lecture 3) 16 / 36

31 : Properties Completeness: yes unless there are an infinite number of nodes with f (n) f (goal) Time complexity: exp. in [relative error in h length of sol.] Space complexity: Optmiality: Theorem: A search is optimal Note: efficiency of A depends on heuristic function Federico Pecora Advanced Artificial Intelligence Problem Solving and Search: Informed Search Strategies (I) (Lecture 3) 16 / 36

32 : Properties Completeness: yes unless there are an infinite number of nodes with f (n) f (goal) Time complexity: exp. in [relative error in h length of sol.] Space complexity: keeps all nodes in memory (i.e., exp. number of nodes in worst case) Optmiality: Theorem: A search is optimal Note: efficiency of A depends on heuristic function Federico Pecora Advanced Artificial Intelligence Problem Solving and Search: Informed Search Strategies (I) (Lecture 3) 16 / 36

33 : Properties Completeness: yes unless there are an infinite number of nodes with f (n) f (goal) Time complexity: exp. in [relative error in h length of sol.] Space complexity: keeps all nodes in memory (i.e., exp. number of nodes in worst case) Optmiality: yes Theorem: A search is optimal Note: efficiency of A depends on heuristic function Federico Pecora Advanced Artificial Intelligence Problem Solving and Search: Informed Search Strategies (I) (Lecture 3) 16 / 36

34 : Properties Completeness: yes unless there are an infinite number of nodes with f (n) f (goal) Time complexity: exp. in [relative error in h length of sol.] Space complexity: keeps all nodes in memory (i.e., exp. number of nodes in worst case) Optmiality: yes Theorem: A search is optimal Note: efficiency of A depends on heuristic function Federico Pecora Advanced Artificial Intelligence Problem Solving and Search: Informed Search Strategies (I) (Lecture 3) 16 / 36

35 : Properties Completeness: yes unless there are an infinite number of nodes with f (n) f (goal) Time complexity: exp. in [relative error in h length of sol.] Space complexity: keeps all nodes in memory (i.e., exp. number of nodes in worst case) Optmiality: yes Theorem: A search is optimal Note: efficiency of A depends on heuristic function Federico Pecora Advanced Artificial Intelligence Problem Solving and Search: Informed Search Strategies (I) (Lecture 3) 16 / 36

36 Theorem: is Optimal Proof (by contradiction): suppose some sub-optimal goal G 2 has been enqueued let n be an unexpanded node on a shortest path to an optimal goal G 1 f (G 2 ) = g(g 2 ) since h(g 2 ) = 0 g(g 2 ) > g(g) since G 2 is sub-optimal g(g) f (n) since h is admissible Since f (G 2 ) > f (n), A could not have expanded G 2 (QED) Federico Pecora Advanced Artificial Intelligence Problem Solving and Search: Informed Search Strategies (I) (Lecture 3) 17 / 36

37 A with Tree-Search Proof of optimality of A implicitly assumes we are using the Tree-Search algorithm S h = 7 g = 2 S g = 4 h = 5 B A g = 1 h = 1 G g = 4 h = 0 fringe = {S} Federico Pecora Advanced Artificial Intelligence Problem Solving and Search: Informed Search Strategies (I) (Lecture 3) 18 / 36

38 A with Tree-Search Proof of optimality of A implicitly assumes we are using the Tree-Search algorithm S h = 7 g = 2 S f = = 5 f = = 7 g = 4 h = 5 B A B g = 1 A h = 1 g = 4 G h = 0 fringe = {A : 5,B : 7} Federico Pecora Advanced Artificial Intelligence Problem Solving and Search: Informed Search Strategies (I) (Lecture 3) 18 / 36

39 A with Tree-Search Proof of optimality of A implicitly assumes we are using the Tree-Search algorithm S h = 7 g = 2 f = = 5 S f = = 7 g = 4 h = 5 B A B g = 1 f = = 8 A h = 1 G g = 4 G h = 0 fringe = {B : 7,G : 8} Federico Pecora Advanced Artificial Intelligence Problem Solving and Search: Informed Search Strategies (I) (Lecture 3) 18 / 36

40 A with Tree-Search Proof of optimality of A implicitly assumes we are using the Tree-Search algorithm S h = 7 g = 2 f = = 5 S f = = 7 g = 4 h = 5 B A B g = 1 f = = 8 f = = 4 A h = 1 G A g = 4 G h = 0 fringe = {A : 4,G : 8} Federico Pecora Advanced Artificial Intelligence Problem Solving and Search: Informed Search Strategies (I) (Lecture 3) 18 / 36

41 A with Tree-Search Proof of optimality of A implicitly assumes we are using the Tree-Search algorithm S h = 7 g = 2 f = = 5 S f = = 7 g = 4 h = 5 B A B g = 1 f = = 8 f = = 4 A h = 1 G A g = 4 f = = 7 G h = 0 G fringe = {G : 7,G : 8} Federico Pecora Advanced Artificial Intelligence Problem Solving and Search: Informed Search Strategies (I) (Lecture 3) 18 / 36

42 A with Tree-Search Proof of optimality of A implicitly assumes we are using the Tree-Search algorithm S h = 7 g = 2 f = = 5 S f = = 7 g = 4 h = 5 B A B g = 1 f = = 8 f = = 4 A h = 1 G A g = 4 f = = 7 G h = 0 G fringe = {G : 7,G : 8} goal test succeeds on G : 7 Federico Pecora Advanced Artificial Intelligence Problem Solving and Search: Informed Search Strategies (I) (Lecture 3) 18 / 36

43 Graph Search Algorithms Tree search with closed list (keeps every node in memory) Closed list can be implemented with hash table to allow efficient lookup Function Graph-Search(problem, fringe) : solution/failure closed /0 fringe insert(make-node(initial-state(problem)), fringe) while true do if fringe = /0 then return failure node remove-front(fringe) if goal-test(problem,state(node)) then return solution(node) if state(node) / closed then closed state(node) fringe insert-all(expand(node, problem), fringe) Federico Pecora Advanced Artificial Intelligence Problem Solving and Search: Informed Search Strategies (I) (Lecture 3) 19 / 36

44 A with Graph-Search If we want to avoid repeated states (e.g., using graph search with closed list) the proof no longer holds! S h = 7 g = 2 S g = 4 h = 5 B A g = 1 h = 1 G g = 4 h = 0 fringe = {S} Federico Pecora Advanced Artificial Intelligence Problem Solving and Search: Informed Search Strategies (I) (Lecture 3) 20 / 36

45 A with Graph-Search If we want to avoid repeated states (e.g., using graph search with closed list) the proof no longer holds! S h = 7 g = 2 S f = = 5 f = = 7 g = 4 h = 5 B A B g = 1 A h = 1 g = 4 G h = 0 fringe = {A : 5,B : 7} Federico Pecora Advanced Artificial Intelligence Problem Solving and Search: Informed Search Strategies (I) (Lecture 3) 20 / 36

46 A with Graph-Search If we want to avoid repeated states (e.g., using graph search with closed list) the proof no longer holds! S h = 7 g = 2 f = = 5 S f = = 7 g = 4 h = 5 B A B g = 1 f = = 8 A h = 1 G g = 4 G h = 0 fringe = {B : 7,G : 8} Federico Pecora Advanced Artificial Intelligence Problem Solving and Search: Informed Search Strategies (I) (Lecture 3) 20 / 36

47 A with Graph-Search If we want to avoid repeated states (e.g., using graph search with closed list) the proof no longer holds! S h = 7 g = 2 f = = 5 S f = = 7 g = 4 h = 5 B A B g = 1 f = = 8 f = = 4 A h = 1 G A g = 4 G h = 0 fringe = {A : 4,G : 8} state A in closed list, do not expand! Federico Pecora Advanced Artificial Intelligence Problem Solving and Search: Informed Search Strategies (I) (Lecture 3) 20 / 36

48 A with Graph-Search If we want to avoid repeated states (e.g., using graph search with closed list) the proof no longer holds! S h = 7 g = 2 f = = 5 S f = = 7 g = 4 h = 5 B A B g = 1 f = = 8 f = = 4 A h = 1 G A g = 4 G h = 0 fringe = {G : 8} goal test succeeds on G : 8 Federico Pecora Advanced Artificial Intelligence Problem Solving and Search: Informed Search Strategies (I) (Lecture 3) 20 / 36

49 Consistent Heuristics Consistent Heuristics and A Optimality So what is the problem? We have defined h in an inconsistent way the relative error of h is not consistent throughout paths this leads to misleading values of f h should ensure that optimal path to any repeated state is always the first one followed Theorem: A with a closed list is optimal if h is consistent h is consistent h(n) c(n,a,n ) + h(n ) i.e., the estimated cost of reaching the goal from n is no greater than the step cost of reaching n plus the estimated cost of reaching the goal from n Federico Pecora Advanced Artificial Intelligence Problem Solving and Search: Informed Search Strategies (I) (Lecture 3) 21 / 36

50 Consistent Heuristics Consistent Heuristics and A Optimality: Proof (I) Lemma 1: If h is consistent, then f cannot decrease along any path f (n ) = g(n ) + h(n ) = g(n) + c(n,a,n ) + h(n ) g(n) + h(n) = f (n) (QED) Key point: h does not become less precise as we go deeper in the search tree Federico Pecora Advanced Artificial Intelligence Problem Solving and Search: Informed Search Strategies (I) (Lecture 3) 22 / 36

51 Consistent Heuristics Consistent Heuristics and A Optimality: Proof (II)... so we know that f is monotone non-decreasing Lemma 2: If h is consistent, then A only expands a node when it has found an optimal path to it we have reached node k, and obtain a new fringe after a few descendants, we reach node k with the same state as k let k be a descendant of some node n since f is non-decreasing, f (k ) f (n) f (k) since h(k) = h(k ) (because they point to the same state), we know that g(k ) g(k) thus the path to k was better! (QED) Federico Pecora Advanced Artificial Intelligence Problem Solving and Search: Informed Search Strategies (I) (Lecture 3) 23 / 36

52 Consistent Heuristics Consistent Heuristics and A Optimality: Proof (III) Theorem: A with a closed list is optimal if h is consistent h is consistent, therefore f is non-decreasing along any path (Lemma 1) the sequence of nodes expanded by A is non-decreasing in f -value whenever A expands (in particular) a goal node, it has found an optimal path to that node (Lemma 2) if a state has already been expanded and then another path to that state is generated, it can safely be discarded, because it cannot be as good as the first path therefore A with a consistent heuristic is optimal (QED) Federico Pecora Advanced Artificial Intelligence Problem Solving and Search: Informed Search Strategies (I) (Lecture 3) 24 / 36

53 Consistent Heuristics f -value Contours in Since h is consistent, A expands outwards along contours of increasing f values A expands all nodes with f (n) < C A expands some nodes with f (n) = C A expands no nodes with f (n) > C Note: in UCS (which is A with h 0), contours are circular around start state; in A they stretch towards goal and become more focused on optimal path Federico Pecora Advanced Artificial Intelligence Problem Solving and Search: Informed Search Strategies (I) (Lecture 3) 25 / 36

54 Admissibility, Consistency and Dominance Admissibility and Consistency Admissibility of h is sufficient for optimality of A if we do not check for repeated states If we do, we need h to be consistent What is the relationship between admissibility and consistency? Any consistent heuristic is admissible Let P = {n,n 1,...,n n,g} be a shortest path from n to goal G Federico Pecora Advanced Artificial Intelligence Problem Solving and Search: Informed Search Strategies (I) (Lecture 3) 26 / 36

55 Admissibility, Consistency and Dominance Admissibility and Consistency Admissibility of h is sufficient for optimality of A if we do not check for repeated states If we do, we need h to be consistent What is the relationship between admissibility and consistency? Any consistent heuristic is admissible Let P = {n,n 1,...,n n,g} be a shortest path from n to goal G h(n) c(n,a,n 1 ) + h(n 1 ) c(n,a,n 1 ) + c(n 1,a,n 2 ) + h(n 2 )... n i,n i+1 P (c(n i,a,n i+1 )) + h(g) = n i,n i+1 P (c(n i,a,n i+1 )) = h (n) Federico Pecora Advanced Artificial Intelligence Problem Solving and Search: Informed Search Strategies (I) (Lecture 3) 26 / 36

56 Admissibility, Consistency and Dominance Admissibility and Consistency Consistency is a stronger requirement than admissibility However it is usually difficult to find admissible heuristics that are not consistent Example: straight line distance to Bucharest (h SLD ) is obviously consistent, as triangle inequality holds Federico Pecora Advanced Artificial Intelligence Problem Solving and Search: Informed Search Strategies (I) (Lecture 3) 27 / 36

57 Admissibility, Consistency and Dominance Admissibile (and Consistent) Heuristics Example: the 8-puzzle h 1 (n) = number of misplaced tiles h 1 (Start State) = 6 h 2 (n) = total Manhattan distance h 2 (Start State) = = 14 Federico Pecora Advanced Artificial Intelligence Problem Solving and Search: Informed Search Strategies (I) (Lecture 3) 28 / 36

58 Admissibility, Consistency and Dominance Admissibile (and Consistent) Heuristics Example: the 8-puzzle h 1 (n) = number of misplaced tiles h 1 (Start State) = 6 h 2 (n) = total Manhattan distance h 2 (Start State) = = 14 Federico Pecora Advanced Artificial Intelligence Problem Solving and Search: Informed Search Strategies (I) (Lecture 3) 28 / 36

59 Admissibility, Consistency and Dominance Admissibile (and Consistent) Heuristics Example: the 8-puzzle h 1 (n) = number of misplaced tiles h 1 (Start State) = 6 h 2 (n) = total Manhattan distance h 2 (Start State) = = 14 Federico Pecora Advanced Artificial Intelligence Problem Solving and Search: Informed Search Strategies (I) (Lecture 3) 28 / 36

60 Admissibility, Consistency and Dominance Admissibile (and Consistent) Heuristics: Dominance If h 2 (n) h 1 (n) for all n (both admissible), then h 2 dominates h 1 Heuristic h(n) = max(h 1 (n),h 2 (n)) is also admissible and dominates both h 1 and h 2 Dominant heuristics are better for search Theorem: Given h 2 (n) h 1 (n) n, if a solution exists, h 1 will lead A to search at least all the nodes searched by h 2 Federico Pecora Advanced Artificial Intelligence Problem Solving and Search: Informed Search Strategies (I) (Lecture 3) 29 / 36

61 Admissibility, Consistency and Dominance Admissibile (and Consistent) Heuristics: Dominance If h 2 (n) h 1 (n) for all n (both admissible), then h 2 dominates h 1 Heuristic h(n) = max(h 1 (n),h 2 (n)) is also admissible and dominates both h 1 and h 2 Dominant heuristics are better for search Theorem: Given h 2 (n) h 1 (n) n, if a solution exists, h 1 will lead A to search at least all the nodes searched by h 2 Some numbers for the 8-puzzle and d = 14 IDS searches 3, 473, 941 nodes A (h 1 ) searches 539 nodes A (h 2 ) searches 113 nodes Federico Pecora Advanced Artificial Intelligence Problem Solving and Search: Informed Search Strategies (I) (Lecture 3) 29 / 36

62 Admissibility, Consistency and Dominance Admissibile (and Consistent) Heuristics: Dominance If h 2 (n) h 1 (n) for all n (both admissible), then h 2 dominates h 1 Heuristic h(n) = max(h 1 (n),h 2 (n)) is also admissible and dominates both h 1 and h 2 Dominant heuristics are better for search Theorem: Given h 2 (n) h 1 (n) n, if a solution exists, h 1 will lead A to search at least all the nodes searched by h 2... and for d = 24 IDS searches 54, 000, 000, 000 nodes A (h 1 ) searches 39,135 nodes A (h 2 ) searches 1,641 nodes Federico Pecora Advanced Artificial Intelligence Problem Solving and Search: Informed Search Strategies (I) (Lecture 3) 29 / 36

63 Admissibility, Consistency and Dominance Admissibile Heuristics from Relaxed Problems Admissible heuristics can be derived from the exact solution cost of a relaxed version of the problem 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 Key point: the optimal solution cost of a relaxed problem is no greater than the optimal solution cost of the real problem Federico Pecora Advanced Artificial Intelligence Problem Solving and Search: Informed Search Strategies (I) (Lecture 3) 30 / 36

64 Admissibility, Consistency and Dominance Admissibile Heuristics from Relaxed Problems Traveling salesperson problem (TSP): find the shortest tour visiting all cities exactly once Spanning tree of a graph: a subgraph which is a tree and connects all the vertices together Minimum spanning tree: a spanning tree with weight less than or equal to the weight of every other spanning tree Weight of MST is lower bound on shortest (open) tour; MST can be computed in close to linear time (e.g., O( E log V ) with Kruskal s algorithm [Kruskal, 1956]) Federico Pecora Advanced Artificial Intelligence Problem Solving and Search: Informed Search Strategies (I) (Lecture 3) 31 / 36

65 Admissibility, Consistency and Dominance Admissibile Heuristics from Relaxed Problems Traveling salesperson problem (TSP): find the shortest tour visiting all cities exactly once Spanning tree of a graph: a subgraph which is a tree and connects all the vertices together Minimum spanning tree: a spanning tree with weight less than or equal to the weight of every other spanning tree Weight of MST is lower bound on shortest (open) tour; MST can be computed in close to linear time (e.g., O( E log V ) with Kruskal s algorithm [Kruskal, 1956]) Federico Pecora Advanced Artificial Intelligence Problem Solving and Search: Informed Search Strategies (I) (Lecture 3) 31 / 36

66 Admissibility, Consistency and Dominance Efficiency of A A is optimally efficient for any heuristic function Why? no other optmial algorithm with the same heuristic expands fewer nodes than A (among algorithms of the same type, i.e., algorithms that extend search paths from root) any algorithm that does not expand all nodes with f (n) < C can miss an optimal solution Federico Pecora Advanced Artificial Intelligence Problem Solving and Search: Informed Search Strategies (I) (Lecture 3) 32 / 36

67 Admissibility, Consistency and Dominance Efficiency of A A is optimally efficient for any heuristic function Why? no other optmial algorithm with the same heuristic expands fewer nodes than A (among algorithms of the same type, i.e., algorithms that extend search paths from root) any algorithm that does not expand all nodes with f (n) < C can miss an optimal solution Federico Pecora Advanced Artificial Intelligence Problem Solving and Search: Informed Search Strategies (I) (Lecture 3) 32 / 36

68 Admissibility, Consistency and Dominance Efficiency of A A is complete, optimal and optimally efficient However, a heuristic cannot always guarantee that there aren t exponential nodes within the goal contour typically the most informed (i.e., perfect) heuristic function h entails the need to solve the search problem itself Heuristics such that h(n) h (n) O(logh (n)) avoid exponential growth error in heuristic function must not grow faster than actual path cost for most heuristics used in practice, error is at least proportional to path cost There is always a tradeoff between heuristic complexity and the pruning power of A Federico Pecora Advanced Artificial Intelligence Problem Solving and Search: Informed Search Strategies (I) (Lecture 3) 33 / 36

69 Admissibility, Consistency and Dominance Relationships Among Search Algorithms f = depth (BFS) h 0 (UCS) h < h A f = g + h Best-first search Depth-first search Generic graph-search algorithms Federico Pecora Advanced Artificial Intelligence Problem Solving and Search: Informed Search Strategies (I) (Lecture 3) 34 / 36

70 Admissibility, Consistency and Dominance Informed Search: Summary Informed algorithms employ a measure of node desirability for node expansion Desirability is defined by a heuristic function h(n) Greedy Best-First search expands nodes with minimal h(n) A search expands nodes with minimal f (n) = g(n) + h(n) complete, optimal and optimally efficient provided that h is admissible (for Tree-Search) or consistent (for Graph-Search) Performance of heuristic search depends on quality of h good heurisitcs can be constructed by relaxing the problem A can be prohibitive due to memory requirements solutions to this will be presented in the next lecture Federico Pecora Advanced Artificial Intelligence Problem Solving and Search: Informed Search Strategies (I) (Lecture 3) 35 / 36

71 Admissibility, Consistency and Dominance Problem Solving and Search: Informed Search Strategies (I) Thank you! Federico Pecora Advanced Artificial Intelligence Problem Solving and Search: Informed Search Strategies (I) (Lecture 3) 36 / 36

72 References References Hart, P., Nilsson, N., and Raphael, B. (1968). A formal basis for the heuristic determination of minimum cost paths. Systems Science and Cybernetics, IEEE Transactions on, 4(2): Kruskal, J. (1956). On the shortest spanning subtree of a graph and the traveling salesman problem. Proceedings of the American Mathematical Society, 7(1): Federico Pecora Advanced Artificial Intelligence Problem Solving and Search: Informed Search Strategies (I) (Lecture 3) 37 / 36

73 References Acknowledgements Portions of this course are inspired, compiled or taken from scientific presentations and teaching material by the following authors: Fahiem Bacchus, Amedeo Cesta, Gary Cottrell, Rina Dechter, Simone Fratini, Geoff Gordon, Russell Greiner, John Harrison, David Kriegman, Sharad Malik, Dana Nau, Peter Norvig, Madhusudan Parthasarathy, Rhys Price Jones, Stuart Russel, Tuomas Sandholm, Stephen F. Smith, Padhraic Smyth, Paolo Traverso, Toby Walsh. Federico Pecora Advanced Artificial Intelligence Problem Solving and Search: Informed Search Strategies (I) (Lecture 3) 37 / 36