Searching MIU

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SearchingMIU

• Graham Kendall

Blind Searching
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Problem Definition - 1

• Initial State
• The initial state of the problem, defined in some
  suitable manner
• Operator
• A set of actions that moves the problem from one
  state to another

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Problem Definition - 1

• Neighbourhood (Successor Function)
• The set of all possible states reachable from a
  given state
• State Space
• The set of all states reachable from the initial
  state

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Problem Definition - 2

• Goal Test
• A test applied to a state which returns if we
  have reached a state that solves the problem
• Path Cost
• How much it costs to take a particular path

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Problem Definition - Example
Initial State
Goal State
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Problem Definition - Example

• States
• A description of each of the eight tiles in each
  location that it can occupy. It is also useful to
  include the blank
• Operators
• The blank moves left, right, up or down

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Problem Definition - Example

• Goal Test
• The current state matches a certain state (e.g.
  one of the ones shown on previous slide)
• Path Cost
• Each move of the blank costs 1

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Problem Definition - Datatype

• Datatype PROBLEM
• Components
• INITIAL-STATE,
• OPERATORS,
• GOAL-TEST,
• PATH-COST-FUNCTION

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How Good is a Solution?

• Does our search method actually find a solution?
• Is it a good solution?
• Path Cost
• Search Cost (Time and Memory)
• Does it find the optimal solution?
• But what is optimal?

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Evaluating a Search

• Completeness
• Is the strategy guaranteed to find a solution?
• Time Complexity
• How long does it take to find a solution?

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Evaluating a Search

• Space Complexity
• How much memory does it take to perform the
  search?
• Optimality
• Does the strategy find the optimal solution where
  there are several solutions?

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Search Trees
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Search Trees

• ISSUES
• Search trees grow very quickly
• The size of the search tree is governed by the
  branching factor
• Even this simple game has a complete search tree
  of 984,410 potential nodes
• The search tree for chess has a branching factor
  of about 35

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Implementing a Search - What we need to store

• State
• This represents the state in the state space to
  which this node corresponds
• Parent-Node
• This points to the node that generated this node.
  In a data structure representing a tree it is
  usual to call this the parent node

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Implementing a Search - What we need to store

• Operator
• The operator that was applied to generate this
  node
• Depth
• The number of nodes from the root (i.e. the
  depth)
• Path-Cost
• The path cost from the initial state to this node

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Implementing a Search - Datatype

• Datatype node
• Components
• STATE,
• PARENT-NODE,
• OPERATOR,
• DEPTH,
• PATH-COST

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Using a Tree The Obvious Solution?

• Advantages
• Its intuitive
• Parents are automatically catered for

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Using a Tree The Obvious Solution?

• But
• It can be wasteful on space
• It can be difficult the implement, particularly
  if there are varying number of children (as in
  tic-tac-toe)
• It is not always obvious which node to expand
  next. We may have to search the tree looking for
  the best leaf node (sometimes called the fringe
  or frontier nodes). This can obviously be
  computationally expensive

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Using a Tree Maybe not so obvious

• Therefore
• It would be nice to have a simpler data
  structure to represent our tree
• And it would be nice if the next node to be
  expanded was an O(1) operation

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General Search

• Function GENERAL-SEARCH(problem, QUEUING-FN)
  returns a solution or failure
• nodes MAKE-QUEUE(MAKE-NODE(INITIAL-STATEproblem
  ))
• Loop do
• If nodes is empty then return failure
• node REMOVE-FRONT(nodes)
• If GOAL-TESTproblem applied to STATE(node)
  succeeds then return node
• nodes QUEUING-FN(nodes,EXPAND(node,OPERATORSpro
  blem))
• End
• End Function

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General Search

• Function GENERAL-SEARCH(problem, QUEUING-FN)
  returns a solution or failure
• nodes MAKE-QUEUE(MAKE-NODE(INITIAL-STATEproblem
  ))
• Loop do
• If nodes is empty then return failure
• node REMOVE-FRONT(nodes)
• If GOAL-TESTproblem applied to STATE(node)
  succeeds then return node
• nodes QUEUING-FN(nodes,EXPAND(node,OPERATORSpro
  blem))
• End
• End Function

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General Search

• Function GENERAL-SEARCH(problem, QUEUING-FN)
  returns a solution or failure
• nodes MAKE-QUEUE(MAKE-NODE(INITIAL-STATEproblem
  ))
• Loop do
• If nodes is empty then return failure
• node REMOVE-FRONT(nodes)
• If GOAL-TESTproblem applied to STATE(node)
  succeeds then return node
• nodes QUEUING-FN(nodes,EXPAND(node,OPERATORSpro
  blem))
• End
• End Function

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General Search

• Function GENERAL-SEARCH(problem, QUEUING-FN)
  returns a solution or failure
• nodes MAKE-QUEUE(MAKE-NODE(INITIAL-STATEproblem
  ))
• Loop do
• If nodes is empty then return failure
• node REMOVE-FRONT(nodes)
• If GOAL-TESTproblem applied to STATE(node)
  succeeds then return node
• nodes QUEUING-FN(nodes,EXPAND(node,OPERATORSpro
  blem))
• End
• End Function

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General Search

• Function GENERAL-SEARCH(problem, QUEUING-FN)
  returns a solution or failure
• nodes MAKE-QUEUE(MAKE-NODE(INITIAL-STATEproblem
  ))
• Loop do
• If nodes is empty then return failure
• node REMOVE-FRONT(nodes)
• If GOAL-TESTproblem applied to STATE(node)
  succeeds then return node
• nodes QUEUING-FN(nodes,EXPAND(node,OPERATORSpro
  blem))
• End
• End Function

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General Search

• Function GENERAL-SEARCH(problem, QUEUING-FN)
  returns a solution or failure
• nodes MAKE-QUEUE(MAKE-NODE(INITIAL-STATEproblem
  ))
• Loop do
• If nodes is empty then return failure
• node REMOVE-FRONT(nodes)
• If GOAL-TESTproblem applied to STATE(node)
  succeeds then return node
• nodes QUEUING-FN(nodes,EXPAND(node,OPERATORSpro
  blem))
• End
• End Function

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General Search

• Function GENERAL-SEARCH(problem, QUEUING-FN)
  returns a solution or failure
• nodes MAKE-QUEUE(MAKE-NODE(INITIAL-STATEproblem
  ))
• Loop do
• If nodes is empty then return failure
• node REMOVE-FRONT(nodes)
• If GOAL-TESTproblem applied to STATE(node)
  succeeds then return node
• nodes QUEUING-FN(nodes,EXPAND(node,OPERATORSpro
  blem))
• End
• End Function

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Blind Searches
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Blind Searches - Characteristics

• Simply searches the State Space
• Can only distinguish between a goal state and a
  non-goal state
• Sometimes called an uninformed search as it has
  no knowledge about its domain

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Blind Searches - Characteristics

• Blind Searches have no preference as to which
  state (node) that is expanded next
• The different types of blind searches are
  characterised by the order in which they expand
  the nodes.
• This can have a dramatic effect on how well the
  search performs when measured against the four
  criteria we defined in an earlier lecture

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Breadth First Search - Method

• Expand Root Node First
• Expand all nodes at level 1 before expanding
  level 2
• OR
• Expand all nodes at level d before expanding
  nodes at level d1

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Breadth First Search - Implementation

• Use a queueing function that adds nodes to the
  end of the queue

• Function BREADTH-FIRST-SEARCH(problem) returns a
  solution or failure
• Return GENERAL-SEARCH(problem,ENQUEUE-AT-END)

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Breadth First Search - Implementation
A
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Evaluating Breadth First Search

• Observations
• Very systematic
• If there is a solution breadth first search is
  guaranteed to find it
• If there are several solutions then breadth first
  search will always find the shallowest goal state
  first and if the cost of a solution is a
  non-decreasing function of the depth then it will
  always find the cheapest solution

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Evaluating Breadth First Search

• Evaluating against four criteria
• Complete? Yes
• Optimal? Yes
• Space Complexity 1 b b2 b3 ... bd i.e
  O(bd)
• Time Complexity 1 b b2 b3 ... bd i.e.
  O(bd)
• Where b is the branching factor and d is the
  depth of the search tree
• Note The space/time complexity could be less as
  the solution could be found anywhere on the dth
  level.

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Exponential Growth

• Exponential growth quickly makes complete state
  space searches unrealistic
• If the branch factor was 10, by level 5 we would
  need to search 100,000 nodes (i.e. 105)

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Exponential Growth
Time and memory requirements for breadth-first
search, assuming a branching factor of 10, 100
bytes per node and searching 1000 nodes/second
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Exponential Growth - Observations

• Space is more of a factor to breadth first search
  than time
• Time is still an issue. Who has 35 years to wait
  for an answer to a level 12 problem (or even 128
  days to a level 10 problem)
• It could be argued that as technology gets faster
  then exponential growth will not be a problem.
  But even if technology is 100 times faster we
  would still have to wait 35 years for a level 14
  problem and what if we hit a level 15 problem!

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Uniform Cost Search (vs BFS)

• BFS will find the optimal (shallowest) solution
  so long as the cost is a function of the depth
• Uniform Cost Search can be used when this is not
  the case and uniform cost search will find the
  cheapest solution provided that the cost of the
  path never decreases as we proceed along the path
• Uniform Cost Search works by expanding the lowest
  cost node on the fringe.

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Uniform Cost Search - Example

• BFS will find the path SAG, with a cost of 11,
  but SBG is cheaper with a cost of 10
• Uniform Cost Search will find the cheaper
  solution (SBG). It will find SAG but will not see
  it as it is not at the head of the queue

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Depth First Search - Method

• Expand Root Node First
• Explore one branch of the tree before exploring
  another branch

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Depth First Search - Implementation

• Use a queueing function that adds nodes to the
  front of the queue

Function DEPTH-FIRST-SEARCH(problem) returns a
solution or failure Return GENERAL-SEARCH(problem
,ENQUEUE-AT-FRONT)
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Depth First Search - Observations

• Only needs to store the path from the root to the
  leaf node as well as the unexpanded nodes. For a
  state space with a branching factor of b and a
  maximum depth of m, DFS requires storage of bm
  nodes
• Time complexity for DFS is bm in the worst case

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Depth First Search - Observations

• If DFS goes down a infinite branch it will not
  terminate if it does not find a goal state.
• If it does find a solution there may be a better
  solution at a lower level in the tree. Therefore,
  depth first search is neither complete nor
  optimal.

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Depth Limited Search (vs DFS)

• DFS may never terminate as it could follow a path
  that has no solution on it
• DLS solves this by imposing a depth limit, at
  which point the search terminates that particular
  branch

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Depth Limited Search - Observations

• Can be implemented by the general search
  algorithm using operators which keep track of the
  depth
• Choice of depth parameter is important
• Too deep is wasteful of time and space
• Too shallow and we may never reach a goal state

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Depth Limited Search - Observations

• If the depth parameter, l, is set deep enough
  then we are guaranteed to find a solution if one
  exists
• Therefore it is complete if lgtd (ddepth of
  solution)
• Space requirements are O(bl)
• Time requirements are O(bl)
• DLS is not optimal

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Map of Romania
On the Romania map there are 20 towns so any town
is reachable in 19 steps
In fact, any town is reachable in 9 steps
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Iterative Deepening Search (vs DLS)

• The problem with DLS is choosing a depth
  parameter
• Setting a depth parameter to 19 is obviously
  wasteful if using DLS
• IDS overcomes this problem by trying depth limits
  of 0, 1, 2, , n. In effect it is combining BFS
  and DFS

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Iterative Deepening Search - Observations

• IDS may seem wasteful as it is expanding the same
  nodes many times. In fact, when b10 only about
  11 more nodes are expanded than for a BFS or a
  DLS down to level d
• Time Complexity O(bd)
• Space Complexity O(bd)
• For large search spaces, where the depth of the
  solution is not known, IDS is normally the
  preferred search method

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Repeated States - Three Methods

• Do not generate a node that is the same as the
  parent nodeOrDo not return to the state you
  have just come from
• Do not create paths with cycles in them. To do
  this we can check each ancestor node and refuse
  to create a state that is the same as this set of
  nodes

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Repeated States - Three Methods

• Do not generate any state that is the same as any
  state generated before. This requires that every
  state is kept in memory (meaning a potential
  space complexity of O(bd))
• The three methods are shown in increasing order
  of computational overhead in order to implement
  them

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Blind Searches Summary
B Branching factor D Depth of solution M
Maximum depth of the search tree L Depth Limit
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G5AIAIIntroduction to AI

• Graham Kendall

End of Blind Searches