Craiova. Dobreta. Eforie. 99 Fagaras. Giurgiu. Hirsova. Iasi. Lugoj. Mehadia. Neamt. Oradea. 97 Pitesti. Sibiu. Urziceni Vaslui.

Transcription

1 Informed search algorithms Chapter 4, Sections 1{2, 4 AIMA Slides cstuart Russell and Peter Norvig, 1998 Chapter 4, Sections 1{2, 4 1

2 Outline } Best-rst search } A search } Heuristics } Hill-climbing } Simulated annealing AIMA Slides cstuart Russell and Peter Norvig, 1998 Chapter 4, Sections 1{2, 4 2

3 A strategy is dened by picking the order of node expansion Review: General search function General-Search( problem, Queuing-Fn) returns a solution, or failure nodes Make-Queue(Make-Node(Initial-State[problem])) loop do if nodes is empty then return failure node Remove-Front(nodes) if Goal-Test[problem] applied to State(node) succeeds then return node nodes Queuing-Fn(nodes, Expand(node, Operators[problem])) end AIMA Slides cstuart Russell and Peter Norvig, 1998 Chapter 4, Sections 1{2, 4 3

4 Best-rst search use an evaluation function for each node Idea: estimate of \desirability" { ) Expand most desirable unexpanded node Implementation: = insert successors in decreasing order of desirability QueueingFn cases: Special search greedy A search AIMA Slides cstuart Russell and Peter Norvig, 1998 Chapter 4, Sections 1{2, 4 4

5 Oradea 71 Neamt Zerind Arad Timisoara 111 Lugoj 70 Mehadia Dobreta Sibiu 99 Fagaras 80 Rimnicu Vilcea 97 Pitesti Bucharest 90 Craiova Giurgiu 87 Iasi Urziceni Vaslui Hirsova 86 Eforie Straight line distance to Bucharest Arad 366 Bucharest 0 Craiova 160 Dobreta 242 Eforie 161 Fagaras 178 Giurgiu 77 Hirsova 151 Iasi 226 Lugoj 244 Mehadia 241 Neamt 234 Oradea 380 Pitesti 98 Rimnicu Vilcea 193 Sibiu 253 Timisoara 329 Urziceni 80 Vaslui 199 Zerind 374 Romania with step costs in km AIMA Slides cstuart Russell and Peter Norvig, 1998 Chapter 4, Sections 1{2, 4 5

6 Greedy search function h(n) (heuristic) Evaluation estimate of cost from n to goal = E.g., h SLD (n) = straight-line distance from n to Bucharest Greedy search expands the node that appears to be closest to goal AIMA Slides cstuart Russell and Peter Norvig, 1998 Chapter 4, Sections 1{2, 4 6

7 Greedy search example Arad 366 AIMA Slides cstuart Russell and Peter Norvig, 1998 Chapter 4, Sections 1{2, 4 7

8 Zerind Sibiu Timisoara AIMA Slides cstuart Russell and Peter Norvig, 1998 Chapter 4, Sections 1{2, 4 8

9 Arad Oradea Fagaras Rimnicu Vilcea AIMA Slides cstuart Russell and Peter Norvig, 1998 Chapter 4, Sections 1{2, 4 9

10 Sibiu Bucharest AIMA Slides cstuart Russell and Peter Norvig, 1998 Chapter 4, Sections 1{2, 4 10

11 Properties of greedy search Complete?? Time?? Space?? Optimal?? AIMA Slides cstuart Russell and Peter Norvig, 1998 Chapter 4, Sections 1{2, 4 11

12 Properties of greedy search No{can get stuck in loops, e.g., Complete??! Neamt! Iasi! Neamt! Iasi Complete in nite space with repeated-state checking Time?? O(b m ), but a good heuristic can give dramatic improvement Space?? O(b m ) keeps all nodes in memory Optimal?? No AIMA Slides cstuart Russell and Peter Norvig, 1998 Chapter 4, Sections 1{2, 4 12

13 Evaluation function f(n) =g(n) +h(n) = cost so far to reach n g(n) = estimated cost to goal from n h(n) A search Idea: avoid expanding paths that are already expensive f(n) = estimated total cost of path through n to goal search uses an admissible heuristic A h(n) h (n) where h (n) is the true cost from n. i.e., E.g., h SLD (n) never overestimates the actual road distance Theorem: A search is optimal AIMA Slides cstuart Russell and Peter Norvig, 1998 Chapter 4, Sections 1{2, 4 13

14 A search example Arad 366 AIMA Slides cstuart Russell and Peter Norvig, 1998 Chapter 4, Sections 1{2, 4 14

15 Zerind Sibiu Timisoara AIMA Slides cstuart Russell and Peter Norvig, 1998 Chapter 4, Sections 1{2, 4 15

16 Arad Oradea Fagaras Rimnicu Vilcea AIMA Slides cstuart Russell and Peter Norvig, 1998 Chapter 4, Sections 1{2, 4 16

17 Craiova Pitesti Sibiu AIMA Slides cstuart Russell and Peter Norvig, 1998 Chapter 4, Sections 1{2, 4 17

18 Rimnicu Vilcea Craiova Bucharest AIMA Slides cstuart Russell and Peter Norvig, 1998 Chapter 4, Sections 1{2, 4 18

19 Sibiu Bucharest AIMA Slides cstuart Russell and Peter Norvig, 1998 Chapter 4, Sections 1{2, 4 19

20 Suppose some suboptimal goal G 2 Optimality of A (standard proof) has been generated and is in the Let n be an unexpanded node on a shortest path to an optimal queue. G 1. goal Start n G G 2 f(g 2 ) = g(g 2 ) since h(g 2 )=0 > g(g 1 ) since G 2 is suboptimal f(n) since h is admissible Since f(g 2 ) >f(n), A will never select G 2 for expansion AIMA Slides cstuart Russell and Peter Norvig, 1998 Chapter 4, Sections 1{2, 4 20

21 adds \f-contours" of nodes (cf. breadth-rst adds layers) Gradually i has all nodes with f = f i, where f i <f i+1 Contour Optimality of A (more useful) Lemma: A expands nodes in order of increasing f value O Z N A S F I V T R L P D M C 420 G B U H E AIMA Slides cstuart Russell and Peter Norvig, 1998 Chapter 4, Sections 1{2, 4 21

22 Properties of A Complete?? Yes, unless there are innitely many nodes with f f(g) Time?? Exponential in [relative error in h length of soln.] Space?? Keeps all nodes in memory Optimal?? Yes cannot expand f i+1 until f i is nished AIMA Slides cstuart Russell and Peter Norvig, 1998 Chapter 4, Sections 1{2, 4 22

23 Proof of lemma: Pathmax For some admissible heuristics, f may decrease along a path n g=5 h=4 f=9 E.g., suppose n 0 is a successor ofn 1 n g =6 h =2 f =8 this throws away information! But =9) true cost of a path through n is 9 f(n) Hence true cost of a path through n 0 is 9 also modication to A : Pathmax of f(n 0 )=g(n 0 )+h(n 0 ), use f(n 0 )=max(g(n 0 )+h(n 0 );f(n)) Instead With pathmax, f is always nondecreasing along any path AIMA Slides cstuart Russell and Peter Norvig, 1998 Chapter 4, Sections 1{2, 4 23

24 1 (n) = number of misplaced tiles h 2 (n) = total Manhattan distance h 1 (S) =?? h 2 (S) =?? h Admissible heuristics E.g., for the 8-puzzle: (i.e., no. of squares from desired location of each tile) Start State Goal State AIMA Slides cstuart Russell and Peter Norvig, 1998 Chapter 4, Sections 1{2, 4 24

25 1 (n) = number of misplaced tiles h 2 (n) = total Manhattan distance h 1 (S) =?? 7 h 2 (S) =?? = 18 h Admissible heuristics E.g., for the 8-puzzle: (i.e., no. of squares from desired location of each tile) Start State Goal State AIMA Slides cstuart Russell and Peter Norvig, 1998 Chapter 4, Sections 1{2, 4 25

26 =14 IDS = 3,473,941 nodes d (h 1 ) = 539 nodes A Dominance h 2 (n) h 1 (n) for all n (both admissible) If h 2 dominates h 1 and is better for search then Typical search costs: (h 2 ) = 113 nodes A =14 IDS = too many nodes d (h 1 ) = 39,135 nodes A (h 2 ) = 1,641 nodes A AIMA Slides cstuart Russell and Peter Norvig, 1998 Chapter 4, Sections 1{2, 4 26

27 heuristics can be derived from the exact Admissible cost of a relaxed version of the problem solution Relaxed problems the rules of the 8-puzzle are relaxed so that a tile can move anywhere, If h 1 (n) gives the shortest solution then the rules are relaxed so that a tile can move to any adjacent square, If h 2 (n) gives the shortest solution then TSP: let path be any structure that connects all cities For minimum spanning tree heuristic =) AIMA Slides cstuart Russell and Peter Norvig, 1998 Chapter 4, Sections 1{2, 4 27

28 Iterative improvement algorithms many optimization problems, path is irrelevant; In goal state itself is the solution the state space = set of \complete" congurations; Then optimal conguration, e.g., TSP nd or, nd conguration satisfying constraints, e.g., n-queens such cases, can use iterative improvement algorithms; In a single \current" state, try to improve it keep Constant space, suitable for online as well as oine search AIMA Slides cstuart Russell and Peter Norvig, 1998 Chapter 4, Sections 1{2, 4 28

29 Example: Travelling Salesperson Problem Find the shortest tour that visits each city exactly once AIMA Slides cstuart Russell and Peter Norvig, 1998 Chapter 4, Sections 1{2, 4 29

30 Example: n-queens n queens on an n n board with no two queens on the same Put column, or diagonal row, AIMA Slides cstuart Russell and Peter Norvig, 1998 Chapter 4, Sections 1{2, 4 30

31 Hill-climbing (or gradient ascent/descent) \Like climbing Everest in thick fog with amnesia" function Hill-Climbing( problem) returns a solution state inputs: problem, a problem local variables: current, a node next, a node current Make-Node(Initial-State[problem]) loop do next a highest-valued successor of current if Value[next] < Value[current] then return current current next end AIMA Slides cstuart Russell and Peter Norvig, 1998 Chapter 4, Sections 1{2, 4 31

32 Hill-climbing contd. Problem: depending on initial state, can get stuck on local maxima value global maximum local maximum states AIMA Slides cstuart Russell and Peter Norvig, 1998 Chapter 4, Sections 1{2, 4 32

33 Simulated annealing escape local maxima by allowing some \bad" moves Idea: gradually decrease their size and frequency but function Simulated-Annealing( problem, schedule) returns a solution state inputs: problem, a problem schedule, a mapping from time to \temperature" local variables: current, a node next, a node T, a \temperature" controlling the probability of downward steps current Make-Node(Initial-State[problem]) for t 1 to 1 do T schedule[t] if T=0 then return current next a randomly selected successor of current E Value[next] {Value[current] if E > 0 then current next else current next only with probability e E =T AIMA Slides cstuart Russell and Peter Norvig, 1998 Chapter 4, Sections 1{2, 4 33

34 Properties of simulated annealing xed \temperature" T, state occupation probability reaches At distribution Boltzman E(x) kt =e p(x) T decreased slowly enough =) always reach best state Is this necessarily an interesting guarantee?? Devised by Metropolis et al., 1953, for physical process modelling Widely used in VLSI layout, airline scheduling, etc. AIMA Slides cstuart Russell and Peter Norvig, 1998 Chapter 4, Sections 1{2, 4 34