Problem Solving: Searching

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Problem Solving Searching
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• Problem
• Collection of information that the agent uses to
  decide what to do.
• Needed
• an initial state
• a set of possible actions (and the states that
  they produce)
• goal test
• path cost (sum of the costs of individual
  actions)
• This is the input to the search algorithm.
• Output is the solution.
• The path from the initial state to the goal
  state.
• Search
• The process of examining different sequence of
  actions to determine which sequence will result
  in the most success.

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Example 8-puzzle

• Initial State
• An (almost) arbitrary distribution of the
  numbered tiles.
• For example
• Goal state

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• Operators
• Move the tile left of the blank right
• Move the tile right of the blank left
• Move the tile above the blank down
• Move the tile below the blank up

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Example A Water Jug Problem

• Initial Situation
• Two empty jugs, a big one and a small one
• The big one holds 3 pints
• The small one holds 1 pint
• Operators
• Fill big jug from the tap
• Fill small jug from the tap
• Fill the small jug from the big one
• Empty the small jug into the big one
• Empty the big jug into the sink
• Empty the small jug into the sink
• Goal
• Small jug is empty
• Big jug contains 2 pints.

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Search Strategies

• Different flavours of search
• Blind Search
• Heuristic Search

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Evaluation of Search Strategies

• Four Criteria
• Completeness
• If a solution exists it is guaranteed to be found
• Time Complexity
• Time taken to find the solution
• Space Complexity
• Memory required to perform search.
• Optimality
• Finding of the highest-quality solution when a
  number exist.

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Blind Search

• No information about the cost or number of steps
  to reach the goal is used to guide the search.
• Two main types of blind search
• Depth-first Search (DFS)
• Breadth-first Search (BFS)
• These strategies differ in the order in which
  nodes are expanded.
• Blind Search quickly leads to search spaces that
  are too large.

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Breadth First Search

• Expands all nodes at one level before moving to
  the next.
• Finds the shallowest goal state.
• Uses a lot of memory because the entire search
  tree must be stored.
• Space complexity is a problem
• Complete.
• Optimal if cost is directly related to depth.
• Modifications
• Uniform Cost Search

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Example Breadth-First Search

• 0 0
• 3 0 0 1
• 3 1 2 1 0 0 3 1 1 0 0 0
• 0 1 3 0 3 0 0 1

2 0
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Uniform Cost Search

• Expands the lowest cost node, rather than the
  lowest depth node.
• Solution is guaranteed to be the cheapest if the
  path cost function is nondecreasing.

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Depth First Search (DFS)

• Expands nodes at the deepest level of the search
  tree.
• Stores only one path from root to leaf.
• Space complexity not a problem
• Time could be wasted going down the wrong branch
  of the tree.
• The above is really bad if there is an infinite,
  or really big search tree.
• Neither complete nor optimal.
• Modifications
• Depth Limited Search
• Iterative Deepening Search

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Example Depth-First Search

• 0 0
• 3 0
• 3 1
• 0 1
• 3 1 1 0
• 1 1

2 0
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• Depth Limited Search
• Set a limit on the depth of the search.
• Addresses the problem of wasting time going down
  the wrong branch.
• The difficulty with this approach is choosing a
  suitable limit.
• Iterative Deepening Search
• Combines DFS and BFS
• Iterative depth limited search
• Overcomes the problem of choosing a suitable
  limit.
• Start with a very low limit and increase
  gradually until a solution is found..

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Informed Searches

• Definition
• A search that uses a heuristic function to direct
  the search towards the goal.
• Heuristic Function (h(n))
• A function that calculates the cost of reaching
  the goal state from the node n.
• Two Basic Approaches
• Greedy Search
• Minimises the estimated cost to reach the goal.
• A Search
• Minimises total path cost.

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Greedy Search

• The node closest to the goal state, as determined
  by h(n), is expanded first.
• Issues
• Finds solution quickly
• Doesnt always find the best solution, since it
  evaluates the immediate best choice, not the long
  term options.
• Can fall prey to false starts
• Expands a node that leads to a dead end.
• Resembles DFS (likes to follow a path to a
  solution or a dead end) so has similar problems.

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Example Greedy Search

• pg. 95 (Russell Norvig)

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A Search

• g(n)
• The path cost from start to node n.
• Admissible Heuristic
• One that never overestimates the cost to reach
  the goal.
• f(n)
• Estimated cost of the cheapest solution through
  n. If h is admissible