Solving problems by searching

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Solving problems by searching

• Chapter 3, part 2
• Modified by Vali Derhami

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Implementation general tree search

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  ???????? ?????.
• ????? ????? ?? ??? ???? ??? ???????.
• ??? ??? ??????? ? ??? ?? ??

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Implementation general tree search

• The operations on a queue are as follows
• MAKE-QUEUE(elemen,...) creates a queue with the
  given element(s).
• EMPTY?(queue) returns true only if there are no
  more elements in the queue.
• FlRST(queue) returns the first element of the
  queue.
• REMOVE-FiRST (queue) returns First(queue) and
  removes it from the queue.
• lNSERT(element, queue) inserts an element into
  the queue and returns the resulting queue. (We
  will see that different types of queues insert
  elements in different orders.)
• Insert- ALL(elements, queue) inserts a set of
  elements into the queue and returns the resulting
  queue.

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Implementation general tree search
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Search strategies

• A search strategy is defined by picking the order
  of node expansion
• Strategies are evaluated along the following
  dimensions
• completeness does it always find a solution if
  one exists?
• time complexity How long does it take to find
  solution (number of nodes generated)
• space complexity How much memory is needed to
  perform the search? maximum number of nodes in
  memory
• optimality does it always find a optimal
  solution?
  
• 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 8)
• ?????? ???? ???? ?? ???? ????

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Uninformed search strategies

• Uninformed search strategies use only the
  information available in the problem definition
  Breadth-first search Uniform-cost search Depth-fir
  st search Depth-limited search Iterative deepening
  search

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Breadth-first search?????? ??? ???

• Expand shallowest unexpanded node
• ????? ???? ???? ???? ??? ??? ????? ??????
• Implementation
• fringe is a FIFO queue, i.e., new successors go
  at end
  

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Breadth-first search?????? ??? ???

• Expand shallowest unexpanded node
• ????? ???? ???? ???? ??? ??? ????? ??????
• Implementation
• fringe is a FIFO queue, i.e., new successors go
  at end
  

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Breadth-first search?????? ??? ???

• Expand shallowest unexpanded node
• ????? ???? ???? ???? ??? ??? ????? ??????
• Implementation
• fringe is a FIFO queue, i.e., new successors go
  at end
  

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Breadth-first search?????? ??? ???

• Expand shallowest unexpanded node
• ????? ???? ???? ???? ??? ??? ????? ??????
• Implementation
• fringe is a FIFO queue, i.e., new successors go
  at end
  

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Properties of breadth-first search

• Complete? Yes (if b is finite)
• Time?
• ????? ??? ??? ????? ???bb2b3 bd b(bd-1)
  O(bd1)
  
• Space? O(bd1) (keeps every node in memory)
  Optimal? Yes (if cost 1 per step)
  
• Space is the bigger problem (more than time)
  ????? ?? ?????? ??????? ????? ?? ?? ???? ?????
  ??? ???? ???? ?????? ????????? ???? ?? ??????

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Uniform-cost search?????? ????? ???????

• Expand least-cost unexpanded nodeImplementation
• fringe queue ordered by path cost
• Equivalent to breadth-first if step costs all
  equal Complete? Yes, if step cost e Time? of n
  odes with g cost of optimal solution,
  O(b1(C/ e)) where C is the cost of the
  optimal solution
• O(b1(C/ e)) can be much greater than O(bd)
• Space? of nodes with g cost of optimal
  solution,
• O(b (C/ e))Optimal? Yes if step cost e

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Depth-first search?????? ??? ???

• Expand deepest unexpanded node ????? ???? ???? ???
  ????? ??????
• Implementation
• fringe LIFO queue, i.e., put successors at
  front
  

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Depth-first search

• Expand deepest unexpanded nodeImplementation
• fringe LIFO queue, i.e., put successors at
  front
  

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Depth-first search

• Expand deepest unexpanded nodeImplementation
• fringe LIFO queue, i.e., put successors at
  front
  

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Depth-first search

• Expand deepest unexpanded nodeImplementation
• fringe LIFO queue, i.e., put successors at
  front
  

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Depth-first search

• Expand deepest unexpanded nodeImplementation
• fringe LIFO queue, i.e., put successors at
  front
  

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Depth-first search

• Expand deepest unexpanded nodeImplementation
• fringe LIFO queue, i.e., put successors at
  front
  

??? ?? ??? ???? ??????? ??? ?? ???.
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Depth-first search

• Expand deepest unexpanded nodeImplementation
• fringe LIFO queue, i.e., put successors at
  front
  

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Depth-first search

• Expand deepest unexpanded nodeImplementation
• fringe LIFO queue, i.e., put successors at
  front
  

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Depth-first search

• Expand deepest unexpanded nodeImplementation
• fringe LIFO queue, i.e., put successors at
  front
  

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Depth-first search

• Expand deepest unexpanded nodeImplementation
• fringe LIFO queue, i.e., put successors at
  front
  

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Depth-first search

• Expand deepest unexpanded nodeImplementation
• fringe LIFO queue, i.e., put successors at
  front
  

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Depth-first search

• Expand deepest unexpanded nodeImplementation
• fringe LIFO queue, i.e., put successors at
  front
  

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Properties of depth-first search

• Complete? No fails in infinite-depth spaces,
  spaces with loops
• Modify to avoid repeated states along path
• ? complete in finite spaces
• Time? O(bm) terrible if m is much larger than d
• m?????? ??? ?? ???
• but if solutions are dense, may be much faster
  than breadth-first
• Space? bm1, O(bm), i.e., linear space!
  Optimal? No
  

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Depth-first search

• Back tracking search A type of Depth- firt
  search
• Back Traking only one successor is generated at
  a time rather than all successors
• Space O(m)

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Depth-limited search

• depth-first search with depth limit l,i.e.,
  nodes at depth l
• ??? dgtl ????? ???? ???? ??? ??? m ????? ????
• ??? dltl ?????? ??? ??? ????? ???.
• Time? O(b l ),
• Space? bl1, O(bl),
• L8 gt Depth firth

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Depth-limited search

• Recursive implementation

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Iterative deepening search?????? ???? ?????
??????
?????? ??? ??? ?? ?????? ??????? ??? ?? ?? ??????
?????? ??????? ?? ????? ?? ?? ??? ??? ??? ????.
?????? ?? ?? ??? ??? ??? ? ?????? ??? ???.
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Iterative deepening search l 0
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Iterative deepening search l 1
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Iterative deepening search l 2
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Iterative deepening search l 3
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Iterative deepening search

• Number of nodes generated in a depth-limited
  search to depth d with branching factor b
• NDLS b1 b2 bd-2 bd-1 bd
• Number of nodes generated in an iterative
  deepening search to depth d with branching factor
  b
• NIDS d b1 (d-1)b2 3bd-2 2bd-1 1bd
• For b 10, d 5,
• NIDS 50 400 3,000 20,000 100,000
  123,450
• NBFS 10 100 1,000 10,000 100,000
  999,990 1,111,100.
• Overhead (123,456 - 111,111)/111,111 11

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Properties of iterative deepening search

• Complete? Yes
• Time? d b1 (d-1)b2 bd O(bd)
• Space? O(bd)
• Optimal? Yes, if step cost 1

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Bidirectional search?????? ?? ????

• ????? ?????? ?????? ??? ?? ???? ????? ?? ??? ???
  ? ????? ?? ??? ?? ??? ???
• ???? ???? ?? ?? ????? ?? ?? ?? ????.
• Time O(bd/2) O(bd/2) O(bd/2) ltlt O(bd)
• Space O(bd/2)
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• Complete and optimal (for uniform step costs) if
  both searches are breadth-first other
  combinations may sacrifice completeness,
  optimality, or both.

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Summary of algorithms
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Repeated states

• Failure to detect repeated states can turn a
  linear problem into an exponential one!
  

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Graph search
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????? ?????

• 1- ????? ?? ?????
• ?? ???? ??? ???? ? ????? ???? ???? Deterministic,
  fully observable
• ???? ??????? ???? (Global information)?? ???? ?
  ??? ?????? ??? ?? ?? ????.
• 2- ????? ???? ?? ?? (?????) (??? ????? ?? ??????
  ?? ???.)
• ?? ??????? ????
• ???? ????? ???? ??? ?? ??????? ??? ??? ?????? ???
  ?? ?? ????.
• 3- ????? ??????? Contingency
• ???? ???? ???? ???? ? ?? ??? ????
• ???? ??? ?????? ??? ?? ??? ????.
• ??? ??????? ???? ?? ????? ???? ???? ????? ??
  ?????? (Adversarial) ?????.
• 4- ????? ??????? exploration Problem
• ?????? ? ??????? ????????

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Example vacuum world

• Single-state, start in 5. Solution?

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Example vacuum world

• Single-state, start in 5. Solution? Right,
  Suck
  
• Sensorless, start in 1,2,3,4,5,6,7,8 e.g.,
  Right goes to 2,4,6,8 Solution?
  

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Example vacuum world

• Sensorless, start in 1,2,3,4,5,6,7,8 e.g.,
  Right goes to 2,4,6,8 Solution?
  Right,Suck,Left,Suck
• Contingency
• Nondeterministic Suck may dirty a clean carpet
• Partially observable location, dirt at current
  location.
• Percept L, Clean, i.e., start in 5 or
  7Solution?

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Example vacuum world

• Sensorless, start in 1,2,3,4,5,6,7,8 e.g.,
  Right goes to 2,4,6,8 Solution?
  Right,Suck,Left,Suck
• Contingency
• Nondeterministic Suck may dirty a clean carpet
• Partially observable location, dirt at current
  location.
• Percept L, Clean, i.e., start in 5 or
  7Solution? Right, if dirt then Suck