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Uninformed Search ECE457 Applied Artificial Intelligence Fall 2007 Lecture #2 Outline Problem-solving by searching Uninformed search techniques Russell & Norvig, chapter 3 ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 2 Problem-solving by searching An agent needs to perform actions to get from its current state to a goal. This process is called searching. Central in many AI systems Theorem proving, VLSI layout, game playing, navigation, scheduling, etc. ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 3 Requirements for searching Define the problem Define a goal Represent the search space by states Define the actions the agent can perform and their cost What is the agent searching for? Define the solution The goal itself? The path (i.e. sequence of actions) to get to the goal? ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 4 Assumptions Goal-based agent Environment Fully observable Deterministic Sequential Static Discrete Single agent ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 5 Formulating problems A well-defined problem has: An initial state A set of actions A goal test A concept of cost ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 6 Well-Defined Problem Example Initial state Action Goal test Move blank left, right, up or down, provided it does not get out of the game Are the tiles in the “goal state” order? Cost Each move costs 1 Path cost is the sum of moves ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 7 Well-Defined Problem Example Travelling salesman problem Initial state Is the agent in the initial city after having visited every city? Concept of cost ECE457 Applied Artificial Intelligence Move to an unvisited city Goal test Any city Set of actions Find the shortest round trip to visit each city exactly once Action cost: distance between cities Path cost: total distance travelled R. Khoury (2007) Page 8 Example: 8-puzzle left left right down ECE457 Applied Artificial Intelligence down left R. Khoury (2007) up down Page 9 Search Tree ParentRoot Child Node (state) Branching factor (b) Expanding a node Edge (action) Maximum depthFringe (m) Leaf ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 10 Properties of Search Algos. Completeness Optimality Is the algorithm guaranteed to find the best goal node, i.e. the one with the cheapest path cost? Time complexity Is the algorithm guaranteed to find a goal node, if one exists? How many nodes are generated? Space complexity What’s the maximum number of nodes stored in memory? ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 11 Types of Search Uninformed Search Only has the information provided by the problem formulation (initial state, set of actions, goal test, cost) Informed Search Has additional information that allows it to judge the promise of an action, i.e. the estimated cost from a state to a goal ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 12 Breath-First Search ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 13 Breath-First Search Complete, if b is finite Optimal, if path cost is equal to depth Guaranteed to return the shallowest goal (depth d) Time complexity = O(bd+1) Space complexity = O(bd+1) ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 14 Breath-First Search Upper-bound case: goal is last node of depth d Number of generated nodes: b+b²+b³+…+bd+(bd+1-b) = O(bd+1) Space & time complexity: all generated nodes ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 15 Uniform-Cost Search Expansion of Breath-First Search Explore the cheapest node first (in terms of path cost) Condition: No zero-cost or negative-cost edges. Minimum cost is є ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 16 Uniform-Cost Search Complete given a finite tree Optimal Time complexity = O(bC*/є) ≥ O(bd+1) Space complexity = O(bC*/є) ≥ O(bd+1) ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 17 Uniform-Cost Search Upper-bound case: goal has path cost C*, all other actions have minimum cost of є Depth explored before taking action C*: C*/є Number of generated nodes: O(bC*/є) Space & time complexity: all generated nodes є C* є є є ECE457 Applied Artificial Intelligence є є є є є є R. Khoury (2007) є є є є Page 18 є Depth-First Search ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 19 Depth-First Search Complete, if m is finite Not optimal Time complexity = O(bm) Space complexity = bm+1 = O(bm) Can be reduced to O(m) with recursive algorithm ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 20 Depth-First Search Upper-bound case for space: goal is last node of first branch After that, we start deleting nodes Number of generated nodes: b nodes at each of m levels Space complexity: all generated nodes = O(bm) ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 21 Depth-First Search Upper-bound case for time: goal is last node of last branch Number of nodes generated: b nodes for each node of m levels (entire tree) Time complexity: all generated nodes O(bm) ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 22 Depth-Limited Search Depth-First Search with depth limit l Avoids problems of Depth-First Search when trees are unbounded Depth-First Search is Depth-Limited Search with l = ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 23 Depth-Limited Search Complete, if l > d Not optimal Time complexity = O(bl) Space complexity = O(bl) ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 24 Depth-Limited Search Upper-bound case for space: goal is last node of first branch After that, we start deleting nodes Number of generated nodes: b nodes at each of l levels Space complexity: all generated nodes = O(bl) ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 25 Depth-Limited Search Upper-bound case for time: goal is last node of last branch Number of nodes generated: b nodes for each node of l levels (entire tree to depth l) Time complexity: all generated nodes O(bl) ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 26 Iterative Deepening Search Depth-First Search with increasing depth limit l Repeat depth-limited search over and over, with l = l + 1 Avoids problems of Depth-First Search when trees are unbounded Avoids problem of Depth-Limited Search when goal depth d > l ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 27 Iterative Deepening Search Complete , if b is finite Optimal, if path cost is equal to depth Guaranteed to return the shallowest goal Time complexity = O(bd) Space complexity = O(bd) Nodes on levels above d are generated multiple times ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 28 Iterative Deepening Search Upper-bound case for space: goal is last node of first branch After that, we start deleting nodes Number of generated nodes: b nodes at each of d levels Space complexity: all generated nodes = O(bd) ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 29 Iterative Deepening Search Upper-bound case for time: goal is last node of last branch Number of nodes generated: b nodes for each node of d levels (entire tree to depth d) Time complexity: all generated nodes O(bd) ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 30 Depth Searches Depth-first Depth-limited Iterative search search deepening search Depth limit m l d Time complexity O(bm) O(bl) O(bd) Space complexity O(bm) O(bl) O(bd) ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 31 Summary of Searches Breath- Uniform Depth Depth- Iterative first Cost -first limited deepening Complete Yes1 Yes1 No4 No5 Yes1 Optimal Yes2 Yes3 No No Yes2 Time O(bd+1) O(bC*/є) O(bm) O(bl) O(bd) Space O(bd+1) O(bC*/є) O(bm) O(bl) O(bd) 1: Assuming b finite (common in trees) 2: Assuming equal action costs 3: Assuming all costs є ECE457 Applied Artificial Intelligence 4: Unless m finite (uncommon in trees) 5: Unless l precisely selected R. Khoury (2007) Page 32 Summary / Example Going from Arad to Bucharest ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 33 Summary / Example Initial state Action Move to a neighbouring city, if a road exists. Goal test Being in Arad Are we in Bucharest? Cost Move cost = distance between cities Path cost = distance travelled since Arad ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 34 Summary / Example Breath-First Search ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 35 Summary / Example Uniform-Cost Search ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 36 Summary / Example Depth-First Search ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 37 Summary / Example Depth-Limited Search, l = 4 ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 38 Summary / Example Iterative Deepening Search ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 39 Repeated States Example: 8-puzzle left left right down ECE457 Applied Artificial Intelligence down left R. Khoury (2007) up down Page 40 Repeated States Unavoidable in problems where Actions are reversible Multiple paths to the same state are possible Can greatly increase the number of nodes in a tree Or even make a finite tree infinite! ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 41 Repeated States A A B B B C C C C C D E D D D D D D D D EEEEEEEE EEEEEEEE ECE457 Applied Artificial Intelligence Each state generates a single child twice 26 different states 225 leaves (i.e. state Z) Over 67M nodes in the tree R. Khoury (2007) Page 42 Repeated States Maintain a closed list of visited states Closed list (for expanded nodes) vs. open list (for fringe nodes) Detect and discard repeated states upon generation Increases space complexity ECE457 Applied Artificial Intelligence R. Khoury (2007) Page 43