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Artificial Intelligence for Games Informed Search (2). Patrick Olivier p.l.olivier@ncl.ac.uk. Heuristic functions. sample heuristics for 8-puzzle: h 1 (n) = number of misplaced tiles h 2 (n) = total Manhattan distance h 1 (S) = ? h 2 (S) = ?. Heuristic functions.

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artificial intelligence for games informed search 2

Artificial Intelligence for GamesInformed Search (2)

Patrick Olivier

p.l.olivier@ncl.ac.uk

heuristic functions
Heuristic functions
  • sample heuristics for 8-puzzle:
    • h1(n) = number of misplaced tiles
    • h2(n) = total Manhattan distance
  • h1(S) = ?
  • h2(S) = ?
heuristic functions1
Heuristic functions
  • sample heuristics for 8-puzzle:
    • h1(n) = number of misplaced tiles
    • h2(n) = total Manhattan distance
  • h1(S) = 8
  • h2(S) = 3+1+2+2+2+3+3+2 = 18
  • dominance:
    • h2(n) ≥ h1(n) for all n (both admissible)
    • h2 is better for search (closer to perfect)
    • less nodes need to be expanded
example of dominance
Example of dominance
  • randomly generate 8-puzzle problems
  • 100 examples for each solution depth
  • contrast behaviour of heuristics & strategies
a enhancements local search
A* enhancements & local search
  • Memory enhancements
    • IDA*: Iterative-Deepening A*
    • SMA*: Simplified Memory-Bounded A*
  • Other enhancements (next lecture)
    • Dynamic weighting
    • LRTA*: Learning Real-time A*
    • MTS: Moving target search
  • Local search (next lecture)
    • Hill climbing & beam search
    • Simulated annealing & genetic algorithms
improving a performance
Improving A* performance
  • Improving the heuristic function
    • not always easy for path planning tasks
  • Implementation of A*
    • key aspect for large search spaces
  • Relaxing the admissibility condition
    • trading optimality for speed
ida iterative deepening a
IDA*: iterative deepening A*
  • reduces the memory constraints of A* without sacrificing optimality
  • cost-bound iterative depth-first search with linear memory requirements
  • expands all nodes within a cost contour
  • store f-cost (cost-limit) for next iteration
  • repeat for next highest f-cost
ida exercise

Start state

Goal state

1 2 3

6 X 4

8 7 5

1 2 3

8 X 4

7 6 5

IDA*: exercise
  • Order of expansion:
    • Move space up
    • Move space down
    • Move space left
    • Move space right
  • Evaluation function:
    • g(n) = number of moves
    • h(n) = misplaced tiles
  • Expand the state space to a depth of 3 and calculate the evaluation function
ida f cost 3

0+3=3

1 2 3

6 X 4

8 7 5

1+4=5

1+3=4

1+4=6

1 3

6 2 4

8 7 5

1 2 3

6 7 4

8 5

1 2 3

6 4 4

8 7 5

1 2 3

X 6 4

8 7 5

1+3=4

IDA*: f-cost = 3

Next f-cost = 4

Next f-cost = 3

Next f-cost = 5

ida f cost 4

0+3=3

1 2 3

6 X 4

8 7 5

1 2 3

6 7 4

8 7 5

1+3=4

1+4=5

1 3

6 2 4

8 7 5

2+3=5

4+0=4

2+2=4

3+3=6

3+1=4

1 2 3

8 4

7 6 5

1 2 3

8 6 4

7 5

1 2 3

8 6 4

7 5

1 2 3

8 6 4

7 5

1 2 3

6 4

8 7 5

IDA*: f-cost = 4

Next f-cost = 4

Next f-cost = 5

simplified memory bounded a
Simplified memory-bounded A*
  • SMA*
    • When we run out of memory drop costly nodes
    • Back their cost up to parent (may need them later)
  • Properties
    • Utilises whatever memory is available
    • Avoids repeated states (as memory allows)
    • Complete (if enough memory to store path)
    • Optimal (or optimal in memory limit)
    • Optimally efficient (with memory caveats)
class exercise
Class exercise
  • Use the state space given in the example
  • Execute the SMA* algorithm over this state space
  • Be sure that you understand the algorithm!
trading optimality for speed
Trading optimality for speed…
  • The admissibility condition guarantees that an optimal path is found
  • In path planning a near-optimal path can be satisfactory
  • Try to minimise search instead of minimising cost:
    • i.e. find a near-optimal path (quickly)
weighting
Weighting…

fw(n) = (1 - w).g(n) + w.h(n)

    • w = 0.0 (breadth-first)
    • w = 0.5 (A*)
    • w = 1.0 (best-first, with f = h)
  • trading optimality for speed
  • weight towards h when confident in the estimate of h