Procedural Game Level Generation

Game Programming
Procedural Content Generation
Nick Prühs, Denis Vaz Alves

Objectives
• To get an overview of the different fields of procedural content
generation
• To learn how to procedurally generate content of different types
• To understand game design implications of using generated content
in games
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Procedural Content Generation
“Procedural content generation (PCG) refers to creating game content
automatically, through algorithmic means.”
- Togelius, Yannakakis, Stanley, Browne
“PCG should ensure that from a few parameters, a large number of
possible types of content can be generated.”
- Doull
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Categories of PCG
• Online Level Generation
• Offline Level Generation
• Fixed Seed Level Generation
• Game Entity Instancing
• User-mediated Content
• Dynamic Systems
• Procedural Puzzles & Plot Generation
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Categories of PCG
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Online Level Generation in Diablo 2

Categories of PCG
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Fixed Seed Level Generation in Elite

Categories of PCG
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Game Entity Instancing in Spore

Categories of PCG
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Dynamic Systems in Left 4 Dead

Opportunities of PCG
• high diversity of the resulting assets
• faster than any human designer could ever be
• significantly reduces production costs
• allows for a mixed-initiative approach to level design
• content automatically implemented in the engine
• can save vital system resources
• players can influence the parameters of the game world
• possibility of automatically analyzing player behavior
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Procedurally Generated Item in Diablo 2

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Map Editor of Age of Empires 2

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Game Parameters of Master of Orion 2

Challenges of PCG
• Satisfying a high number of constraints (e.g. full connectivity)
▪ Finding these constraints and tweaking unintuitive parameters of
the system can degenerate into trial and error
• Produce aesthetically pleasing results
▪ Levels can become too similar to each other
• Maximize the expressive range (variety of results)
▪ Can decrease co-op multiplayer playability
• May require spending too much time on inventing a sophisticated
level generator
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Challenges of PCG
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Generation Mistakes in Infinite Mario
(Picture by Mawhoter, Mateas)

“Random” Numbers
• Computers are deterministic – thus, producing “random” numbers
seems to be conceptually impossible.
• Pseudo-random numbers in computers are generated by applying a
fixed rule to a given number, generating the next number in the
sequence.
• The first number of the sequence is often called the seed of the
random number generator.
• The length of the sequence before repeating numbers is called its
period.
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“Good” “Random” Numbers
“What is random enough for one application may not be random
enough for another.”
For games:
• Don’t use generators with a period less than 264!
• Don’t use built-in language generators!
• Don’t use overengineered generators!
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A Good Pseudo-Random Number Generator
• Should combine at least two unrelated generation methods, in order
to mitigate the flaws of each.
• Should be an object instead of a static class, as it maintains an
internal state (the current number in the sequence).
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64-bit XOR Shift Method
Max. Period: 264 – 1
Operations: 3x XOR, 3x shift
Algorithm:
1. x = x ^ (x >> A1)
2. x = x ^ (x << A2)
3. x = x ^ (x >> A3)
with A1 = 21, A2 = 35, A3 = 4 as full-period triple.
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MLCG Module 264
Max. Period: 262 (suffers from serial correlation)
Operations: 1x multiplication, 1x modulo
Algorithm:
x = (x * A) mod 264
with A = 2685821657736338717 as recommended multiplier.
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Combined Generator
Period: 1.8 x 1019
Algorithm:
1. x = x ^ (x >> 21)
2. x = x ^ (x << 35)
3. x = x ^ (x >> 4)
4. x = (x * 2685821657736338717 ) mod 264
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Level Generation Algorithms
• Context-free grammars
• Reinforcement learning
• Genetic algorithms
• Chunk-based approach
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Context-Free Grammar
• Approach for generating levels for platform games
• Originally presented by Ince in 1999
▪ Later improved on by Compton and Mateas in 2006
• Generation model is represented as a context-free grammar
▪ Level as start symbol
▪ Most basic units out of which the levels are constructed as
terminal symbols
▪ Sequences of these units as nonterminal symbols
▪ Connections between these sequences as productions
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Reinforcement Learning
• Context-free grammar approach requires programmers to find the
generational rules, which can be extremely difficult
• In 2009, Laskov interpreted level generation as a maintenance task
▪ Level building agent performs certain actions
▪ Goal is to generate a level that satisfies user-specified
parameters at all times
▪ Enforced by a reward function
o Punishes fail-states like unplayable levels
o Rewards branches and game elements like enemies or treasures
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Genetic Algorithms
• Sorenson and Pasquier were among the first to propose an algorithm
that is able to generate platform game levels and 2D adventure game
levels alike
• Feasible-infeasible two-population algorithm
▪ Considers levels which do not yet satisfy all given constraints as
infeasible population
o Evolved towards minimising the number of violated constraints
▪ Considers all others as feasible population
o Subject to a fitness function that rewards levels based on the criteria
specified by the level designers
• Generating Roguelike levels on an average machine took “less than an
hour”...
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Chunk-based Approach
• Used in Infinite Mario Bros and Torchlight
• Creates a game level by assembling pre-authored level chunks
• Requires applying some post-processing algorithms afterwards
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Chunk-based Approach
• Very intuitive for designers
• Applicable for both 2D and 3D
• Easy to increase the variety of different levels just by extending the
chunk library
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Definition (Game Element)
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A game element is any domain-specific game object a player can
interact with (i.e. enemies, items, levers).

Definition (Chunk)
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A chunk is the most basic building block of a level. It contains
information about its extents, its position and rotation as well as about
where to align it to the existing level and where to add game elements.

Definition (Chunk Library)
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A chunk library holds a set of chunk templates and is needed by the
level generator to have a specific repertoire of chunks that may be used
during the generation process.

Definition (Context)
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Every chunk contains at least one single context describing the relative
position at which it may be aligned to other chunks.

Definition (Anchor)
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Every chunk may contain one or more tagged anchors describing the
relative position at which game elements can be added.

Definition (Level)
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A level is a bounded space containing a limited number of level chunks.

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Definition (Level)
Procedurally Generated Game Level

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Level Generation Framework
UML Class Diagram of the ByChance Framework

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Main Routine
Precondition: C is a non-empty chunk library. L is a bounded level
which contains chunks of C, only, and none of these chunks overlap or
exceed the level bounds.

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Main Routine
1. While L contains at least one non-blocked context:
1. Select a non-blocked context for expanding L.
2. Find all compatible chunk candidates as follows: For each chunk in C, and
for each of its contexts:
1. If contexts are compatible and new chunk wouldn't overlap or exceed the level
bounds, add to list of candidates.
2. Else if the chunk is allowed to be rotated and hasn't already been rotated by 360
degrees in every direction, rotate it and try again.
3. Else, reject the chunk.
3. If candidate list is empty, block selected context and continue.
4. Else, select a compatible chunk from candidate list and add it to L, aligning
the selected contexts.
2. Perform post-processing.

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Main Routine
Postcondition: All contexts of L are blocked. No chunk exceeds the
level bounds, and no two different level chunks overlap.

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Chunk Selection
• Three attributes influence the probability of picking a particular
chunk:
▪ Weight
▪ Quantity
▪ Tags
• Tuning these three attributes is key to producing enjoyable levels.

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Post-Processing
Cluster of floors.
Cluster of rooms.

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Post-Processing
Dead ends.

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Post-Processing
Connected contexts.

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Level Layouts
Special level layouts.
(Picture by Compton, Mateas)

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Editor Support
ByChance Chunk Template Editor in Unity

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Editor Support
ByChance Scene View in Unity

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Editor Support
ByChance Game View in Unity

Perlin Noise
• Originally created by Ken Perlin in 1983
• Mapping from ℝn to [-1; 1]
• Can be used to assign a greyscale value to each pixel of a bitmap
• Bitmap can be used as heightmap for 3D terrain
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Perlin Noise
• Perlin Noise is coherent
▪ For any two points A, B, the value of the noise function changes
smoothly as you move from A to B.
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Non-coherent noise (left) vs. coherent noise (right)
(Picture by Matt Zucker)

Creating Perlin Noise
• Well-known approach is creating non-coherent noise and smooth
(blur) it
• Original approach by Perlin is different, mathematically well-defined
and more efficient
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Wanted:
noise: ℝn [-1; 1]
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Wanted:
noise: ℝ2 [-1; 1]
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On a bounded space of size Size x Size,
Size > 0
impose a grid of size GridSize x GridSize,
Size >= GridSize > 0
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• Grid points are defined for each whole number.
• Any number with a fractional part (i.e. 3.14) lies between grid points.
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Picture by Matt Zucker.

Step 0: Assign a pseudorandom gradient of length 1 to each grid point.
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Note: The gradient of each grid point must not change after it has been
computed once (e.g. don’t compute random gradients every time
computing noise(x,y)).
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Step 1: Find the grid points surrounding (x, y). In ℝ2, we have 4 of
them, which we will call (x0,y0), (x0, y1), (x1, y0), and (x1, y1).
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Step 2: Find the vectors going from each grid point to (x, y).
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Step 3: Compute the influence of each gradient by performing a dot
product of the gradient and the vector going from its associated grid
point to (x, y).
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s = g(x0, y0) · ((x, y) -
(x0, y0))
t = g(x1, y0) · ((x, y) - (x1, y0))
u = g(x0, y1) · ((x, y) - (x0, y1))
v = g(x1, y1) · ((x, y) - (x1, y1))

Step 4: Ease the position of the point, exaggerating its proximity to
zero or one.
For inputs that are sort of close to zero, output a number really close to
zero. For inputs close to one, output a number really close to one.
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f(p) = 3p2 – 2p3
(Picture by Matt Zucker)

Step 4: Ease the position of the point, exaggerating its proximity to
zero or one.
For inputs that are sort of close to zero, output a number really close to
zero. For inputs close to one, output a number really close to one.
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Sx = 3(x - x0)² - 2(x - x0)³
Sy = 3(y - y0)² - 2(y - y0)³

Step 5: Linearly interpolate between the influences of the gradients.
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a = s + Sx(t - s)
b = u + Sx(v - u)
noise(x, y) = a + Sy(b - a)

In order to use noise as greyscale value, you might want to transform it
to a more useful interval.
Full source code is available at https://github.com/npruehs/perlin-noise.
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noise(x, y) = a + Sy(b - a)
[-1; 1]
transformedNoise(x, y) = (noise(x, y) + 1) / 2 [0; 1]
greyscale(x, y) = transformedNoise(x, y) * 255 [0; 255]

Markov Chains
• Named after Andrey Markov
• State space with random transitions
• Usually memory-less
▪ Next state only depends on current state
▪ Thus the name
• Usually doesn’t terminate
▪ There is always a next state
• Generally impossible to predict the state at a given point in the future
▪ Statistical properties can be predicted (and are more interesting in
most cases)
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Markov Chains
• Example: Drunkard’s Walk
▪ One-dimensional state space
▪ Position may change by +1 or -1 with equal probability
▪ Two possible transitions from each state
▪ Transition probability only depends on the current state
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Markov Chains
• Example: Drunkard’s Walk
▪ One-dimensional state space
▪ Position may change by +1 or -1 with equal probability
▪ Two possible transitions from each state
▪ Transition probability only depends on the current state
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Markov Chains
• Example: Creature Diet
▪ Creature eats only grapes, cheese, or lettuce
▪ Eats exactly once a day
▪ If it ate cheese today, tomorrow it will eat lettuce or grapes with
equal probability.
▪ If it ate grapes today, tomorrow it will eat grapes with probability
1/10, cheese with probability 4/10 and lettuce with probability
5/10.
▪ If it ate lettuce today, tomorrow it will eat grapes with probability
4/10 or cheese with probability 6/10.
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Markov Chains
• Example: Creature Diet
▪ Can be modeled with a Markov chain since its choice tomorrow
depends solely on what it ate today
o not what it ate yesterday
o not what it ate any other time in the past
▪ Statistical property that could be calculated is the expected
percentage, over a long period, of the days on which the
creature will eat grapes
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Markov Chain – Definition
A Markov chain is a sequence of random variables X1, X2, X3, ... with
the Markov property, namely that, given the present state, the future
and past states are independent:
Pr 𝑋1 = 𝑥1, … , 𝑋 𝑛 = 𝑥 𝑛 > 0 ⇒
Pr 𝑋 𝑛+1 = 𝑥 𝑋1= 𝑥1, 𝑋2 = 𝑥2, … . , 𝑋 𝑛 = 𝑥 𝑛 = Pr 𝑋 𝑛+1 = 𝑥 𝑋 𝑛 = 𝑥 𝑛)
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Markov Chains – Description
• Directed graph
▪ Edges are labeled by the probabilities of going from one state at
time n to the other states at time n+1
• Transition matrix
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Stationary Markov Chains
Stationary Markov chains are processes where the probability of a
transition is independent of n.
∀𝑛 ∈ 𝑁:
Pr 𝑋 𝑛+1 = 𝑥 𝑋 𝑛 = 𝑦) = Pr(𝑋 𝑛 = 𝑥 | 𝑋 𝑛−1 = 𝑦)
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Higher Order Markov Chains
A Markov chain of order 𝑚 ∈ 𝑁 is a process where the future state
depends on the past m states.
∀𝑛 ∈ 𝑁, 𝑛 > 𝑚:
Pr 𝑋 𝑛+1 = 𝑥 𝑋 𝑛 = 𝑥 𝑛, 𝑋 𝑛−1 = 𝑥 𝑛−1, … , 𝑋1 = 𝑥1) =
Pr 𝑋 𝑛+1 = 𝑥 𝑋 𝑛 = 𝑥 𝑛, 𝑋 𝑛−1 = 𝑥 𝑛−1, … , 𝑋 𝑛−𝑚 = 𝑥 𝑛−𝑚)
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Random Name Generation
Given an input set of feasible existing names, a new random name can be
generated using an m-order Markov chain as follows:
1. Pick any existing name, with equal probability.
2. Take the first m letters of that name.
3. Find all existing names containing these m letters.
4. In all of these existing names, check the following letter. (Consider end-of-word
as letter here.) Count the occurrences of the same following letters.
5. Pick the next letter of the generated name with probability of the distribution in
existing names.
6. If the next letter is not end-of-word, start over from step 3, always considering
the last m letters of the current name.
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Example (m = 2)
1. Pick any existing name, with equal probability.
LILY
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Example (m = 2, name = “LI”, current = “LI”)
2. Take the first m letters of that name.
LI
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AMELIA, OLIVIA, LILY, ALICE, ELISABETH, LILAH, JULIET,
CAROLINE, EVANGELINE, MADELINE, NATELIE, ROSALIE, LILLIAN,
ELISE, ADELINE, DELILAH, ELIANA, FELICITY, JULIA
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4. In all of these existing names, check the following letter. (Consider
end-of-word as letter here.) Count the occurrences of the same
following letters.
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Next Letter Occurrences Probability
A 3 16 %
V 1 5 %
L 4 21 %
C 2 11 %
S 2 11 %
E 3 16 %
N 4 21 %

Example (m = 2, name = “LIN”, current = “LI”)
5. Pick the next letter of the generated name with probability of the
distribution in existing names.
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A 3 16 %
V 1 5 %
L 4 21 %
C 2 11 %
S 2 11 %
E 3 16 %
N 4 21 %

Example (m = 2, name = “LIN”, current = “IN”)
2. Take the last m letters of the generated name.
IN
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CAROLINE, EVANGELINE, MADELINE, JOSEPHINE, ADELINE,
QUINN
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following letters.
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E 5 83 %
N 1 17 %

Example (m = 2, name = “LINN”, current = “IN”)
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E 5 83 %
N 1 17 %

Example (m = 2, name = “LINN”, current = “NN”)
2. Take the last m letters of the generated name.
NN
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ARIANNA, HANNAH, ANNA, SAVANNAH, ANNABELLE, QUINN,
SIENNA
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following letters.
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A 6 86 %
End-of-word 1 14 %

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A 6 86 %
End-of-word 1 14 %

Example (m = 2)
Generated Name:
LINN
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References
• Togelius, Yannakakis, Stanley, Browne. Search-based procedural content generation. In
EvoApplications Workshop, volume 2024 of LNCS, pages 141150, November 2010.
• Andrew Doull. The death of the level designer: Procedural content generation in games.
http://roguelikedeveloper.blogspot.de/2008/01/death-of-level-designer-procedural.html, January
2008.
• Peter Mawhorter and Michael Mateas. Procedural level generation using occupancy-regulated
extension. In CIG, pages 351358, 2010.
• Press, Teukolsky, Vetterling, Flannery. Numerical Recipes 3rd Edition: The Art of Scientic
Computing. Cambridge University Press, New York, NY, USA, 2007.
• Ince. Automatic Dynamic Content Generation for Computer Games. PhD thesis, University of
Sheeld, 1999.
• Compton, Mateas. Procedural level design for platform games. In Proceedings of the 2nd Articial
Intelligence and Interactive Digital Entertainment Conference (AIIDE), pages 109111, Marina del
Rey, California, June 2006.

References
• Laskov. Level generation system for platform games based on a reinforcement learning approach.
Master's thesis, The University of Edinburgh, School of Informatics, 2009.
• Sorenson, Pasquier. Towards a generic framework for automated video game level creation. In
EvoApplications (1), pages 131140, 2010.
• Prühs, Vaz Alves. Towards a Generic Framework for Procedural Generation of Game Levels.
Master’s Thesis, Hamburg University of Applied Sciences, 2011.
• Perlin. Noise and Turbulence. http://www.mrl.nyu.edu/~perlin/doc/oscar.html, 1997.
• Perlin. Making Noise. http://www.noisemachine.com/talk1/index.html, December 9, 1999.
• Zucker. The Perlin noise math FAQ. http://webstaff.itn.liu.se/~stegu/TNM022-
2005/perlinnoiselinks/perlin-noise-math-faq.html#toc-algorithm, February 2001.
• Wikipedia. Markow chain. http://en.wikipedia.org/wiki/Markov_chain, September 6, 2014.
• Silicon Commader Games. Markow Name Generator.
http://www.siliconcommandergames.com/MarkovNameGenerator.htm, May 2016.

Thank you!
http://www.npruehs.de
https://github.com/npruehs
@npruehs
nick.pruehs@daedalic.com

5 Minute Review Session
• Which categories of procedural content generation do you know?
• Name a few opportunities of PCG!
• Name a few challenges of using generated content!
• In a few words, explain the chunk-based level generation approach!
• What is coherent noise, and what can it be used for?
• What are Markov chains, and what can they be used for?

Procedural Game Level Generation

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