Summary. Search CSL302 - ARTIFICIAL INTELLIGENCE 1

Transcription

1 Summary Search CSL302 - ARTIFICIAL INTELLIGENCE 1

2 Informed Search CHAPTER 3

3 Informed (Heuristic) Search qheuristic problem-specific knowledge ofinds solutions more efficiently qnew terms o!(#): cost from initial state to the state at node n oh # : estimated cost from state at node n to the closest goal ØDepends only in the state at node # Øh(#) = 0, if # is a goal node o* # : evaluation function; cost estimate from the initial state to the closest goal state through the state at node n qgenre of Best First Search (BFS) ogreedy BFS oa* search Informed Search CSL302 ARTIFICIAL INTELLIGENCE 3

4 Greedy Best First Search (GBFS) qexpand the most desirable node odesirability is measured through the evaluation function!(#) and here! # = h(#) qimplementation: priority queue based on h(#) Informed Search CSL302 ARTIFICIAL INTELLIGENCE 4

5 GBFS - Analysis qcompleteness: No, can get stuck in loops ocan be made complete with repeated state checking qoptimality: No qtime Complexity:! " # qspace Complexity:!(" # ) okeeps all nodes in memory. Neamt 87 Iasi 92 Vaslui Arad Bucharest Craiova Drobeta Eforie Fagaras Giurgiu Hirsova Iasi Lugoj Mehadia Neamt Oradea Pitesti Rimnicu Vilcea Sibiu Timisoara Urziceni Vaslui Zerind ti Urziceni Bucharest 90 Giurgiu Hirsova 86 Eforie Informed Search CSL302 ARTIFICIAL INTELLIGENCE 5

6 A* search qidea: avoid expanding paths that are already expensive. qimplementation: priority queue based on the evaluation function!(#)! # = & # + h # o& # : cost so far to reach the node # oh(#): estimated cost to goal from # o!(#): estimated total cost of path through # to goal. qidentical to uniform cost search(ucs) except that we use & + h instead of only & Informed Search CSL302 ARTIFICIAL INTELLIGENCE 6

7 A* - Example Informed Search CSL302 ARTIFICIAL INTELLIGENCE 7

8 A* search - Analysis qcompleteness: Yes, unless there are infinitely many nodes with!!($) qoptimality:??? Informed Search CSL302 ARTIFICIAL INTELLIGENCE 8

9 Admissible Heuristic qa* uses an admissible heuristic h(#) h # h #, # oh (#) is the true cost to the goal node from # oh # 0, # 71 Oradea Neamt Arad Zerind 140 Timisoara 151 Sibiu 99 Fagaras 80 Rimnicu Vilcea 87 Iasi 92 Vaslui 111 Lugoj 97 Pitesti Drobeta Mehadia Urziceni 138 Bucharest 90 Craiova Giurgiu Hirsova 86 Eforie Informed Search CSL302 ARTIFICIAL INTELLIGENCE 9

10 Admissible Heuristic qa* uses an admissible heuristic h(#) h # h #, # oh (#) is the true cost to the goal node from # oh # 0, # Search nodes Heuristic value h* h_1 h_2 h_3 h_4 h_5 Informed Search CSL302 ARTIFICIAL INTELLIGENCE 10

11 Example Heuristic Functions (1) q8-puzzle problem qexamples onumber of misplaced tiles ototal Manhattan distance Informed Search CSL302 ARTIFICIAL INTELLIGENCE 11

12 Example Heuristic Functions (2) qromania 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. qexamples ostraight line distance never overestimates the actual road distance Informed Search CSL302 ARTIFICIAL INTELLIGENCE 12

13 A* search - optimality qproof by contradiction olet! be the goal node A* outputs and suppose there is another goal node!. Then #! #(! & ) oassume to the contrary #! & < #(!) owhen we picked! for expansion, either! or an ancestor of! -! must have been on the queue. Since we picked! for expansion o)! )! && implies #! + h! #! && + h! && o#! #! && + h! && For a goal node h! = 0 o#! & = #! && +./01! &,! &&, h! && h! && =./01! &,! && oso #! & #! && + h! && (2) ofrom (1) and (2) #! #! & - contradiction Informed Search CSL302 ARTIFICIAL INTELLIGENCE 13

14 A* search - Analysis qcompleteness: Yes, unless there are infinitely many nodes with!!($) qoptimality: Yes qspace Complexity: Keeps all nodes in memory qtime Complexity: exponential in [relative error in h * length of the solution] oa* expands all nodes with! < ( oa* expands some nodes with! = ( oa* expands no nodes with! > ( Informed Search CSL302 ARTIFICIAL INTELLIGENCE 14

15 Visualizing A* Search qa* expands nodes of increasing! value Uniform cost search Informed Search CSL302 ARTIFICIAL INTELLIGENCE 15

16 Admissibility, Monotonicity, Pathmax Correction qis orange h admissible? qis green h admissible? qdoes "($) make sense? o"(&)= = 8.9 o"($)=.2+0 = 0.2 qpath cost estimate reduces othis doesn t make sense since we are reducing the estimate of the actual cost of the path qto make "(. ) monotonic along a path, we say " ) = max "./01)2, 4 ) + h ) o Also referred to as Pathmax correction B C D A G Informed Search CSL302 ARTIFICIAL INTELLIGENCE 16

17 Monotonic Heuristic qconsistent Heuristic qa heuristic is monotonic if h " $ ", &, " ' + h(" ' ) qif h is monotonic, we have + " ' = - " ' + h " ' = - " + $ ", &, " ' + h(" ' ) - " + h " = + " qi.e., +(")is monotonic along any path Triangle Inequality Informed Search CSL302 ARTIFICIAL INTELLIGENCE 17

18 Effect of Heuristic Accuracy on Performance (1) qtotal number of nodes generated by A* search! qsolution depth is " qeffective branching factor - # - branching factor of a uniform tree of depth " with! + 1 nodes! + 1 = 1 + # + # ) + + # + qwhat should be the value of # to perform search efficiently? Informed Search CSL302 ARTIFICIAL INTELLIGENCE 18

19 Effect of Heuristic Accuracy on Performance (2) q8-puzzle problem qtwo heuristics onumber of misplaced tiles h " ototal Manhattan distance - h # Informed Search CSL302 ARTIFICIAL INTELLIGENCE 19

20 Effect of Heuristic Accuracy on Performance (3) q8-puzzle problem 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. Informed Search CSL302 ARTIFICIAL INTELLIGENCE 20

21 Effect of Heuristic Accuracy on Performance (4) qfor two heuristics - h " and h # qif h # $ h " $, $ qthen h # dominates h " qdomination ~ efficiency of search in terms of number of nodes expanded oh # will never expand more nodes than A* using h " (except for some nodes with ( $ = * ) Informed Search CSL302 ARTIFICIAL INTELLIGENCE 21

22 Generating Admissible Heuristic Relaxed Problems qshortest Path Problem on the plane I I h " G circular abstraction I G h # G Polygonal abstraction Actual h I G Informed Search CSL302 ARTIFICIAL INTELLIGENCE 22

23 Heuristic Functions - Abstractions Total cost incurred in search h_0 h_d h_c h_p h* > Reduced level of abstraction cost of searching with the heuristic cost of computing the heuristic Informed Search CSL302 ARTIFICIAL INTELLIGENCE 23

24 Generating Admissible Heuristics Pattern Databases qstore the exact solution costs for every sub problem instance qadmissible heuristic will be cost for solving the corresponding sub problem. ocan be generated by working backwards from the goal state Informed Search CSL302 ARTIFICIAL INTELLIGENCE 24

25 Combining Heuristics qcan we add the heuristics obtained from and databases? owould it still be an admissible heuristic? qdisjoint pattern databases qhow will you combine admissible heuristics? qwhat is the effect of using inadmissible heuristics? Informed Search CSL302 ARTIFICIAL INTELLIGENCE 25

26 Iterative Deepening A* (IDA*)Search qessentially IDDS, that uses! as the cost threshold, instead of depth. qimplementation: add child to the queue if! "h$%& < (h)*+h,%&. o Start with! cutoff equal to the! value of the root node o Loop until solution is found o Generate and search all nodes whose! values are the current threshold o Use DFS to search the trees in each iteration o Keep track of the node which has the smallest! value that is > the current threshold. o If a goal node is found terminate, else set the threshold to be next highest! value and loop back. Informed Search CSL302 ARTIFICIAL INTELLIGENCE 26

27 IDA* Search - Example Informed Search CSL302 ARTIFICIAL INTELLIGENCE 27

28 IDA* Search - Analysis qcompleteness: yes qoptimality: yes qtime Complexity: worst case:! " owhen all nodes have distinct # values qspace Complexity: linear -!$ Informed Search CSL302 ARTIFICIAL INTELLIGENCE 28

29 Additional Readings qrecursive Best First Search qsimplified memory-bounded A* qbeam Search Informed Search CSL302 ARTIFICIAL INTELLIGENCE 29