Search

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Search
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Introduction

• In the past, most problems you encountered were
  such that the way to solve them was very clear
  and straight forward
• Solving quadratic equation.
• Solving linear equation.
• Matching input to a regular expression.
• Most of the problems in AI are of different kind.
• NP-Hard problems.
• Require searching in a large solution space.

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Searching Problems

• A Search Problem is defined as
• A set S of possible states the search space.
• Initial State sI in S.
• Operators - which take us from one state to
  another (S?S functions)
• Goal Conditions.
• The Search Space defined by a Search Problem is a
  graph G(V,E) where
• Vv v in S
• Eltv,ugt exists o in Operators such that
  o(v)u
• The goal of search is to find a path in the
  search space from sI to a state s where the Goal
  Conditions are valid.
• Sometimes we only care of s, and not the path
• Sometimes we only care whether s exists.
• Sometimes we value is associated with states, and
  we want the best s.
• Sometimes cost is associated with operators, and
  we want the cheapest path

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Search Problem Example

• Finding a solution for Rubik's Cube.
• SI Some state of the cube.
• Operators 90o rotation of any of the 9 plates.
• Goal Conditions Same color for each of the
  cube's sides.
• The Search Space will include
• A node for any possible state of the cube.
• An edge between two nodes if you can reach from
  one to the other by a 90o rotation of some plate.

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Another Example
Initial State
Goal State

• States Locations of Tiles.
• Operators Move blank right/left/down/up.
• Goal Conditions As in the picture.
• Note optimal solution of n-Puzzle family
  isNP-hard

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Solving Search Problems

• Apply the Operators on States in order to expand
  (produce) new States, utill we find one that
  satisfies the Goal Conditions.
• At any moment we may have different options to
  proceed (multiple Operators may apply)
• Q In what order should we expand the States?
• Our choice affects
• Computation time
• Space used
• Whether we reach an optimal solution?
• Whether we are guaranteed to find a solution if
  one exists (Completeness)

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Search Tree

• You can think of the search as spanning a tree
• Root The initial state.
• Children of node v are all states reachable from
  v by applying an operator.
• We can reach the same state via different paths.
• In such case, we ignore this duplicated state,
  and treat it as a leaf node
• Important parameters affecting performance
• b Average branching factor.
• d Depth of the closest solution.
• m Maximal depth of the tree.
• Fringe set of current leaf (unexpanded) nodes

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Search Methods

• Different search methods differ in the order in
  which they visit/expand the nodes.
• It is important to distinguish between
• The search space.
• The order by which we scan that space.
• Blind Search
• No additional information about search states is
  used.
• Informed Search
• Additional information used to improve efficiency
  search efficiency

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Relation to Planning

• Simplest way to solve planning problem is to
  search from the initial state to a goal
• Search states are states of the world
• Operators are possible actions
• But there are other formulations (for instance,
  search states plans)
• Search is a very general technique very
  important for many applications

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Blind Search
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Blind Search

• Main algorithms
• DFS Depth First Search
• Expand the deepest unexpanded node.
• Fringe is a LIFO.
• BFS Breath First Search
• Expand the shallowest unexpanded node.
• Fringe is a FIFO.
• IDS Iterative Deepening Search
• Combine the advantages of both methods.
• Avoid the disadvantages of each method.
• There are other variants

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Breadth First Search
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Properties of BFS

• Complete?Yes (if b is finite).
• Time?1bb2b3b(bd-1)O(bd1(
• Space? O(bd( (keeps every node on the fringe).
• Optimal?Yes (if cost is 1 per step).
• Space is a BIG problem

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Depth First Search
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Properties of DFS

• Complete?
• No given infinite branches (e.g., when we have
  loops)
• Easy to modify check for repeat states along
  path
• Complete in finite spaces!
• May require exponential space in that case
• Time?
• Worst case O(bm(.
• Terrible if m is much larger then d.
• But if solutions occur often or in the left
  part of the tree, may be much faster then BFS.
• Space?
• O(m( Linear Space!
• Optimal?
• No.

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Depth Limited Search

• DFS with a depth limit l, i.e., nodes at depth l
  have no successors.

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Iterative Deepening Search

• Increase the depth limit of the DLS with each
  Iteration

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Iterative Deepening Search
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Complexity of IDS

• Number of nodes generated in a depth-limited
  search to depth d with branching factor b
• NDLS b0 b1 b2 bd-2 bd-1 bd
• Number of nodes generated in an iterative
  deepening search to depth d with branching factor
  b
• NIDS (d1)b0 db1 (d-1)b2 3bd-2 2bd-1
  1bd
• For b 10, d 5
• NDLS 1 10 100 1,000 10,000 100,000
  111,111
  
• NIDS 6 50 400 3,000 20,000 100,000
  123,456
  
• Overhead (123,456 - 111,111)/111,111 11

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Properties of IDS

• Complete?Yes.
• Time?(d1)b0 db1 (d-1)b2 bd O(bd)
• Space?O(d)
• Optimal?Yes. If step cost is 1.

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Comparison of the Algorithms
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Bidirectional Search

• Assumptions
• There is a small number of states satisfying the
  goal conditions.
• It is possible to reverse the operators.
• Simultaneously run BFS from both directions.
• Stop when reaching a state from both directions.
• Under the assumption that validating this overlap
  takes constant time, we get
• Time O(bd/2)
• Space O(bd/2)

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Informed Search
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Informed/Heuristic Search

• Heuristic Function hS?R
• For every state s, h(s) is an estimation of the
  minimal distance/cost from s to a solution.
• Distance is only one way to set a price.
• How to produce h? later on
• Cost Function gS?R
• For every state s, g(s) is the minimal cost to s
  from the initial state.
• fgh, is an estimation of the cost from the
  initial state to a solution.

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Best First Search

• Greedy on h values.
• Fringe stored in a queue ordered by h values.
• In every step, expand the best node so far,
  i.e., the one with the best h value.

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Romania with step cost in KMs

• How to reach from Arad to Bucharest?

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Solution by Best First
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Properties of Best First

• Complete?
• No, can get into loops
• Time?
• O(bm). But a good heuristic can give dramatic
  improvement.
• Space?
• O(bm). Keeps all nodes in memory.
• Optimal
• No.

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A Search

• Idea Avoid expanding paths that are already
  expensive.
• Greedy on f values.
• Fringe is stored in a queue ordered by f values.
• Recall, f(n)g(n)h(n), where
• g(n) cost so far to reach n.
• h(n) estimated cost from n to goal.
• f(n) estimated total cost of path through n to
  goal.

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Solution by A
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Solution by A
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Admissible Heuristics

• A heuristic h(n) is admissible if for every node
  n, h(n) h(n), where h(n) is the real cost to
  reach the goal state from n.
  
• An admissible heuristic never overestimates the
  cost to reach the goal, i.e., it is optimistic
  
• Example hSLD(n). As it never overestimates the
  actual road distance.
  
• TheoremIf h(n) is admissible, A using
  Tree-Search is optimal.

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Optimality of A (Proof)

• Suppose some suboptimal goal G2 has been
  generated and is in the fringe. Let n be an
  unexpanded node in the fringe such that n is on a
  shortest path to an optimal goal G.

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Optimality of A (Proof)

• f(G2) g(G2) since h(G2) 0.
• f(G) g(G) since h(G) 0.
• g(G2) gt g(G) since G2 is suboptimal.
• f(G2) gt f(G) by 1,2,3.
• h(n) h(n) since h is admissible.
• g(n)h(n) g(n)h(n) by 5.
• f(n) f(G) by definition of f.
• f(G2) gt f(n) by 4,7.
• Hence A will never select G2 for expansion.

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Consistent Heuristics

• A heuristic is consistent (monotonic) if for
  every node n, every successor n' of n generated
  by any action a,h(n) c(n,a,n') h(n')
  
• If h is consistent, we have f(n') g(n') h(n'
  )
• g(n) c(n,a,n') h(n')
• g(n) h(n)
• f(n)
• i.e., f(n) is non-decreasing along any path.
• TheoremIf h(n) is consistent, A using
  Graph-Search is optimal.

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Optimality of A

• A expands nodes in order of increasing f value.
• Gradually adds "f-contours" of nodes .
• Contour i has allnodes with ffi, where fi lt
  fi1.

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Properties of A

• Complete?Yes, unless there are infinitely many
  nodes with f f(G).
• Time?Exponential.
• Space?Keeps all nodes in memory.
• Optimal?Yes.

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Iterative Deepening A (IDA)

• Use Iterative Deepening with A in order to
  overcome the space problem.
• A simple idea
• Define an upper bound U for f values.
• If f(n)gtU, n is not enqueued.
• If no solution was found for some U, increase it.

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Branch Bound

• Used when searching for optimal node (i.e., in
  problems where there are multiple possible
  solutions, but some are better than others)
• Attempts to prune entire sub-trees
• Applicable in the context of different search
  methods (BFS, DFS, ).
• Assume an upper and lower bound U on every node n
• They bound the value of the best descendent of n
• If for some node n there exists a node n such
  that U(n)ltI(n), we can prune n.

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Local search algorithms

• In many problems, the path to the goal is
  irrelevant the goal state itself is all we care
  about.
  
• Exampe n-queens problems
• More generally constraint-satisfaction problems
• Local search algorithms keep a single "current"
  state, and try to improve it.

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Hill-climbing search

• Like climbing mount Everest in thick fog with
  amnesia
• Improve while you can, i.e. stop when reaching a
  maxima (minima).

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Hill-climbing search

• Problem depending on initial state, can get
  stuck in local maxima.

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N-Queens Problem

• Situate n queens on a nn board such that no two
  queens threat one another. i.e., no two queens
  share row, column or a diagonal.

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Solution by Hill-climbing
A

• h number of pairs of queens that are attacking
  each other, either directly or indirectly.
• Moves change position of a single queen
• On A, h17.
• On B, h1. local minimum.

B
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Simulated annealing search

• Idea escape local maxima by allowing some "bad"
  moves, but gradually decrease their frequency.

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Properties of SAS

• One can prove If T decreases slowly enough, then
  SAS will find a global optimum with probability
  approaching 1.
• Widely used in VLSI layout, airline scheduling,
  etc.

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Local beam search

• Keep track of k states rather than just one.
• Start with k randomly generated states.
• At each iteration, all the successors of all k
  states are generated.
  
• If any one is a goal state, stop else select the
  k best successors from the complete list and
  repeat.