Heuristic search

1
Heuristic search

• Reading AIMA 2nd ed., Ch. 4.1-4.3

2
(Blind) Tree search so far
function TreeSearch(problem, fringe) n
InitialState(problem) fringe Insert(n) while
(1) If Empty(fringe) return failure n
RemoveFront(fringe) If Goal(problem,n) return
n fringe Insert( Expand(problem,n) ) end end

• Strategies
• BFS fringe FIFO
• DFS fringe LIFO
• Strategies defined by the order of node expansion

3
Example of BFS and DFS in a maze
G
G
s0
s0
4
Informed searches best-first search

• Idea Add domain-specific information to select
  the best path to continue searching along.
• How to do it?
• Use evaluation function, f(n), which selects the
  most desirable (best-first) node to expand.
• Fringe is a queue sorted in decreasing order of
  desirability.
• Ideal case f(n) true cost to goal state t(n)
• Of course, t(n) is expensive to compute (BFS!)
• Use an estimate (heuristic) instead

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maze example

• f(n) straight-line distance to goal

2
1
G
3
2
1
1
4
3
2
2
2
s0
3
3
3
4
4
4
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Romania example
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Greedy search

• Evaluation function f(n) h(n) --- heuristic
  estimate of cost from n to closest goal
• Greedy search expands nodes that appear to be
  closest to goal

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Example of greedy search
Arad h 366
Arad h 366
Sibiu h 253
Fagaras h 176
Bucharest h 0
Is this the optimal solution?
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Properties of greedy search

• Completeness
• Not complete, can get stuck in loops, e.g., Iasi
  -gt Neamt -gt Iasi -gt Neamt -gt
• Can be made complete in finite spaces with
  repeated-state checking
• Time complexity
• O(bm), but good heuristic can give excellent
  improvement
• Space complexity
• O(bm), keeps all nodes in memory
• Optimality
• No, can wander off to suboptimal goal states

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A-search

• Improve greedy search by discouraging
  wandering-off
• f(n) g(n) h(n)g(n) cost of path to reach
  node nh(n) estimated cost from node n to goal
  nodef(n) estimated total cost through node n
• Search along most promising path, not node
• Is A-search optimal?

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Optimality of A-search

• Theorem Let h(n) be at most ? higher than
  t(n), for all n. Than, the A-search will be at
  most ? steps longer than the optimal search.
• Proof If n is a node on the path found by
  A-search, then f(n) g(n) h(n) ? g(n) t(n)
  ? optimal ?

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A search

• How to make A-search optimal?
• Make ? 0.
• This means heuristic must always underestimate
  the true cost to goal. h(n) has to be an
  Optimistic estimate.
• 0 ? h(n) ? t(n) is called an admissible
  heuristic.
• Theorem Tree A-search is optimal if h(n) is
  admissible.

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Example of A search
Arad h 366
Arad
Sibiu
Fagaras
Rimnicu V.
Pitesti
Bucharest f 4180
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A optimality proof

• Suppose there is a suboptimal goal G on the
  queue. Let n be an unexpanded node on a shortest
  path to optimal goal G.

• f(G) g(G) h(G) g(G) since h(G)0
  gt g(G) since G is suboptimal ?
  f(n) g(n) h(n) since h is admissible
• Hence, f(G) gt f(n), and f(G) will never be
  expanded
• Remember example on the previous page?

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Consistency of A

• Consistency conditionh(n) - c(n,n) ?
  h(n) where n is any successor of n
• Guarantees optimality of graph search (remember,
  the proof was for tree search!)
• Also called monotonicityf(n) g(n) h(n)
  g(n) c(n,n) h(n) ? g(n) h(n)
  f(n)f(n) ? f(n)
• f(n) is monotonically non-decreasing

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Example of consistency/monotonicity

• A expands nodes in order of increasing f-value
• Adds f-contours of nodes
• BFS adds layers

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Properties of A

• Completeness
• Yes (for finite graphs)
• Time complexity
• Exponential in h(n) - t(n) x length of
  solution
• Memory requirements
• Keeps all nodes in memory
• Optimal
• Yes

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Admissible heuristics h(n)

• Straight-line distance for the maze and Romanian
  examples
• 8-puzzle
• h1(n) number of misplaced tiles
• h2(n) number of squares from desired location
  of each tile (Manhattan distance)

• h1(S) 7
• h2(S) 40331021 14

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Dominance

• If h1(n) and h2(n) are both admissible and h1(n)
  ? h2(n), then h2(n) dominates over h1(n)
• Which one is better for search?
• h2(n), because it is closer to t(n)
• Typical search costsd 14, IDS 3,473,941
  nodes A(h1) 539 A(h2) 113d 24, IDS
  54 billion nodes A(h1) 39135 A(h2)
  1641

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Relaxed problems

• How to derive admissible heuristics?
• Can be derived from exact solutions to problems
  that are simpler (relaxed) versions of the
  problem one is trying to solve
• Examples
• 8-puzzle
• Tile can move anywhere from initial position (h1)
• Tiles can occupy same square but have to move one
  square at a time (h2)
• maze
• Can move any distance and over any obstacles
• ImportantThe cost of optimal solution to
  relaxed problems is no greater than the optimal
  solution to the real problem.

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Local search

• In many problems one does not care about the
  path, rather one wants to find the goal state
  (based on a goal condition).
• Use local search / iterative improvement
  algorithms
• Keep a single current state, try to improve it.

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Example

• N-queens problemfrom initial configuration
  move to other configurations such that the number
  of conflicts is reduced

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Hill-climbingGradient ascent / descent
function HillClimbing(problem) n
InitialState(problem) while (1) neighbors
Expand(problem,n) n arg max
Value(neighbors) If Value(n) lt Value(n), return
n n n end end

• Goaln arg max Value( n )

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Hill climbing (contd)

• ProblemDepending on initial state, can get
  stuck in local maxima (minima), ridges, plateaus

25
Beam search

• Problem of local minima (maxima) in hill-climbing
  can be alleviated by starting HC from multiple
  random starting points
• Or make it stochastic (Stochastic HC) by choosing
  successors at random, based on how good they
  are
• Local beam search somewhat similar to
  random-restart HC
• Start from N initial states.
• Expand all N states and keep N best successors.
• Stochastic beam search stochastic version of
  LBS, similar to SHC.

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Simulated annealing

• Idea
• Allow bad moves, initially more, later fewer
• Analogy with annealing in metallurgy

function SimulatedAnnealing(problem,schedule) n
InitialState(problem) t 1 while (1) T
schedule(t) neighbors Expand(problem,n) n
Random(neighbors) ?V Value(n) - Value(n) If
?V gt 0, n n Else n n with probability
exp(?V/T) t t 1 end end
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Properties of simulated annealing

• At fixed temperature T, state occupation
  probability reaches Boltzman distribution, exp(
  V(n)/T )
• Devised by Metropolis et al. in 1953

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Genetic algorithms