Heuristic Informed Search

1
Heuristic (Informed) Search
RN Chap. 4, Sect. 4.13
2
Iterative Deepening A (IDA)

• Idea Reduce memory requirement of A by applying
  cutoff on values of f
• Consistent heuristic function h
• Algorithm IDA
• Initialize cutoff to f(initial-node)
• Repeat
• Perform depth-first search by expanding all nodes
  N such that f(N) ? cutoff
• Reset cutoff to smallest value f of non-expanded
  (leaf) nodes

3
8-Puzzle
f(N) g(N) h(N) with h(N) number of
misplaced tiles
Cutoff4
4
8-Puzzle
f(N) g(N) h(N) with h(N) number of
misplaced tiles
Cutoff4
5
8-Puzzle
f(N) g(N) h(N) with h(N) number of
misplaced tiles
Cutoff4
6
8-Puzzle
f(N) g(N) h(N) with h(N) number of
misplaced tiles
Cutoff4
7
8-Puzzle
f(N) g(N) h(N) with h(N) number of
misplaced tiles
Cutoff4
8
8-Puzzle
f(N) g(N) h(N) with h(N) number of
misplaced tiles
Cutoff5
9
8-Puzzle
f(N) g(N) h(N) with h(N) number of
misplaced tiles
Cutoff5
10
8-Puzzle
f(N) g(N) h(N) with h(N) number of
misplaced tiles
Cutoff5
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8-Puzzle
f(N) g(N) h(N) with h(N) number of
misplaced tiles
Cutoff5
12
8-Puzzle
f(N) g(N) h(N) with h(N) number of
misplaced tiles
Cutoff5
13
8-Puzzle
f(N) g(N) h(N) with h(N) number of
misplaced tiles
Cutoff5
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8-Puzzle
f(N) g(N) h(N) with h(N) number of
misplaced tiles
Cutoff5
5
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Advantages/Drawbacks of IDA

• Advantages
• Still complete and optimal
• Requires less memory than A
• Avoid the overhead to sort the fringe
• Drawbacks
• Cant avoid revisiting states not on the current
  path
• Essentially a DFS
• Available memory is poorly used (?
  memory-bounded search, see RN p. 101-104)

16
Another approach

• Local Search Algorithms
• Hill-climbing or Gradient descent
• Simulated Annealing
• Genetic Algorithms, others

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Local Search

• Light-memory search method
• No search tree only the current state is
  represented!
• Only applicable to problems where the path is
  irrelevant (e.g., 8-queen), unless the path is
  encoded in the state
• Many similarities with optimization techniques

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Hill-climbing search

• If there exists a successor s for the current
  state n such that
• h(s) lt h(n)
• h(s) lt h(t) for all the successors t of n,
• then move from n to s. Otherwise, halt at n.
• Looks one step ahead to determine if any
  successor is better than the current state if
  there is, move to the best successor.
• Similar to Greedy search in that it uses h, but
  does not allow backtracking or jumping to an
  alternative path since it doesnt remember
  where it has been.
• Not complete since the search will terminate at
  "local minima," "plateaus," and "ridges."

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Hill climbing on a surface of states

• Height Defined by Evaluation Function

20
Robot Navigation
Local-minimum problem
f(N) h(N) straight distance to the goal
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Drawbacks of hill climbing

• Problems
• Local Maxima peaks that arent the highest point
  in the space
• Plateaus the space has a broad flat region that
  gives the search algorithm no direction (random
  walk)
• Ridges flat like a plateau, but with dropoffs to
  the sides steps to the North, East, South and
  West may go down, but a step to the NW may go up.
• Remedy
• Introduce randomness
• Random restart.
• Some problem spaces are great for hill climbing
  and others are terrible.

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Examples of problems with HC

• http//www.ndsu.nodak.edu/instruct/juell/vp/cs724s
  00/hill_climbing/hill_climbing.html

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Hill climbing example
start
h 0
goal
h -4
-2
-5
-5
h -3
h -1
-4
-3
h -2
h -3
-4
f(n) -(number of tiles out of place)
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Example of a local maximum
-5
-5
start
goal
4
0
-5
5
-4
-5
7
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Steepest Descent

• S ? initial state
• Repeat
• S ? arg minS?SUCCESSORS(S)h(S)
• if GOAL?(S) return S
• if h(S) ? h(S) then S ? S else return failure
• Similar to
• - hill climbing with h
• - gradient descent over continuous space

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Application 8-Queen

• Repeat n times
• Pick an initial state S at random with one queen
  in each column
• Repeat k times
• If GOAL?(S) then return S
• Pick an attacked queen Q at random
• Move Q in its column to minimize the number of
  attacking queens ? new S min-conflicts
  heuristic
• Return failure

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Application 8-Queen

• Why does it work ???
• There are many goal states that are
  well-distributed over the state space
• If no solution has been found after a few
  steps, its better to start it all over again
• Building a search tree would be much less
  efficient because of the high branching factor
• Running time almost independent of the number
  of queens

• Repeat n times
• Pick an initial state S at random with one queen
  in each column
• Repeat k times
• If GOAL?(S) then return S
• Pick an attacked queen Q at random
• Move Q it in its column to minimize the number of
  attacking queens is minimum ? new S

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Steepest Descent

• S ? initial state
• Repeat
• S ? arg minS?SUCCESSORS(S)h(S)
• if GOAL?(S) return S
• if h(S) ? h(S) then S ? S else return failure
• may easily get stuck in local minima
• Random restart (as in n-queen example)
• Monte Carlo descent

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Monte Carlo Descent

• S ? initial state
• Repeat k times
• If GOAL?(S) then return S
• S ? successor of S picked at random
• if h(S) ? h(S) then S ? S
• else
• Dh h(S)-h(S)
• with probability exp(?Dh/T), where T is called
  the temperature, do S ? S
  Metropolis criterion
• Return failure
• Simulated annealing lowers T over the k
  iterations.
• It starts with a large T and slowly decreases T

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Simulated annealing

• Simulated annealing (SA) exploits an analogy
  between the way in which a metal cools and
  freezes into a minimum-energy crystalline
  structure (the annealing process) and the search
  for a minimum or maximum in a more general
  system.
• SA can avoid becoming trapped at local minima.
• SA uses a random search that accepts changes that
  increase objective function f, as well as some
  that decrease it.
• SA uses a control parameter T, which by analogy
  with the original application is known as the
  system temperature.
• T starts out high and gradually decreases toward
  0.
• Applet http//www.heatonresearch.com/articles/64/p
  age1.html

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Simulated annealing (cont.)

• A bad move from A to B is accepted with a
  probability
• (f(B)-f(A)/T)
• e
• The higher the temperature, the more likely it is
  that a bad move can be made.
• As T tends to zero, this probability tends to
  zero, and SA becomes more like hill climbing
• If T is lowered slowly enough, SA is complete and
  admissible.

32
The simulated annealing algorithm
33
Parallel Local Search Techniques

• They perform several local searches concurrently,
  but not independently
• Beam search
• Genetic algorithms
• See RN, pages 115-119

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Local Beam Search

• Idea Keep track of k states rather than just one
• Start with k randomly generated states
• Repeat
• 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

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Local Beam Search

• Not the same as k searches run in parallel!
• Searches that find good states recruit other
  searches to join them
• Problem
• quite often, all k states end up on same local
  hill
• Solution
• choose k successors randomly biased towards good
  ones
• Close analogy to natural selection

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Genetic Algorithm (GA)

• GAstochastic local beam search generate
  successors from pairs of states
• Statea string over a finite alphabet (e.g, a
  string of 0 and 1)
• E.g, for 8-queen, the position of the queen in
  each column is denoted by a number
• Cross over and mutation
• http//www.heatonresearch.com/articles/65/page1.ht
  ml

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Genetic Algorithm (GA)
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Genetic Algorithm (GA)

• Crossover helps iff substrings are meaningful
  components

39
Search problems
Blind search
Heuristic search best-first and A
Variants of A
Construction of heuristics
Local search