Solve day 9 part 2 using a sweep line algorithm

Author: Ben Rampling

src/year2025/day09.rs

@@ -1,13 +1,5 @@
-use crate::util::grid::*;
-use crate::util::hash::*;
 use crate::util::iter::*;
 use crate::util::parse::*;
-use crate::util::point::*;
-use std::collections::VecDeque;
-
-const OUTSIDE: i64 = 0;
-const INSIDE: i64 = 1;
-const UNKNOWN: i64 = 2;
 
 type Tile = [u64; 2];
 
@@ -30,75 +22,140 @@ pub fn part1(tiles: &[Tile]) -> u64 {
 }
 
 pub fn part2(tiles: &[Tile]) -> u64 {
-    let size = tiles.len();
-    let shrink_x = shrink(tiles, 0);
-    let shrink_y = shrink(tiles, 1);
-    let shrunk: Vec<_> = tiles.iter().map(|&[x, y]| (shrink_x[&x], shrink_y[&y])).collect();
+    let mut tiles = tiles.to_vec();
 
-    let mut area = 0;
-    let mut todo = VecDeque::from([ORIGIN]);
-    let mut grid = Grid::new(shrink_x.len() as i32, shrink_y.len() as i32, UNKNOWN);
+    tiles.sort_unstable_by_key(|&[x, y]| (y, x));
 
-    for i in 0..size {
-        let (x1, y1, x2, y2) = minmax(shrunk[i], shrunk[(i + 1) % size]);
+    let tiles = tiles;
 
-        for x in x1..x2 + 1 {
-            for y in y1..y2 + 1 {
-                grid[Point::new(x, y)] = INSIDE;
-            }
-        }
+    // Track the largest area so far during scanning:
+    let mut largest_area: u64 = 0;
+
+    // Each red tile (`x`, `y`) becomes a candidate for being a top corner of the largest area, and during the 
+    // scan the `interval` containing the maximum possible width is updated:
+    struct Candidate {
+        x: u64,
+        y: u64,
+        interval: Interval,
     }
 
-    while let Some(point) = todo.pop_front() {
-        for next in ORTHOGONAL.map(|o| point + o) {
-            if grid.contains(next) && grid[next] == UNKNOWN {
-                grid[next] = OUTSIDE;
-                todo.push_back(next);
+    let mut candidates: Vec<Candidate> = Vec::with_capacity(512);
+    
+    // Maintain an ordered list of descending edges, i.e. [begin_interval_0, end_interval_0, begin_interval_1, end_interval_1, ...]:
+    let mut descending_edges: Vec<u64> = vec![];
+    let mut intervals_from_descending_edges = vec![];
+
+    // Invariants on the input data (defined by the puzzle) result in points arriving in pairs on the same y line:
+    let mut it = tiles.into_iter();
+
+    while let (Some([x0, y]), Some([x1, y1])) = (it.next(), it.next()) {
+        debug_assert_eq!(y, y1);
+    
+        // Update the descending edges; since we are scanning from top to bottom, and within each line left to right,
+        // when we, starting from outside of the region, hit a corner tile it is either:
+        //
+        // - The corner of two edges, one going right and one going down. In this case, the `descending_edges` won't contain
+        //   the `x` coordinate, and we should "toggle" it on to denote that there is a new descending edge.
+        // - The corner of two edges, one going right and one going up. The `descending_edges` will contain an `x` coordinate, 
+        //   that should be "toggled" off.
+        //
+        // Simular arguments work for when we are scanning inside the edge and we hit the corner that ends the edge; this is also
+        // why corners always arrive in pairs.
+        //
+        // Do the update:
+        for x in [x0, x1] {
+            toggle_value_membership_in_ordered_list(&mut descending_edges, x);
+        }
+    
+        // Every pair of descending edges in the ordered list defines a region; find the resulting intervals on this line:
+        update_intervals_from_descending_edges(&descending_edges, &mut intervals_from_descending_edges);
+    
+        // Check the rectangles this red tile could be a bottom tile for, with the current candidates:
+        for candidate in candidates.iter() {
+            for x in [x0, x1] {
+                if candidate.interval.contains(x) {
+                    largest_area = largest_area.max((candidate.x.abs_diff(x) + 1) * (candidate.y.abs_diff(y) + 1));
+                }
+            }
+        }
+    
+        // Update candidates when their interval shrinks due to descending edge changes, and drop them when their interval becomes empty:
+        candidates.retain_mut(|candidate| {
+            if let Some(intersection_containing_x) = intervals_from_descending_edges.iter().find(|i| i.contains(candidate.x)) {
+                candidate.interval = intersection_containing_x.intersection(candidate.interval);
+
+                true
+            } else {
+                false
+            }
+        });
+        
+        // Add any new candidates:
+        for x in [x0, x1] {
+            if let Some(&containing) = intervals_from_descending_edges.iter().find(|i| i.contains(x)) {
+                candidates.push(Candidate { x, y, interval: containing });
             }
         }
     }
 
-    for y in 1..grid.height {
-        for x in 1..grid.width {
-            let point = Point::new(x, y);
-            let value = i64::from(grid[point] != OUTSIDE);
-            grid[point] = value + grid[point + UP] + grid[point + LEFT] - grid[point + UP + LEFT];
-        }
+    largest_area
+}
+
+/// The set { x in u64 | l <= x <= r }.
+#[derive(Clone, Copy)]
+struct Interval {
+    l: u64,
+    r: u64,
+}
+
+impl Interval {
+    fn new(l: u64, r: u64) -> Self {
+        debug_assert!(l <= r);
+
+        Interval { l, r }
     }
 
-    for i in 0..size {
-        for j in i + 1..size {
-            let (x1, y1, x2, y2) = minmax(shrunk[i], shrunk[j]);
-
-            let expected = (x2 - x1 + 1) as i64 * (y2 - y1 + 1) as i64;
-            let actual = grid[Point::new(x2, y2)]
-                - grid[Point::new(x1 - 1, y2)]
-                - grid[Point::new(x2, y1 - 1)]
-                + grid[Point::new(x1 - 1, y1 - 1)];
-
-            if expected == actual {
-                let [x1, y1] = tiles[i];
-                let [x2, y2] = tiles[j];
-                let dx = x1.abs_diff(x2) + 1;
-                let dy = y1.abs_diff(y2) + 1;
-                area = area.max(dx * dy);
-            }
-        }
+    fn intersects(self, other: Self) -> bool {
+        other.l <= self.r && self.l <= other.r
     }
 
-    area
+    fn intersection(self, other: Self) -> Self {
+        debug_assert!(self.intersects(other));
+
+        Interval::new(self.l.max(other.l), self.r.min(other.r))
+    }
+
+    fn contains(self, x: u64) -> bool {
+        self.l <= x && x <= self.r
+    }
 }
 
-fn shrink(tiles: &[Tile], index: usize) -> FastMap<u64, i32> {
-    let mut axis: Vec<_> = tiles.iter().map(|tile| tile[index]).collect();
-    axis.push(u64::MIN);
-    axis.push(u64::MAX);
-    axis.sort_unstable();
-    axis.dedup();
-    axis.iter().enumerate().map(|(i, &n)| (n, i as i32)).collect()
+// Adds `value` if it isn't in `ordered_list`, removes it if it is, maintaining the order.
+fn toggle_value_membership_in_ordered_list(ordered_list: &mut Vec<u64>, value: u64) {
+    let mut i = 0;
+
+    while i < ordered_list.len() && ordered_list[i] < value {
+        i += 1;
+    }
+    
+    if i == ordered_list.len() || ordered_list[i] != value {
+        ordered_list.insert(i, value);
+    } else {
+        ordered_list.remove(i);
+    }
 }
 
+// Changes the list of descending edges, [begin_interval_0, end_interval_0, begin_interval_1, end_interval_1, ...],
+// into a vector containing the intervals.
 #[inline]
-fn minmax((x1, y1): (i32, i32), (x2, y2): (i32, i32)) -> (i32, i32, i32, i32) {
-    (x1.min(x2), y1.min(y2), x1.max(x2), y1.max(y2))
+fn update_intervals_from_descending_edges(descending_edges: &[u64], to_update: &mut Vec<Interval>) {
+    debug_assert!(descending_edges.len().is_multiple_of(2));
+
+    to_update.clear();
+
+    let mut it = descending_edges.iter();
+
+    while let (Some(&l), Some(&r)) = (it.next(), it.next()) {
+        to_update.push(Interval::new(l, r));
+    }
 }