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Artificial Intelligence for Games Informed Search (2)PowerPoint Presentation

Artificial Intelligence for Games Informed Search (2)

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Heuristic functions

- sample heuristics for 8-puzzle:
- h1(n) = number of misplaced tiles
- h2(n) = total Manhattan distance

- h1(S) = ?
- h2(S) = ?

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

- randomly generate 8-puzzle problems
- 100 examples for each solution depth
- contrast behaviour of heuristics & strategies

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 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*

- 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

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

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 = 3Next f-cost = 4

Next f-cost = 3

Next f-cost = 5

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 = 4Next f-cost = 4

Next f-cost = 5

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

- 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…

- 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… trading optimality for speed weight towards h when confident in the estimate of h

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)

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