Uninformed Search B. Navigating through a search tree. Navigating through a search tree. Navigating through a search tree
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1 Uninformed Search Russell and Norvig chap. 3 Following this, the pong paddle went on a mission to destroy tari headquarters and, due to a mixup, found himself inside the game The Matrix Reloaded. oy, was TT ever hard to explain to him. D E 1
2 D E F G J 2
3 Unexpanded s: the frontier t every point in the search process we keep track of a list of s that haven t been expanded yet: the frontier Tree search Tree search nitial state D E function TREE-SER(problem) return a solution or failure nitialize frontier using the initial state of problem do if the frontier is empty then return failure choose leaf from the frontier if is a goal state then return solution else expand the and add resulting s to the frontier function TREE-SER(problem) return a solution or failure nitialize frontier using the initial state of problem do if the frontier is empty then return failure choose leaf from the frontier if is a goal state then return solution else expand the and add resulting s to the frontier What s in a When the search graph is not a tree State Parent ction (the action that got us from the parent) Depth Path-ost (total cost to get to the ) Why do we need the parent and action information? 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. ncreases memory requirements (especially for DFS): bounded by the size of the state space. 3
4 Graph Search function GRP-SER(problem) return a solution or failure initialize the frontier using the initial state of problem initialize the explored set to be empty loop do if the frontier is empty then return failure choose a from the frontier if is a goal state then return the corresponding solution add the to the explored set expand the, adding the resulting s to the frontier (only if not in the frontier or explored set) Search strategies function GRP-SER(problem) return a solution or failure initialize the frontier using the initial state of problem initialize the explored set to be empty loop do if the frontier is empty then return failure choose a from the frontier if is a goal state then return the corresponding solution add the to the explored set expand the, adding the resulting s to the frontier (only if not in the frontier or explored set) Search strategies differ in how a is chosen from the frontier lgorithms that forget their history are doomed to repeat it Uninformed search strategies Metrics for comparing search strategies a.k.a. blind search = use only information available in problem definition. q When strategies can determine whether one non-goal state is better than another informed search. Search algorithms are defined by the expansion method: q readth-first search q Uniform-cost search q Depth-first search q Depth-limited search q terative deepening search. q idirectional search strategy is defined by the order of expansion. Problem-solving performance is measured in four ways: q ompleteness: Does it always find a solution if one exists? q Optimality: Does it always find the least-cost solution? q Time omplexity: Number of s generated/expanded. q Space omplexity: Number of s stored in memory during search. Time and space complexity are measured in terms of: q b - maximum branching factor of the search tree q d - depth of the least-cost solution q m - maximum depth of the state space (may be ) readth-first Search (FS) all s at depth d before proceeding to depth d+1. Return the first goal found mplementation: queue (FFO). Evaluation of FS ompleteness: q Does it always find a solution if one exists? 4
5 Evaluation of FS Evaluation of FS ompleteness: q Does it always find a solution if one exists? YES (if shallowest goal is at some finite depth d) ompleteness: Time complexity: q 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 = O(b d ) Evaluation of FS Evaluation of FS ompleteness: Time complexity: q Total number of s generated: b + b 2 + b b d = O(b d ) ompleteness: Time complexity: q Total number of s generated: b + b 2 + b b d = O(b d ) Space complexity: q Same, if each is retained in memory Space complexity: q Same, if each is retained in memory Optimality: q Does it always find the least-cost solution? YES (if all actions have the same cost) 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: q with lowest path cost mplementation: fringe = priority 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/ 5
6 Uniform cost search ompleteness: Time complexity: q ssume * the cost of the optimal solution. q ssume that every action costs at least ε q Worst-case: O(b */ε ) Space complexity: q Same as time complexity Optimality: (s expanded in order of increasing path cost) D E D D E 6
7 D E F G D E F G DFS evaluation D E F G ompleteness: q Does it always find a solution if one exists? mplementation: fringe is a stack (FO) DFS evaluation DFS evaluation ompleteness: q Does it always find a solution if one exists? NO unless search space is finite (also beware of loops if using graph search) ompleteness: q NO (unless search space is finite). Time complexity: q Terrible if m (depth of search space) is much larger than d (depth of optimal solution) q ut if many solutions, then faster than FS 7
8 DFS evaluation ompleteness: q NO unless search space is finite. Time complexity: O(b m ) Space complexity: O(bm) q Possible to use even less (expand one successor instead of all b). DFS evaluation ompleteness: q NO unless search space is finite. Time complexity: O(b m ) Space complexity: O(bm) Optimallity: No Depth-limited search DFS with depth limit l. q Treat s at depth l as if they have no successors. Solves the infinite-path problem. f l < d (depth of least cost solution) then incomplete. f l > d then not optimal. Time complexity: 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 = O(b d ) terative deepening search function TERTVE_DEEPENNG_SER(problem) return a solution or failure for depth = 0 to do result DEPT-MTED_SER(problem, depth) if result cutoff then return result Evaluation of DS ompleteness: (no infinite paths) Note: depth-limited_search returns cutoff when it has reached the given depth without finding a solution 8
9 Evaluation of DS Evaluation of DS ompleteness: Time complexity: O(b d ) ompleteness: Time complexity: O(b d ) Space complexity: O(bd) q Same as DFS Evaluation of DS ompleteness: Time complexity: O(b d ) Space complexity: O(bd) q Same as DFS Optimality: if step cost is 1. terative Deepening Search nalogous to FS: explores a complete layer of s before proceeding to the next one. ombines benefits of DFS and FS. idirectional Search idirectional Search Two simultaneous searches from start and goal. q 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). Need to keep at least one of the search trees in memory Time complexity: O(b d/2 ). Space complexity: same. omplete and optimal (for uniform step costs) if both searches are FS ssues in applying: The predecessor of each should be efficiently computable. q When actions are easily reversible. Goal : sometimes not unique or known explicitly (e.g. in chess). Memory consumption 9
10 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* 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 10
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