Blind Search Russell and Norvig: Chapter 3, Sections 3.4 – 3.6 CS121 – Winter 2003

Simple Agent Algorithm Problem-Solving-Agent 1. initial-state  sense/read state 2. goal  select/read goal 3. successor  select/read action models 4. problem  (initial-state, goal, successor) 5. solution  search(problem) 6. perform(solution)

Search of State Space  search tree

Basic Search Concepts Search tree Search node Node expansion Search strategy: At each stage it determines which node to expand

Search Nodes  States The search tree may be infinite even when the state space is finite The search tree may be infinite even when the state space is finite

Node Data Structure STATE PARENT ACTION COST DEPTH If a state is too large, it may be preferable to only represent the initial state and (re-)generate the other states when needed If a state is too large, it may be preferable to only represent the initial state and (re-)generate the other states when needed

Fringe Set of search nodes that have not been expanded yet Implemented as a queue FRINGE INSERT(node,FRINGE) REMOVE(FRINGE) The ordering of the nodes in FRINGE defines the search strategy

Search Algorithm 1. If GOAL?(initial-state) then return initial-state 2. INSERT(initial-node,FRINGE) 3. Repeat: If FRINGE is empty then return failure n  REMOVE(FRINGE) s  STATE(n) For every state s’ in SUCCESSORS(s)  Create a node n’ as a successor of n  If GOAL?(s’) then return path or goal state  INSERT(n’,FRINGE)

Performance Measures Completeness Is the algorithm guaranteed to find a solution when there is one? Probabilistic completeness: If there is a solution, the probability that the algorithms finds one goes to 1 “quickly” with the running time

Performance Measures Completeness Is the algorithm guaranteed to find a solution when there is one? Optimality Is this solution optimal? Time complexity How long does it take? Space complexity How much memory does it require?

Important Parameters Maximum number of successors of any state  branching factor b of the search tree Minimal length of a path in the state space between the initial and a goal state  depth d of the shallowest goal node in the search tree

Blind vs. Heuristic Strategies Blind (or un-informed) strategies do not exploit any of the information contained in a state Heuristic (or informed) strategies exploits such information to assess that one node is “more promising” than another

Example: 8-puzzle Goal state N1 N2 STATE For a blind strategy, N1 and N2 are just two nodes (at some depth in the search tree) For a heuristic strategy counting the number of misplaced tiles, N2 is more promising than N1

Important Remark Some problems formulated as search problems are NP-hard problems (e.g., (n 2 -1)-puzzle We cannot expect to solve such a problem in less than exponential time in the worst-case But we can nevertheless strive to solve as many instances of the problem as possible

Blind Strategies Breadth-first Bidirectional Depth-first Depth-limited Iterative deepening Uniform-Cost Step cost = 1 Step cost = c(action)   > 0

Breadth-First Strategy New nodes are inserted at the end of FRINGE FRINGE = (1)

Breadth-First Strategy New nodes are inserted at the end of FRINGE FRINGE = (2, 3)

Breadth-First Strategy New nodes are inserted at the end of FRINGE FRINGE = (3, 4, 5)

Breadth-First Strategy New nodes are inserted at the end of FRINGE FRINGE = (4, 5, 6, 7)

Evaluation b: branching factor d: depth of shallowest goal node Complete Optimal if step cost is 1 Number of nodes generated: 1 + b + b 2 + … + b d = (b d+1 -1)/(b-1) = O(b d ) Time and space complexity is O(b d )

Big O Notation g(n) is in O(f(n)) if there exist two positive constants a and N such that: for all n > N, g(n)  a  f(n)

Time and Memory Requirements d#NodesTimeMemory msec11 Kbytes 411,1111 msec1 Mbyte 6~ sec100 Mb 8~ sec10 Gbytes 10~ hours1 Tbyte 12~ days100 Tbytes 14~ years10,000 Tb Assumptions: b = 10; 1,000,000 nodes/sec; 100bytes/node

Time and Memory Requirements d#NodesTimeMemory msec11 Kbytes 411,1111 msec1 Mbyte 6~ sec100 Mb 8~ sec10 Gbytes 10~ hours1 Tbyte 12~ days100 Tbytes 14~ years10,000 Tb Assumptions: b = 10; 1,000,000 nodes/sec; 100bytes/node

Bidirectional Strategy 2 fringe queues: FRINGE1 and FRINGE2 Time and space complexity = O(b d/2 ) << O(b d )

Depth-First Strategy New nodes are inserted at the front of FRINGE FRINGE = (1)

Depth-First Strategy New nodes are inserted at the front of FRINGE FRINGE = (2, 3)

Depth-First Strategy New nodes are inserted at the front of FRINGE FRINGE = (4, 5, 3)

Depth-First Strategy New nodes are inserted at the front of FRINGE

Depth-First Strategy New nodes are inserted at the front of FRINGE

Depth-First Strategy New nodes are inserted at the front of FRINGE

Depth-First Strategy New nodes are inserted at the front of FRINGE

Depth-First Strategy New nodes are inserted at the front of FRINGE

Depth-First Strategy New nodes are inserted at the front of FRINGE

Depth-First Strategy New nodes are inserted at the front of FRINGE

Depth-First Strategy New nodes are inserted at the front of FRINGE

Evaluation b: branching factor d: depth of shallowest goal node m: maximal depth of a leaf node Complete only for finite search tree Not optimal Number of nodes generated: 1 + b + b 2 + … + b m = O(b m ) Time complexity is O(b m ) Space complexity is O(bm) or O(m)

Depth-Limited Strategy Depth-first with depth cutoff k (maximal depth below which nodes are not expanded) Three possible outcomes: Solution Failure (no solution) Cutoff (no solution within cutoff)

Iterative Deepening Strategy Repeat for k = 0, 1, 2, …: Perform depth-first with depth cutoff k Complete Optimal if step cost =1 Time complexity is: (d+1)(1) + db + (d-1)b 2 + … + (1) b d = O(b d ) Space complexity is: O(bd) or O(d)

Calculation db + (d-1)b 2 + … + (1) b d = b d + 2b d-1 + 3b d-2 +… + db = b d (1 + 2b b -2 + … + db -d )  b d (  i=1,…,   ib (1-i) ) = b d ( b/(b-1) ) 2

Comparison of Strategies Breadth-first is complete and optimal, but has high space complexity Depth-first is space efficient, but neither complete nor optimal Iterative deepening is asymptotically optimal

Repeated States 8-queens No assembly planning Few puzzle and robot navigation Many search tree is finitesearch tree is infinite

Avoiding Repeated States Requires comparing state descriptions Breadth-first strategy: Keep track of all generated states If the state of a new node already exists, then discard the node

Avoiding Repeated States Depth-first strategy: Solution 1:  Keep track of all states associated with nodes in current path  If the state of a new node already exists, then discard the node  Avoids loops Solution 2:  Keep track of all states generated so far  If the state of a new node has already been generated, then discard the node  Space complexity of breadth-first

Detecting Identical States Use explicit representation of state space Use hash-code or similar representation

Revisiting Complexity Assume a state space of finite size s Let r be the maximal number of states that can be attained in one step from any state In the worst-case r = s-1 Assume breadth-first search with no repeated states Time complexity is O(rs). In the worst case it is O(s 2 )

Example s = n x x n y r = 4 or 8 Time complexity is O(s)

Uniform-Cost Strategy Each step has some cost   > 0. The cost of the path to each fringe node N is g(N) =  costs of all steps. The goal is to generate a solution path of minimal cost. The queue FRINGE is sorted in increasing cost. S 0 1 A 5 B 15 C SG A B C G 11 G 10

Modified Search Algorithm 1. INSERT(initial-node,FRINGE) 2. Repeat: If FRINGE is empty then return failure n  REMOVE(FRINGE) s  STATE(n) If GOAL?(s) then return path or goal state For every state s’ in SUCCESSORS(s)  Create a node n’ as a successor of n  INSERT(n’,FRINGE)

Exercises Adapt uniform-cost search to avoid repeated states while still finding the optimal solution Uniform-cost looks like breadth-first (it is exactly breadth first if the step cost is constant). Adapt iterative deepening in a similar way to handle variable step costs

Summary Search tree  state space Search strategies: breadth-first, depth- first, and variants Evaluation of strategies: completeness, optimality, time and space complexity Avoiding repeated states Optimal search with variable step costs