Blind Search

1
Blind Search

• Russell and Norvig Chapter 3, Sections 3.4
  3.6

2
?? ????? ????

• ???? ????? ??? ??? ??, ??? ???
• ?? ?? ??? kn?? ???
• ???? Polynomial complexity
• ?? ????? ?? ? ???, ??? ?? ??
• ????
• ???? ??? ? ???, ?????? ?? polynomial?. ??? ?? ??
  ??? ?? ? ??. ?? ???? ????? ??? ?? ??
• ???? ?? NP ???
• ??? ?(feasible solution)? ??. ????? ??? ?? ? ? ??.

3
Simple Agent Algorithm

• Problem-Solving-Agent
• initial-state ? sense/read state
• goal ? select/read goal
• ? goal formulation and test a function for goal
  testing
• successor ? select/read action models
• ? successor function to calculate next states
• problem ? (initial-state, goal, successor)
• solution ? search(problem)
• perform(solution)

4
Search of State Space
? search tree
5
Basic Search Concepts

• Search tree
• Search node
• Node expansion
• Search strategy At each stage it determines
  which node to expand

6
Search Nodes ? States
The search tree may be infinite even when the
state space is finite
7
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
8
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

9
Search Algorithm

• If GOAL?(initial-state) then return initial-state
• INSERT(initial-node,FRINGE)
• 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)

10
Performance Measures

• CompletenessIs 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
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Performance Measures

• CompletenessIs the algorithm guaranteed to find
  a solution when there is one?
• OptimalityIs this solution optimal?
• Time complexityHow long does it take?
• Space complexityHow much memory does it require?

12
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

13
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

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Example 8-puzzle
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
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Important Remark

• Some problems formulated as search problems are
  NP-hard problems (e.g., (n2-1)-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

16
Blind Strategies

• Breadth-first
• Bidirectional
• Depth-first
• Depth-limited
• Iterative deepening
• Uniform-Cost

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Breadth-First Strategy

• New nodes are inserted at the end of FRINGE

FRINGE (1)
18
Breadth-First Strategy

• New nodes are inserted at the end of FRINGE

FRINGE (2, 3)
19
Breadth-First Strategy

• New nodes are inserted at the end of FRINGE

FRINGE (3, 4, 5)
20
Breadth-First Strategy

• New nodes are inserted at the end of FRINGE

FRINGE (4, 5, 6, 7)
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Evaluation

• b branching factor
• d depth of shallowest goal node
• Complete
• Optimal if step cost is 1
• Number of nodes generated 1 b b2 bd
  (bd1-1)/(b-1)
  O(bd)
• Time and space complexity is O(bd)

22
Big O Notation

• g(n) is in O(f(n)) if there exist two positive
  constants a and N such that
• for all n gt N, g(n) ? a?f(n)

23
Time and Memory Requirements
d Nodes Time Memory
2 111 .01 msec 11 Kbytes
4 11,111 1 msec 1 Mbyte
6 106 1 sec 100 Mb
8 108 100 sec 10 Gbytes
10 1010 2.8 hours 1 Tbyte
12 1012 11.6 days 100 Tbytes
14 1014 3.2 years 10,000 Tb
Assumptions b 10 1,000,000 nodes/sec
100bytes/node
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Time and Memory Requirements
d Nodes Time Memory
2 111 .01 msec 11 Kbytes
4 11,111 1 msec 1 Mbyte
6 106 1 sec 100 Mb
8 108 100 sec 10 Gbytes
10 1010 2.8 hours 1 Tbyte
12 1012 11.6 days 100 Tbytes
14 1014 3.2 years 10,000 Tb
Assumptions b 10 1,000,000 nodes/sec
100bytes/node
25
Bidirectional Strategy
2 fringe queues FRINGE1 and FRINGE2
Time and space complexity O(bd/2) ltlt O(bd)
26
Depth-First Strategy

• New nodes are inserted at the front of FRINGE

1
27
Depth-First Strategy

• New nodes are inserted at the front of FRINGE

1
28
Depth-First Strategy

• New nodes are inserted at the front of FRINGE

1
29
Depth-First Strategy

• New nodes are inserted at the front of FRINGE

1
30
Depth-First Strategy

• New nodes are inserted at the front of FRINGE

1
31
Depth-First Strategy

• New nodes are inserted at the front of FRINGE

1
32
Depth-First Strategy

• New nodes are inserted at the front of FRINGE

1
33
Depth-First Strategy

• New nodes are inserted at the front of FRINGE

1
34
Depth-First Strategy

• New nodes are inserted at the front of FRINGE

1
35
Depth-First Strategy

• New nodes are inserted at the front of FRINGE

1
36
Depth-First Strategy

• New nodes are inserted at the front of FRINGE

1
37
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 b2 bm
  O(bm)
• Time complexity is O(bm)
• Space complexity is O(bm) or O(m)

38
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)

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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 (d1)(1) db (d-1)b2
  (1) bd O(bd)
• Space complexity is O(bd) or O(d)

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Calculation

• db (d-1)b2 (1) bd
• bd 2bd-1 3bd-2 db
• bd(1 2b-1 3b-2 db-d)
• ? bd(Si1,,? ib(1-i))
• bd (b/(b-1))2

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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

42
Repeated States
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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
• The cost to find the repeated state (space and
  time)

44
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

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Detecting Identical States

• Use explicit representation of state space
• Use hash-code or similar representation

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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(s2)

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Example

• s nx x ny
• r 4 or 8
• Time complexity is O(s)

48
Uniform-Cost Strategy

• Each step has some cost ? ? gt 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.

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Modified Search Algorithm

• INSERT(initial-node,FRINGE)
• 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)

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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

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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