CO2301 Games Development 1 Week 15 Search Methods

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CO2301 - Games Development 1Week 15Search
Methods

• Gareth Bellaby

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Topics

• Search Methods in Traditional Games - Tower of
  Hanoi example
• Combinatorial Explosion
• Chess

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• Search Methods in Traditional Games - Tower of
  Hanoi example

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Traditional Games

• The approach described in this lecture is not one
  that is used much within computer games.
• Useful for games such as Chess. I've called these
  "traditional" games for want of a better phrase.
• The approach may be applicable to certain
  problems in modern computer games.

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Towers of Hanoi (1)

• A simple game played with three pegs and a pile
  of disks of different sizes. The pegs are stacked
  on a peg in order of size.
• The object is to stack the disks onto a new peg.
• Only one disk may be moved at a time.
• A disk can never be placed on top of a smaller
  disk.

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Towers of Hanoi (2)
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Lists

• The disks can be represented as numbers
• the small disk is 1
• the medium sized disk is 2
• the large disk is 3.
• This will allow us to easily describe the rule
  about "A disk can never be placed on top of a
  smaller disk".
• A useful (and common) representation of a problem
  is the list structure.
• A empty list is given by

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Lists

• The disks on a single peg can be represented as a
  list. One common convention is to use square
  brackets for a list. The elements of the list are
  separated by commands. A peg with the small disk
  on top, the medium sized disk in the middle and
  the large disk on the bottom would therefore be
  1, 2, 3
• The state of all three pegs would be represented
  by a list of lists 2,3, 1,

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Towers of Hanoi (4)

• 1,2,3
• / \
• 2,31 2,3 1
• 312 321

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Towers of Hanoi (5)

• 1,2,3
• / \
• 2,31 2,3 1
• 312 321
• 3 1,2 31,2
• 31,2 1,23
• 132 123
• 12,3 1 2,3
• 1,2,3 1,2,3

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• Search trees

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Searches

• Search is a basic problem-solving technique.
• Grow a tree from the start position to the goal.

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Trees Again...

• The Tower of Hanoi problem can be represented as
  a set of states.
• A new state is generated by applying a rule.
• The states form a tree structure.
• A route through the tree to a goal state
  represents a solution.
• Search is a basic problem-solving technique.

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Search Trees

• A tree is
• A representation of a problem
• A way to solve the problem using a search
  technique, e.g. breadth-first search, best-first
  search, etc.

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Conceptional Search Pathfinding

• Pathfinding searches a map. The map is just a
  representation.
• A Conceptional Search searches a problem space.
  The space is just a representation of states
  within the problem.
• Dijkstra's algorithm uses a graph. In other
  algorithms the nodes, linked together, can form a
  tree-like structure.

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Rules

• Rules are used to generate new nodes. For
  example, in the pathfinding exercise you used a
  grid and rules to move north, south, east and
  west.
• For the Towers of Hanoi a good representation of
  the problem state is
• 1,2,3
• A rule would be "Can only remove the topmost
  disk" and, in this representation, this would be
  the leftmost number in the list.

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Search

• A state space (or problem space) represents a
  problem.
• Indeed the full tree represents the whole of the
  problem in every possible permutation.
• A state space is a graph whose nodes correspond
  to problem situations. A given problem is a path
  in this graph.

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Search Tree (Graph)

• The nodes, linked together, form a tree-like
  structure.
• The whole tree is described as the search state
  (or problem state)
• Lots of generate and search techniques have been
  developed.

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Search Tree (Graph)

• Nodes correspond to situations, e.g. a node
  represents a move, or position, in a game.
• A link (or arc) between nodes represents a
  legitimate move, or action, according to the
  rules.

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Search Tree (Graph)

• A specific problem is defined by
• state space
• start node (root node)
• goal condition (may be more than one node)

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Evaluation Function

• The Tower of Hanoi problem was represented using
  lists. Lists are a commonly used technique in AI
  for representing a problem state.
• Another common technique is to describe a problem
  using a number.
• A number is an evaluation of a state, e.g. a node
  (a move or position) in a game.
• Numbers are useful because they mean that states
  can be compared with one another.

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• Combinatorial Explosion

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Combinatorial Explosion

• Combinatorial Explosion
• A rapid increase in the amount of possible
  outcomes due to the large number of combinations.
• Towers of Hanoi is a simple problem, but most
  problems are far more complex.
• A directed search is used to overcome the
  problem of combinatorial explosion. A heuristic
  is used to guide the search and hopefully remove
  the need to evaluate every possible permutation
  of the problem.

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Travelling Salesman (1)

• An example of combinatorial explosion.
• Plan the quickest route for a salesman visiting
  different cities.
• Initially appears a simple problem.

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Travelling Salesman (2)
nottingham
20
leicester
15
37
derby
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Travelling Salesman (3)
nottingham
20
leicester
30
15
37
40
derby
18
sheffield
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• Chess

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Chess

• Consider chess. How would you
• Build a chess game?
• Represent a move?
• What are the rules?
• Create an evaluation function?
• Wwhat are the characteristics of the game of
  chess?
• The same approach as solving the Tower of Hanoi
  problem
• Each node of the tree represents one move (or
  position) in the game
• Each layer of the tree is all of the moves either
  Black or White could take that turn

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Chess and Combinatorial Explosion

• Average number of moves in a single turn of chess
  is 35.
• The number of moves in an average game is around
  50 for each player.
• Therefore, full evaluation of a game requires
  35100 moves

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Heuristics

• Heuristics can be used to guide search.
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• non-heuristic blind search
• heuristic directed search
•  
• Chess heuristic put pawns into the center
  squares.
• Employ a search algorithm.

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Chess

• 2-player game
• Perfect information
• Certain outcomes, e.g. success and failure
  clearly defined for the game and for a move.
• What about computer games?

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Computer Games

• n-player games. This does not mean that there
  are necessarily lots of people playing the game
  but that lots of units or NPCs are moving at the
  same time (contrast to one piece moving in Chess
  or Draughts).
• Imperfect information - outcomes are
  unpredicatable.
• Success and failure may not be clearly defined
  for the game or an instance within the game (The
  player has cleared an area of enemies but how
  much life and ammo has been lost?).