Search is a universal problem-solving method

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Search Chapter 3
Search is a universal problem-solving
method Search algorithms based on Problem-Space
Model The task Find a sequence of operations
that map initial state to goal state. Search
methods Breadth-first, Depth-first, A
(heuristic)
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Problem space model

• Problem space is the environment in which the
  search takes place.
• A set of states of the problem
• Operators
• Problem instance
• Problem space,
• initial state,
• goal state

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Eight Puzzle
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Vacuum world problem (state) space graph
states? actions? Initial? goal?
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The Russian Farmer Problem
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(F,W,G,C) (l, l, l, l)
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Tree versus Graph

• Most problem spaces correspond to graphs with
  more than one paths between two nodes
• Duplicate nodes increasing the size of the tree
• For simplicity they are represented as a tree
  with initial state as root
• The cost
• Duplicate nodes
• Increased tree size
• Benefit
• Absence of cycles simplifying most search
  algorithms

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What differenciates AI search algorithms from
other graph based search algorithms is the size
of the graph. The Entire Chess graph has over 1040

• State space of AI problems never represented
  explicitly by listing each state.
• Size not expressed in terms of number of nodes
• Instead
• Branching factor (b)- average number of children
• Solution depth (d)- length of shortest path from
  initial to goal
• Or shortest
  sequence of operators

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Brute-Force Search Uninformed search strategies
use only the information available in the problem
definition

• State description
• Set of legal operators
• Initial state
• Description of goal state

• Important brute force techniques
• Breadth first
• Uniform-cost
• Depth-first
• Interactive deepening
• Bidirectional

Terminology Generate node Create a DS
corresponding to node Expand node Generate all
children of the node
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Breadth First

• Expand nodes in order of their distance from root
  one level at a time until solution found

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Implementation

• Queue (FIFO) of nodes initially containing the
  root
• Remove the node at the head of the queue and
  expand
• Add children to the tail of the queue

Space bound and exhaust memory in a few minutes.
Always finds the shortest path to a goal Time
spent Proportional to the number of nodes
generated(constant and f of b d) Number of
nodes at level d is bd Total number of nodes
generated in worst case bb2 b3 bd which
is O (bd)
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Uniform-Cost Search

• Edges have different costs
• Instead of expanding nodes in order of their
  depth from root expand in order of cost from root
• At each step, expand a node whose cost g(n) is
  lowest where g(n) is sum of costs of edges from
  root to node n
• Nodes are stored in a priority queue
• Worst case time complexity
• O (bc/m) where c is the cost of an optimal
  solution and m is the minimum edge cost

Problem Memory- similar to breadth first
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Depth-First Search

• Remedies space limitation of breadth first by
  always generating a child of the deepest
  unexpanded node.

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Implementation

• Maintain a list of unexpanded nodes treated as a
  LIFO stack (as opposed to FIFO for breadth first
• Usually implemented recrsively- with the
  recursion stack taking the place of an explicit
  node stack
• Time complexity is O (bd)-Practically DF is time
  limited and BF is space limited
• Disadvantage does not terminate if tree infinite
  or cycles in graph
• Solution cutoff depth (d)
• d small?
• d large?

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Depth First Interactive Deepening
Combines best features of DF and BF
Performs a depth first to depth 1 then starts
over executing a complete depth first to depth 2
and continues to run depth first searches to
successive greater depths until a solution is
found.
Optimal in terms of time and space among all
brute force algorithms on a tree.