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Addressing PR issues.

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This commit is contained in:
Joshua Yanovski 2019-08-22 23:48:13 +02:00
parent ba1a146cd5
commit 2ff192021c
3 changed files with 170 additions and 159 deletions
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1
world/src/lib.rs
View File

@ -1,6 +1,5 @@
#![deny(unsafe_code)]
#![feature(
box_syntax,
const_generics,
euclidean_division,
bind_by_move_pattern_guards,

167
world/src/sim/mod.rs
View File

@ -1,9 +1,13 @@
mod util;
mod location;
mod settlement;
// Reexports
pub use self::location::Location;
pub use self::settlement::Settlement;
use self::util::{
cdf_irwin_hall, InverseCdf, uniform_idx_as_vec2, uniform_noise,
};
use crate::{
all::ForestKind,
@ -27,161 +31,6 @@ use vek::*;
pub const WORLD_SIZE: Vec2<usize> = Vec2 { x: 1024, y: 1024 };
/// Computes the cumulative distribution function of the weighted sum of k independent,
/// uniformly distributed random variables between 0 and 1. For each variable i, we use weights[i]
/// as the weight to give samples[i] (the weights should all be positive).
///
/// If the precondition is met, the distribution of the result of calling this function will be
/// uniformly distributed while preserving the same information that was in the original average.
///
/// For N > 33 the function will no longer return correct results since we will overflow u32.
///
/// NOTE:
///
/// Per [1], the problem of determing the CDF of
/// the sum of uniformly distributed random variables over *different* ranges is considerably more
/// complicated than it is for the same-range case. Fortunately, it also provides a reference to
/// [2], which contains a complete derivation of an exact rule for the density function for
/// this case. The CDF is just the integral of the cumulative distribution function [3],
/// which we use to convert this into a CDF formula.
///
/// This allows us to sum weighted, uniform, independent random variables.
///
/// At some point, we should probably contribute this back to stats-rs.
///
/// 1. https://www.r-bloggers.com/sums-of-random-variables/,
/// 2. Sadooghi-Alvandi, S., A. Nematollahi, & R. Habibi, 2009.
/// On the Distribution of the Sum of Independent Uniform Random Variables.
/// Statistical Papers, 50, 171-175.
/// 3. hhttps://en.wikipedia.org/wiki/Cumulative_distribution_function
fn cdf_irwin_hall<const N: usize>(weights: &[f32; N], samples: [f32; N]) -> f32 {
// Let J_k = {(j_1, ... , j_k) : 1 ≤ j_1 < j_2 < ··· < j_k ≤ N }.
//
// Let A_N = Π{k = 1 to n}a_k.
//
// The density function for N ≥ 2 is:
//
// 1/(A_N * (N - 1)!) * (x^(N-1) + Σ{k = 1 to N}((-1)^k *
// Σ{(j_1, ..., j_k) ∈ J_k}(max(0, x - Σ{l = 1 to k}(a_(j_l)))^(N - 1))))
//
// So the cumulative distribution function is its integral, i.e. (I think)
//
// 1/(product{k in A}(k) * N!) * (x^N + sum(k in 1 to N)((-1)^k *
// sum{j in Subsets[A, {k}]}(max(0, x - sum{l in j}(l))^N)))
//
// which is also equivalent to
//
// (letting B_k = { a in Subsets[A, {k}] : sum {l in a} l }, B_(0,1) = 0 and
// H_k = { i : 1 ≤ 1 ≤ N! / (k! * (N - k)!) })
//
// 1/(product{k in A}(k) * N!) * sum(k in 0 to N)((-1)^k *
// sum{l in H_k}(max(0, x - B_(k,l))^N))
//
// We should be able to iterate through the whole power set
// instead, and figure out K by calling count_ones(), so we can compute the result in O(2^N)
// iterations.
let x: f32 = weights
.iter()
.zip(samples.iter())
.map(|(weight, sample)| weight * sample)
.sum();
let mut y = 0.0f32;
for subset in 0u32..(1 << N) {
// Number of set elements
let k = subset.count_ones();
// Add together exactly the set elements to get B_subset
let z = weights
.iter()
.enumerate()
.filter(|(i, _)| subset & (1 << i) as u32 != 0)
.map(|(_, k)| k)
.sum::<f32>();
// Compute max(0, x - B_subset)^N
let z = (x - z).max(0.0).powi(N as i32);
// The parity of k determines whether the sum is negated.
y += if k & 1 == 0 { z } else { -z };
}
// Divide by the product of the weights.
y /= weights.iter().product::<f32>();
// Remember to multiply by 1 / N! at the end.
y / (1..=N as i32).product::<i32>() as f32
}
/// First component of each element of the vector is the computed CDF of the noise function at this
/// index (i.e. its position in a sorted list of value returned by the noise function applied to
/// every chunk in the game). Second component is the cached value of the noise function that
/// generated the index.
type InverseCdf = Box<[(f32, f32); WORLD_SIZE.x * WORLD_SIZE.y]>;
/// Computes the position Vec2 of a SimChunk from an index, where the index was generated by
/// uniform_noise.
fn uniform_idx_as_vec2(idx: usize) -> Vec2<i32> {
Vec2::new((idx / WORLD_SIZE.x) as i32, (idx % WORLD_SIZE.x) as i32)
}
/// Compute inverse cumulative distribution function for arbitrary function f, the hard way. We
/// pre-generate noise values prior to worldgen, then sort them in order to determine the correct
/// position in the sorted order. That lets us use `(index + 1) / (WORLDSIZE.y * WORLDSIZE.x)` as
/// a uniformly distributed (from almost-0 to 1) regularization of the chunks. That is, if we
/// apply the computed "function" F⁻¹(x, y) to (x, y) and get out p, it means that approximately
/// (100 * p)% of chunks have a lower value for F⁻¹ than p. The main purpose of doing this is to
/// make sure we are using the entire range we want, and to allow us to apply the numerous results
/// about distributions on uniform functions to the procedural noise we generate, which lets us
/// much more reliably control the *number* of features in the world while still letting us play
/// with the *shape* of those features, without having arbitrary cutoff points / discontinuities
/// (which tend to produce ugly-looking / unnatural terrain).
///
/// As a concrete example, before doing this it was very hard to tweak humidity so that either most
/// of the world wasn't dry, or most of it wasn't wet, by combining the billow noise function and
/// the computed altitude. This is because the billow noise function has a very unusual
/// distribution that is heavily skewed towards 0. By correcting for this tendency, we can start
/// with uniformly distributed billow noise and altitudes and combine them to get uniformly
/// distributed humidity, while still preserving the existing shapes that the billow noise and
/// altitude functions produce.
///
/// f takes an index, which represents the index corresponding to this chunk in any any SimChunk
/// vector returned by uniform_noise, and (for convenience) the float-translated version of those
/// coordinates.
/// f should return a value with no NaNs. If there is a NaN, it will panic. There are no other
/// conditions on f.
///
/// Returns a vec of (f32, f32) pairs consisting of the percentage of chunks with a value lower than
/// this one, and the actual noise value (we don't need to cache it, but it makes ensuring that
/// subsequent code that needs the noise value actually uses the same one we were using here
/// easier).
fn uniform_noise(f: impl Fn(usize, Vec2<f64>) -> f32) -> InverseCdf {
let mut noise = (0..WORLD_SIZE.x * WORLD_SIZE.y)
.map(|i| {
(
i,
f(
i,
(uniform_idx_as_vec2(i) * TerrainChunkSize::SIZE.map(|e| e as i32))
.map(|e| e as f64),
),
)
})
.collect::<Vec<_>>();
// sort_unstable_by is equivalent to sort_by here since we include the index in the
// comparison. We could leave out the index, but this might make the order not
// reproduce the same way between different versions of Rust (for example).
noise.sort_unstable_by(|f, g| (f.1, f.0).partial_cmp(&(g.1, g.0)).unwrap());
// Construct a vector that associates each chunk position with the 1-indexed
// position of the noise in the sorted vector (divided by the vector length).
// This guarantees a uniform distribution among the samples.
let mut uniform_noise = box [(0.0, 0.0); WORLD_SIZE.x * WORLD_SIZE.y];
let total = (WORLD_SIZE.x * WORLD_SIZE.y) as f32;
for (noise_idx, (chunk_idx, noise_val)) in noise.into_iter().enumerate() {
uniform_noise[chunk_idx] = ((1 + noise_idx) as f32 / total, noise_val);
}
uniform_noise
}
/// Calculates the smallest distance along an axis (x, y) from an edge of
/// the world. This value is maximal at WORLD_SIZE / 2 and minimized at the extremes
/// (0 or WORLD_SIZE on one or more axes). It then divides the quantity by cell_size,
@ -197,6 +46,9 @@ fn map_edge_factor(posi: usize) -> f32 {
.min(1.0)
}
/// A structure that holds cached noise values and cumulative distribution functions for the input
/// that led to those values. See the definition of InverseCdf for a description of how to
/// interpret the types of its fields.
struct GenCdf {
humid_base: InverseCdf,
temp_base: InverseCdf,
@ -666,7 +518,7 @@ impl SimChunk {
const HUMID_WEIGHTS: [f32; 2] = [1.0, 1.0];
let humidity = cdf_irwin_hall(&HUMID_WEIGHTS, [humid_base, 1.0 - alt_uniform]);
let (temp_base, temp_old) = gen_cdf.temp_base[posi];
let (temp_base, _) = gen_cdf.temp_base[posi];
// We also correlate temperature negatively with altitude using different weighting than we
// use for humidity.
@ -695,8 +547,6 @@ impl SimChunk {
let logistic_2_base = 3.0f32.sqrt().mul(f32::consts::FRAC_2_PI);
// Assumes μ = 0, σ = 1
let logistic_cdf = |x: f32| x.div(logistic_2_base).tanh().mul(0.5).add(0.5);
// Weighted logit sum.
let f = |humidity, density| logistic_cdf(logit(humidity) + 0.5 * logit(density));
// No trees in the ocean or with zero humidity (currently)
let tree_density = if alt <= CONFIG.sea_level + 5.0 {
@ -716,6 +566,7 @@ impl SimChunk {
} else if humidity >= 1.0 || tree_density >= 1.0 {
1.0
} else {
// Weighted logit sum.
logistic_cdf(logit(humidity) + 0.5 * logit(tree_density))
}
// rescale to (-0.9, 0.9)

161
world/src/sim/util.rs Normal file
View File

@ -0,0 +1,161 @@
use common::{
terrain::TerrainChunkSize,
vol::VolSize,
};
use super::WORLD_SIZE;
use vek::*;
/// Computes the cumulative distribution function of the weighted sum of k independent,
/// uniformly distributed random variables between 0 and 1. For each variable i, we use weights[i]
/// as the weight to give samples[i] (the weights should all be positive).
///
/// If the precondition is met, the distribution of the result of calling this function will be
/// uniformly distributed while preserving the same information that was in the original average.
///
/// For N > 33 the function will no longer return correct results since we will overflow u32.
///
/// NOTE:
///
/// Per [1], the problem of determing the CDF of
/// the sum of uniformly distributed random variables over *different* ranges is considerably more
/// complicated than it is for the same-range case. Fortunately, it also provides a reference to
/// [2], which contains a complete derivation of an exact rule for the density function for
/// this case. The CDF is just the integral of the cumulative distribution function [3],
/// which we use to convert this into a CDF formula.
///
/// This allows us to sum weighted, uniform, independent random variables.
///
/// At some point, we should probably contribute this back to stats-rs.
///
/// 1. https://www.r-bloggers.com/sums-of-random-variables/,
/// 2. Sadooghi-Alvandi, S., A. Nematollahi, & R. Habibi, 2009.
/// On the Distribution of the Sum of Independent Uniform Random Variables.
/// Statistical Papers, 50, 171-175.
/// 3. hhttps://en.wikipedia.org/wiki/Cumulative_distribution_function
pub fn cdf_irwin_hall<const N: usize>(weights: &[f32; N], samples: [f32; N]) -> f32 {
// Let J_k = {(j_1, ... , j_k) : 1 ≤ j_1 < j_2 < ··· < j_k ≤ N }.
//
// Let A_N = Π{k = 1 to n}a_k.
//
// The density function for N ≥ 2 is:
//
// 1/(A_N * (N - 1)!) * (x^(N-1) + Σ{k = 1 to N}((-1)^k *
// Σ{(j_1, ..., j_k) ∈ J_k}(max(0, x - Σ{l = 1 to k}(a_(j_l)))^(N - 1))))
//
// So the cumulative distribution function is its integral, i.e. (I think)
//
// 1/(product{k in A}(k) * N!) * (x^N + sum(k in 1 to N)((-1)^k *
// sum{j in Subsets[A, {k}]}(max(0, x - sum{l in j}(l))^N)))
//
// which is also equivalent to
//
// (letting B_k = { a in Subsets[A, {k}] : sum {l in a} l }, B_(0,1) = 0 and
// H_k = { i : 1 ≤ 1 ≤ N! / (k! * (N - k)!) })
//
// 1/(product{k in A}(k) * N!) * sum(k in 0 to N)((-1)^k *
// sum{l in H_k}(max(0, x - B_(k,l))^N))
//
// We should be able to iterate through the whole power set
// instead, and figure out K by calling count_ones(), so we can compute the result in O(2^N)
// iterations.
let x: f32 = weights
.iter()
.zip(samples.iter())
.map(|(weight, sample)| weight * sample)
.sum();
let mut y = 0.0f32;
for subset in 0u32..(1 << N) {
// Number of set elements
let k = subset.count_ones();
// Add together exactly the set elements to get B_subset
let z = weights
.iter()
.enumerate()
.filter(|(i, _)| subset & (1 << i) as u32 != 0)
.map(|(_, k)| k)
.sum::<f32>();
// Compute max(0, x - B_subset)^N
let z = (x - z).max(0.0).powi(N as i32);
// The parity of k determines whether the sum is negated.
y += if k & 1 == 0 { z } else { -z };
}
// Divide by the product of the weights.
y /= weights.iter().product::<f32>();
// Remember to multiply by 1 / N! at the end.
y / (1..=N as i32).product::<i32>() as f32
}
/// First component of each element of the vector is the computed CDF of the noise function at this
/// index (i.e. its position in a sorted list of value returned by the noise function applied to
/// every chunk in the game). Second component is the cached value of the noise function that
/// generated the index.
pub type InverseCdf = Box<[(f32, f32); WORLD_SIZE.x * WORLD_SIZE.y]>;
/// Computes the position Vec2 of a SimChunk from an index, where the index was generated by
/// uniform_noise.
pub fn uniform_idx_as_vec2(idx: usize) -> Vec2<i32> {
Vec2::new((idx / WORLD_SIZE.x) as i32, (idx % WORLD_SIZE.x) as i32)
}
/// Compute inverse cumulative distribution function for arbitrary function f, the hard way. We
/// pre-generate noise values prior to worldgen, then sort them in order to determine the correct
/// position in the sorted order. That lets us use `(index + 1) / (WORLDSIZE.y * WORLDSIZE.x)` as
/// a uniformly distributed (from almost-0 to 1) regularization of the chunks. That is, if we
/// apply the computed "function" F⁻¹(x, y) to (x, y) and get out p, it means that approximately
/// (100 * p)% of chunks have a lower value for F⁻¹ than p. The main purpose of doing this is to
/// make sure we are using the entire range we want, and to allow us to apply the numerous results
/// about distributions on uniform functions to the procedural noise we generate, which lets us
/// much more reliably control the *number* of features in the world while still letting us play
/// with the *shape* of those features, without having arbitrary cutoff points / discontinuities
/// (which tend to produce ugly-looking / unnatural terrain).
///
/// As a concrete example, before doing this it was very hard to tweak humidity so that either most
/// of the world wasn't dry, or most of it wasn't wet, by combining the billow noise function and
/// the computed altitude. This is because the billow noise function has a very unusual
/// distribution that is heavily skewed towards 0. By correcting for this tendency, we can start
/// with uniformly distributed billow noise and altitudes and combine them to get uniformly
/// distributed humidity, while still preserving the existing shapes that the billow noise and
/// altitude functions produce.
///
/// f takes an index, which represents the index corresponding to this chunk in any any SimChunk
/// vector returned by uniform_noise, and (for convenience) the float-translated version of those
/// coordinates.
/// f should return a value with no NaNs. If there is a NaN, it will panic. There are no other
/// conditions on f.
///
/// Returns a vec of (f32, f32) pairs consisting of the percentage of chunks with a value lower than
/// this one, and the actual noise value (we don't need to cache it, but it makes ensuring that
/// subsequent code that needs the noise value actually uses the same one we were using here
/// easier).
pub fn uniform_noise(f: impl Fn(usize, Vec2<f64>) -> f32) -> InverseCdf {
let mut noise = (0..WORLD_SIZE.x * WORLD_SIZE.y)
.map(|i| {
(
i,
f(
i,
(uniform_idx_as_vec2(i) * TerrainChunkSize::SIZE.map(|e| e as i32))
.map(|e| e as f64),
),
)
})
.collect::<Vec<_>>();
// sort_unstable_by is equivalent to sort_by here since we include the index in the
// comparison. We could leave out the index, but this might make the order not
// reproduce the same way between different versions of Rust (for example).
noise.sort_unstable_by(|f, g| (f.1, f.0).partial_cmp(&(g.1, g.0)).unwrap());
// Construct a vector that associates each chunk position with the 1-indexed
// position of the noise in the sorted vector (divided by the vector length).
// This guarantees a uniform distribution among the samples.
let mut uniform_noise = Box::new([(0.0, 0.0); WORLD_SIZE.x * WORLD_SIZE.y]);
let total = (WORLD_SIZE.x * WORLD_SIZE.y) as f32;
for (noise_idx, (chunk_idx, noise_val)) in noise.into_iter().enumerate() {
uniform_noise[chunk_idx] = ((1 + noise_idx) as f32 / total, noise_val);
}
uniform_noise
}