Evaluation function Cost function g g Evaluation function Cost - - PDF document

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Evaluation function Evaluation function

• expand vertex minimizing

expand vertex minimizing f f( (v v) = ) = g g( (s s ~>

~> v

v) + ) + h h( (v v ~>

~> r

r) )

• g

g( (s s ~>

~> v

v) estimates the minimum cost from the ) estimates the minimum cost from the start vertex to start vertex to v v

• h

h( (v v ~>

~> r

r) estimates (heuristically) the cost from ) estimates (heuristically) the cost from v v to the goal vertex to the goal vertex

• if we had exact evaluation function

if we had exact evaluation function f f *

*, we could

, we could solve the problem without expanding any solve the problem without expanding any unnecessary vertices unnecessary vertices

Cost function Cost function g g

• actual cost from

actual cost from s s to to v v along the cheapest path along the cheapest path found so far found so far

• exact cost if

exact cost if G G is a tree is a tree

• can never underestimate the cost if

can never underestimate the cost if G G is a general is a general graph graph

• f

f( (v v) = ) = g g( (s s ~>

~> v

v) and unit cost ) and unit cost → breadth → breadth-

• first search

first search

• f

f( (v v) = ) = – –g g( (s s ~>

~> v

v) and unit cost ) and unit cost → depth → depth-

• first search

first search

Heuristic function Heuristic function h h

• carries information from outside the graph

carries information from outside the graph

• defined for the problem domain

defined for the problem domain

• the closer to the actual cost, the less superfluous

the closer to the actual cost, the less superfluous vertices are expanded vertices are expanded

• f

f( (v v) = ) = g g( (s s ~>

~> v

v) → cheapest ) → cheapest-

• first search

first search

• f

f( (v v) = ) = h h( (v v ~>

~> r

r) → best ) → best-

• first search

first search

Admissibility Admissibility

• let Algorithm A be a best

let Algorithm A be a best-

• first search using the

first search using the evaluation function evaluation function f f

• search algorithm is

search algorithm is admissible admissible if it finds the if it finds the minimal path (if it exists) minimal path (if it exists)

• if

if f f = = f f *

*, Algorithm A is admissible

, Algorithm A is admissible

• Algorithm A* = Algorithm A using an estimate

Algorithm A* = Algorithm A using an estimate function function h h

• A* is admissible, if

A* is admissible, if h h does not overestimate the does not overestimate the actual cost actual cost

Monotonicity Monotonicity

• h

h is locally admissible → is locally admissible → h h is monotonic is monotonic

• monotonic heuristic is also admissible

monotonic heuristic is also admissible

• actual cost is never less than the heuristic cost

actual cost is never less than the heuristic cost → → f f will never decrease will never decrease

• monotonicity → A* finds the shortest path to

monotonicity → A* finds the shortest path to any vertex the first time it is expanded any vertex the first time it is expanded

• if a vertex is rediscovered, path will not be shorter

if a vertex is rediscovered, path will not be shorter

• simplifies implementation

simplifies implementation

Optimality Optimality

• Optimality theorem: The first path from

Optimality theorem: The first path from s s to to r r found by A* is optimal. found by A* is optimal.

• Proof: lecture notes pp. 94

Proof: lecture notes pp. 94-

• 95

95

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Informedness Informedness

• the more closely

the more closely h h approximates approximates h h*

*, the better

, the better A* performs A* performs

• if A

if A1

1 using

using h h1

1 will never expand a vertex that is

will never expand a vertex that is not also expanded by A not also expanded by A2

2 using

using h h2

2, A

, A1

1 is more

is more informed that A informed that A2

2

• informedness → no other search strategy with

informedness → no other search strategy with the same amount of outside knowledge the same amount of outside knowledge can do less can do less work than A* and be sure of finding the optimal work than A* and be sure of finding the optimal solution solution

Algorithm A* Algorithm A*

• because of monotonicity

because of monotonicity

• all weights must be positive

all weights must be positive

• closed list can be omitted

closed list can be omitted

• the path is constructed from the mapping

the path is constructed from the mapping π π starting from the goal vertex starting from the goal vertex

• s

s → … → → … → π π( (π π( (π π( (r r))) → ))) → π π( (π π( (r r)) → )) → π π( (r r) → ) → r r

Practical considerations Practical considerations

• computing

computing h h

• despite the extra vertices expanded, less informed

despite the extra vertices expanded, less informed h h may yield computationally less intensive may yield computationally less intensive implementation implementation

• suboptimal solutions

suboptimal solutions

• by allowing overestimation A* becomes

by allowing overestimation A* becomes inadmissible, but the results may be good enough for inadmissible, but the results may be good enough for practical purposes practical purposes

Search algorithms Search algorithms

depth depth-

• first

first best best-

• first

first A* A* breadth breadth-

• first

first

Realizing the movement Realizing the movement

• movement through the waypoints

movement through the waypoints

• unrealistic: does not follow the game world geometry

unrealistic: does not follow the game world geometry

• aesthetically displeasing: straight lines and sharp

aesthetically displeasing: straight lines and sharp turns turns

• improvements

improvements

• line

line-

• of
• f-
• sight testing

sight testing

• bstacle avoidance
• bstacle avoidance
• combining path finding to user

combining path finding to user-

• interface

interface

• real

real-

• time response

time response

Recapitulation Recapitulation

1. 1.

discretization of the game world discretization of the game world

• grid, navigation mesh

grid, navigation mesh

• waypoints, connections, costs

waypoints, connections, costs

2. 2.

path finding in a graph path finding in a graph

• Algorithm A*

Algorithm A*

3. 3.

realizing the movement realizing the movement

• geometric corrections

geometric corrections

• aesthetic improvements

aesthetic improvements

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Alternatives? Alternatives?

• Although this is the

Although this is the de facto de facto approach in approach in (commercial) computer games, are there (commercial) computer games, are there alternatives? alternatives?

• possible answers

possible answers

• AI processors (unrealistic?)

AI processors (unrealistic?)

• robotics: reactive agents (unintelligent?)

robotics: reactive agents (unintelligent?)

• analytical approaches (inaccessible?)

analytical approaches (inaccessible?)