Uninformed Search B. Navigating through a search tree. Navigating through a search tree. Navigating through a search tree

Transcription

1 Uninformed Search Russell and Norvig chap. 3 D E 1

2 Unexpanded s: the fringe Tree search nitial state t every point in the search process we keep track of a list of s that haven t been expanded yet: the fringe function TREE-SER(problem, strategy) return a solution or failure nitialize search tree to the initial state of the problem do if no candidates for expansion then return failure choose leaf for expansion according to strategy if contains goal state then return solution else expand the and add resulting s to the search tree enddo 2

3 Tree search What s in a function TREE-SER(problem, strategy) return a solution or failure nitialize search tree to the initial state of the problem do if no candidates for expansion then return failure choose leaf for expansion according to strategy if contains goal state then return solution else expand the and add resulting s to the search tree enddo State Parent ction (the action that got us from the parent) Depth Path-ost Tree search Metrics for comparing search strategies function TREE-SER(problem,fringe) return a solution or failure fringe NSERT(ME-NODE(NT-STTE[problem]), fringe) loop do if EMPTY(fringe) then return failure REMOVE-FRST(fringe) if GO-TEST[problem] applied to STTE[] succeeds then return SOUTON() fringe NSERT-(EXPND(, problem), fringe) strategy is defined by the order of expansion. Problem-solving performance is measured in four ways: ompleteness: Does it always find a solution if one exists? Optimality: Does it always find the least-cost solution? Time omplexity: Number of s generated/expanded. Space omplexity: Number of s stored in memory during search. Time and space complexity 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 ) Uninformed search strategies readth-first Search (FS) a.k.a. blind search = use only information available in problem definition. When strategies can determine whether one non-goal state is better than another informed search. Search algorithms are defined by the expansion method: readth-first search Uniform-cost search Depth-first search Depth-limited search terative deepening search. idirectional search all s at depth d before proceeding to depth d+1. mplementation: queue (FFO). 3

4 Evaluation of FS Evaluation of FS ompleteness: Does it always find a solution if one exists? YES (if shallowest goal is at some finite depth d) ompleteness: YES ssume a state space where every state has b successors. ssume solution is at depth d Worst case: expand all but the last at depth d Total number of s expanded: b + b 2 + b b d + (b d +1 b) = O(b d +1 ) Evaluation of FS Evaluation of FS ompleteness: YES Total number of s generated: ompleteness: YES Total number of s generated: b + b 2 + b b d + (b d +1 b) = O(b d +1 ) Space complexity: Same, if each is retained in memory b + b 2 + b b d + (b d +1 b) = O(b d +1 ) Space complexity: Same, if each is retained in memory Optimality: Does it always find the least-cost solution? n general YES FS evaluation Memory requirements are a bigger problem than its execution time. Exponential complexity search problems cannot be solved by uninformed search methods for any but the smallest instances. DEPT NODES TME MEMORY seconds 1 megabyte seconds 106 megabytes minutes 10 gigabytes hours 1 terabyte days 101 terabytes years 10 petabytes years 1 exabyte Uniform cost search Extension of FS: with lowest path cost mplementation: fringe = queue ordered by path cost. Same as FS when all step-costs are equal. Time and memory requirements for FS for b=10, 10,000 s/sec; 1000 bytes/ 4

5 Uniform cost search ompleteness: YES, if step-cost > ε (smal positive constant) ssume * the cost of the optimal solution. ssume that every action costs at least ε Worst-case: O(b */ε ) Space complexity: Same as time complexity Optimality: s expanded in order of increasing path cost. YES, if complete. D E D D E 5

6 D E F G D E F G DFS evaluation D E F G ompleteness: Does it always find a solution if one exists? NO unless search space is finite and no loops are possible. mplementation: fringe is a stack (FO) DFS evaluation DFS evaluation ompleteness: NO unless search space is finite. Terrible if m (depth of search space) is much larger than d (depth of optimal solution) ut if many solutions, then faster than FS ompleteness: NO unless search space is finite. O(b m ) Space complexity: O(bm) Possible to use even less (expand one successor instead of all b). 6

7 DFS evaluation ompleteness: NO unless search space is finite. O(b m ) Space complexity: O(bm) Optimallity: No Same issues as completeness Depth-limited search DFS with depth limit l. i.e. s at depth l have no successors. Problem knowledge can be used. Solves the infinite-path problem. f l < d (depth of least cost solution) then incomplete f l > d then not optimal. O(b l ) Space complexity: O(bl) terative Deepening Search (DS) strategy to find best depth limit l. Depth-imited Search to depth 1, 2, s from the root each time. ppears very wasteful, but combinatorics can be counter intuitive: N(DS) = b + b b d-1 + b d = O(b d ) N(DS) = db + (d-1)b b d-1 + b d = O(b d ) N(FS) = b + b b d + b d+1 = O(b d+1 ) Example: For b = 10 and d = 5 N(DS) = 111,111 N(DS) = 123,456 N(FS) = 1,111,100. terative deepening search function TERTVE_DEEPENNG_SER(problem) return a solution or failure inputs: problem for depth 0 to do result DEPT-MTED_SER(problem, depth) if result cuttoff then return result Evaluation of DS ompleteness: YES (no infinite paths) Evaluation of DS ompleteness: YES O(b d ) 7

8 Evaluation of DS Evaluation of DS ompleteness: YES O(b d ) Space complexity: O(bd) Same as DFS ompleteness: YES O(b d ) Space complexity: O(bd) Same as DFS Optimality: YES if step cost is 1. terative Deepening Search idirectional Search nalogous to FS: explores a complete layer of s before proceeding to the next one. ombines benefits of DFS and FS. Two simultaneous searches from start and goal. Motivation: b d/2 + b d/2 much less than b d efore a is expanded it is checked if it is in the fringe of the other search (can be done in constant time using a hash table). O(b d/2 ). Space complexity: same. omplete and optimal (for uniform step costs) if both searches are FS idirectional Search omparison of search strategies riterion readth- First Depth-First Uniformcost Depthlimited terative deepening idirection al search omplete? YES* YES* NO YES, if l d YES YES* ssues in applying: The predecessor of each should be efficiently computable. When actions are easily reversible. Goal : sometimes not known explicitly (e.g. in chess). Time b d+1 b */e b m b l b d b d/2 Space b d+1 b */e bm bl bd b d/2 Optimal? YES* YES* NO NO YES YES 8

9 When the search graph is not a tree Need to avoid repeated states! appens in problems with reversible operators Examples: missionaries and cannibals problem, sliding blocks puzzles, route finding problems. Detection: compare a to be expanded to those already expanded. Those are kept in the closed list. ncreases memory requirements (especially for DFS): bounded by the size of the state space. Graph Search function GRP-SER(problem,fringe) return a solution or failure closed an empty set fringe NSERT(ME-NODE(NT-STTE[problem]), fringe) loop do if EMPTY?(fringe) then return failure REMOVE-FRST(fringe) if GO-TEST[problem] applied to STTE[] succeeds then return SOUTON() if STTE[] is not in closed then add STTE[] to closed fringe NSERT-(EXPND(, problem), fringe) 9