Document Year 2021 Day 22

Author: maneatingape

src/year2021/day22.rs

@@ -1,6 +1,35 @@
+//! # Reactor Reboot
+//!
+//! The key to solving this problem efficiently is the
+//! [inclusion-exclusion principle ](https://en.wikipedia.org/wiki/Inclusion%E2%80%93exclusion_principle).
+//!
+//! Looking at a two dimensional example
+//! ```none
+//!    ┌──────────────┐A            Volume of A: 144
+//!    │              │             Volume of B: 66
+//!    │ ┌─────────┐B │             Volume of C: 18
+//!    │ │         │  │
+//!    │ │ ┌────┐C │  │
+//!    │ │ │    │  │  │
+//!    │ │ └────┘  │  │
+//!    │ └─────────┘  │
+//!    └──────────────┘
+//! ```
+//!
+//! Using the inclusion-exclusion principle the remaining size of A is:
+//!
+//! 144 (initial size) - 66 (overlap with B) - 18 (overlap with C) + 18
+//! (overlap between B and C) = 78
+//!
+//! If there were any triple overlaps we would subtract those, add quadruple, and so on until
+//! there are no more overlaps remaining.
+//!
+//! The complexity of this approach depends on how many cubes overlap. In my input most
+//! cubes overlapped with zero others, a few with one and rarely with more than one.
 use crate::util::iter::*;
 use crate::util::parse::*;
 
+/// Wraps a cube with on/off information.
 pub struct RebootStep {
     on: bool,
     cube: Cube,
@@ -14,6 +43,8 @@ impl RebootStep {
     }
 }
 
+/// Technically this is actually a [rectangular cuboid](https://en.wikipedia.org/wiki/Cuboid#Rectangular_cuboid)
+/// but that was longer to type!
 #[derive(Clone, Copy)]
 pub struct Cube {
     x1: i32,
@@ -25,6 +56,8 @@ pub struct Cube {
 }
 
 impl Cube {
+    /// Keeping the coordinates in ascending order per axis makes calculating intersections
+    /// and volume easier.
     fn from(points: [i32; 6]) -> Cube {
         let [a, b, c, d, e, f] = points;
         let x1 = a.min(b);
@@ -36,6 +69,7 @@ impl Cube {
         Cube { x1, x2, y1, y2, z1, z2 }
     }
 
+    /// Returns a `Some` of the intersection if two cubes overlap or `None` if they don't.
     fn intersect(&self, other: &Cube) -> Option<Cube> {
         let x1 = self.x1.max(other.x1);
         let x2 = self.x2.min(other.x2);
@@ -46,6 +80,7 @@ impl Cube {
         (x1 <= x2 && y1 <= y2 && z1 <= z2).then_some(Cube { x1, x2, y1, y2, z1, z2 })
     }
 
+    /// Returns the volume of a cube, converting to `i64` to prevent overflow.
     fn volume(&self) -> i64 {
         let w = (self.x2 - self.x1 + 1) as i64;
         let h = (self.y2 - self.y1 + 1) as i64;
@@ -60,6 +95,8 @@ pub fn parse(input: &str) -> Vec<RebootStep> {
     first.zip(second).map(RebootStep::from).collect()
 }
 
+/// We re-use the logic between part one and two, by first intersecting all cubes with
+/// the specified range. Any cubes that lie completely outside the range will be filtered out.
 pub fn part1(input: &[RebootStep]) -> i64 {
     let region = Cube { x1: -50, x2: 50, y1: -50, y2: 50, z1: -50, z2: 50 };
 
@@ -76,26 +113,39 @@ pub fn part1(input: &[RebootStep]) -> i64 {
 pub fn part2(input: &[RebootStep]) -> i64 {
     let mut total = 0;
     let mut candidates = Vec::new();
+    // Only "on" cubes contribute to volume.
+    // "off" cubes are considered when subtracting volume
     let on_cubes = input.iter().enumerate().filter_map(|(i, rs)| rs.on.then_some((i, rs.cube)));
 
     for (i, cube) in on_cubes {
+        // Only consider cubes after this one in input order.
+        // Previous cubes have already had all possible intersections subtracted from their
+        // volume, so no longer need to be considered.
+        // We check both "on" and "off" cubes when calculating overlaps to subtract volume.
         input[(i + 1)..]
             .iter()
             .filter_map(|rs| cube.intersect(&rs.cube))
             .for_each(|next| candidates.push(next));
 
+        // Apply the inclusion/exclusion principle recursively, considering overlaps of
+        // increasingly higher order until there are no more overlaps remaining.
         total += cube.volume() + subsets(&cube, -1, &candidates);
         candidates.clear();
     }
 
     total
 }
 
+// Apply inclusion/exclusion principle. The sign of the result alternates with each level,
+// so that we subtract single overlaps, then add double, subtract triple, and so on...
 fn subsets(cube: &Cube, sign: i64, candidates: &[Cube]) -> i64 {
     let mut total = 0;
 
     for (i, other) in candidates.iter().enumerate() {
         if let Some(next) = cube.intersect(other) {
+            // Subtle nuance here. Similar to the main input we only need to consider higher level
+            // overlaps of inputs *after* this one, as any overlaps with previous cubes
+            // have already been considered.
             total += sign * next.volume() + subsets(&next, -sign, &candidates[(i + 1)..]);
         }
     }