Heuristic Search

1
Heuristic Search

• Chapter 4

2
Outline

• Heuristic function
• Greedy Best-first search
• Admissible heuristic and A
• Properties of A Algorithm
• IDA

3
Heuristic Search

• Heuristic A rule of thumb generally based on
  expert experience, common sense to guide
  problem-solving process
• In search, use a heuristic function that
  estimates how far we are from a goal.
• How do we use heuristics?

4
Romania with step costs in km
5
Greedy best-first search example
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Greedy best-first search example
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Greedy best-first search example
8
Greedy best-first search example
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Robot Navigation
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Robot Navigation
f(N) h(N), with h(N) Manhattan distance to
the goal
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Properties of greedy best-first search

• Complete? No can get stuck in loops Yes, if we
  can avoid repeated states
• Time? O(bm), but a good heuristic can give
  dramatic improvement
• Space? O(bm) -- keeps all nodes in memory
• Optimal? No

12
A Search

• A search combines Uniform-cost and Greedy
  Best-first Search
• Evaluation function f(N) g(N)
  h(N) where
• g(N) is the cost of the best path found so far to
  N
• h(N) is an admissible heuristic
• f(N) is the estimated cost of cheapest solution
  
• through N
• 0 lt ? ? c(N,N) (no negative cost steps).
• Order the nodes in the fringe in increasing
  values of f(N)

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A search example
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A search example
15
A search example
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A search example
17
A search example
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A search example
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Admissible heuristic

• Let h(N) be the cost of the optimal path from N
  to a goal node
• Heuristic h(N) is admissible if 0
  ? h(N) ? h(N)
• An admissible heuristic is always optimistic

20
8-Puzzle
N
goal

• h1(N) number of misplaced tiles 6 is
  admissible

• h2(N) sum of distances of each tile to goal
  13 is admissible

• What heuristics are overestimates?

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Completeness Optimality of A

• Claim
• If there is a path from the initial to a goal
  node, A using TREE-SEARCH terminates by finding
  the best path, hence is
• complete
• optimal

22
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.
• We can show f(n) lt f(G2), so A would not have
  selected G2.

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Optimality of A (proof)

• Contd
• f(G2) g(G2) since h(G2) 0
• g(G2) gt g(G) since G2 is suboptimal
• f(G) g(G) since h(G) 0
• f(G2) gt f(G) from above

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Optimality of A (proof)

• Contd
• f(G2) gt f(G) from above
• h(n) h(n) since h is admissible
• g(n) h(n) g(n) h(n)
• f(n) f(G)
• Hence f(G2) gt f(n), and A will never select
  G2 for expansion

25
Exampel Graph Search returns a suboptimal
solution
h6
2
1
B
A
5
S
G
h7
h1
h0
4

• h is admissible

26
Exampel Graph Search returns a suboptimal
solution
h6
2
1
B
A
5
S
G
h7
h1
h0
4

• h is admissible

27
Exampel Graph Search returns a suboptimal
solution
h6
2
1
B
A
5
S
G
h7
h1
h0
4

• h is admissible

28
Exampel Graph Search returns a suboptimal
solution
h6
2
1
B
A
5
S
G
h7
h1
h0
4

• h is admissible

29
Exampel Graph Search returns a suboptimal
solution
h6
2
1
B
A
5
S
G
h7
h1
h0
4

• h is admissible

30
Consistent Heuristic

• The admissible heuristic h is consistent (or
  satisfies the monotone restriction) if for every
  node N and every successor N of Nh(N) ?
  c(N,N) h(N)(triangular inequality)
• A consisteny heuristic is admissible.

31
Exampel Graph Search returns a suboptimal
solution
h6
2
1
B
A
5
S
G
h7
h1
h0
4

• h is admissible but not consistent e.g.
• h(S)7 ? c(S,A) h(A) 5 ?
• No.

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8-Puzzle
N
goal
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Claims

• If h is consistent, then the function f alongany
  path is non-decreasing f(N) g(N) h(N)
  f(N) g(N) c(N,N) h(N) h(N) ? c(N,N)
  h(N) f(N) ? f(N)
• If h is consistent, then whenever A expands a
  node it has already found an optimal path to the
  state associated with this node

34
Optimality of A

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

35
Avoiding Repeated States in A

• If the heuristic h is consistent, then
• Let CLOSED be the list of states associated with
  expanded nodes
• When a new node N is generated
• If its state is in CLOSED, then discard N
• If it has the same state as another node in the
  fringe, then discard the node with the largest f

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Heuristic Accuracy

• h(N) 0 for all nodes is admissible and
  consistent. Hence, breadth-first and uniform-cost
  are particular A !!!
• Let h1 and h2 be two admissible and consistent
  heuristics such that for all nodes N h1(N) ?
  h2(N).
• Then, every node expanded by A using h2 is
  also expanded by A using h1.
• h2 is more informed than h1
• h2 dominates h1
• Which heuristic for 8-puzzle is better?

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Complexity of A

• Time exponential
• Space can keep all nodes in memory
• If we want save space, use IDA

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Iterative Deepening A (IDA)

• Use f(N) g(N) h(N) with admissible and
  consistent h
• Each iteration is depth-first with cutoff on the
  value of f of expanded nodes

39
8-Puzzle
f(N) g(N) h(N) with h(N) number of
misplaced tiles
Cutoff4
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8-Puzzle
f(N) g(N) h(N) with h(N) number of
misplaced tiles
Cutoff4
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8-Puzzle
f(N) g(N) h(N) with h(N) number of
misplaced tiles
Cutoff4
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8-Puzzle
f(N) g(N) h(N) with h(N) number of
misplaced tiles
Cutoff4
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8-Puzzle
f(N) g(N) h(N) with h(N) number of
misplaced tiles
Cutoff4
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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
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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
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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
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8-Puzzle
f(N) g(N) h(N) with h(N) number of
misplaced tiles
Cutoff5
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About Heuristics

• Heuristics are intended to orient the search
  along promising paths
• The time spent computing heuristics must be
  recovered by a better search
• After all, a heuristic function could consist of
  solving the problem then it would perfectly
  guide the search
• Deciding which node to expand is sometimes called
  meta-reasoning
• Heuristics may not always look like numbers and
  may involve large amount of knowledge

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When to Use Search Techniques?

• The search space is small, and
• There are no other available techniques, or
• It is not worth the effort to develop a more
  efficient technique
• The search space is large, and
• There is no other available techniques, and
• There exist good heuristics

53
Summary

• Heuristic function
• Greedy Best-first search
• Admissible heuristic and A
• A is complete and optimal
• Consistent heuristic and repeated states
• Heuristic accuracy
• IDA