diff --git a/crates/trace/src/centerline/mod.rs b/crates/trace/src/centerline/mod.rs
index 7999f655f..332c4499c 100644
--- a/crates/trace/src/centerline/mod.rs
+++ b/crates/trace/src/centerline/mod.rs
@@ -65,6 +65,9 @@ pub(crate) fn lines(
     let mask = |i: usize| labels.get(i) == Some(&label);
     let mut seen = vec![false; w * h];
     let mut stack = vec![];
+    // Thin shapes nested in one another have overlapping boxes: the cells of all of them together
+    // are bounded, so the work is too (a component past it stays as it is, not a line).
+    let mut cells_left = n.saturating_mul(8).max(1 << 22);
     for start in 0..w * h {
         if !mask(start) || seen.get(start).copied().unwrap_or(false) {
             continue;
@@ -86,6 +89,8 @@ pub(crate) fn lines(
             }
         }
         let Some(g) = Grid::from_component(&comp, w) else { continue };
+        let Some(left) = cells_left.checked_sub(g.w.saturating_mul(g.h)) else { break };
+        cells_left = left;
         let Some(dt) = distance::squared(&g.on, g.w, g.h) else { continue };
         let widest = 2.0 * f64::from(dt.iter().copied().fold(0.0f32, f32::max)).sqrt() - 1.0;
         if widest > max_width {
diff --git a/crates/trace/src/group.rs b/crates/trace/src/group.rs
index d5b615de6..50b165f1a 100644
--- a/crates/trace/src/group.rs
+++ b/crates/trace/src/group.rs
@@ -73,6 +73,9 @@ fn inside_or_near(p: Point, o: &Outer) -> bool {
     inside
 }
 
+/// The most paths grouped.
+const MAX_GROUPED_PATHS: usize = 20_000;
+
 /// For each of `paths` (bottom-most first), the path it lies inside, if any: the smallest filled
 /// path whose outer contour holds all of its own. The path that covers the whole `w` × `h` image
 /// (the paper) is nobody's parent: everything would be inside it.
@@ -81,6 +84,10 @@ fn inside_or_near(p: Point, o: &Outer) -> bool {
 /// `paths` paints in its own order: where the paths can overlap (`stacked`), a path only gets a parent below it, and only
 /// if no path that would then come after it instead of before it shares any of its box.
 pub(crate) fn parents(paths: &[TracedPath], w: usize, h: usize, stacked: bool) -> Vec<Option<usize>> {
+    // Each path is tested against the others (quadratic): past this many, they stay ungrouped.
+    if paths.len() > MAX_GROUPED_PATHS {
+        return vec![None; paths.len()];
+    }
     let outers: Vec<Option<Outer>> = paths.iter().map(outer).collect();
     let paper = Rect::new(1.0, 1.0, w as f64 - 1.0, h as f64 - 1.0);
     let mut parent: Vec<Option<usize>> = vec![None; paths.len()];
diff --git a/crates/trace/src/lib.rs b/crates/trace/src/lib.rs
index d9dfeab21..1e3849bf4 100644
--- a/crates/trace/src/lib.rs
+++ b/crates/trace/src/lib.rs
@@ -516,7 +516,7 @@ pub fn trace(img: &Raster, params: &TraceParams) -> TraceResult {
 /// colour layers are traced in parallel where threads are available.
 pub fn trace_within(img: &Raster, params: &TraceParams, max_anchors: usize) -> Result<TraceResult, TraceError> {
     let (w, h) = (img.width as usize, img.height as usize);
-    if w == 0 || h == 0 || img.rgba.len() != w * h * 4 {
+    if w == 0 || h == 0 || w.checked_mul(h).and_then(|n| n.checked_mul(4)) != Some(img.rgba.len()) {
         return Ok(TraceResult::default());
     }
     let mut out = trace_paths(img, params, max_anchors)?;
diff --git a/crates/trace/src/logo/pipeline.rs b/crates/trace/src/logo/pipeline.rs
index 887989b5d..6004af7f4 100644
--- a/crates/trace/src/logo/pipeline.rs
+++ b/crates/trace/src/logo/pipeline.rs
@@ -342,7 +342,7 @@ fn choose_palette(params: &TraceParams, pix: &[Px], w: usize, h: usize) -> Resul
 /// Trace a flat-colour logo.
 pub(crate) fn trace_logo(img: &Raster, params: &TraceParams, max_anchors: usize) -> Result<TraceResult, TraceError> {
     let (w0, h0) = (img.width as usize, img.height as usize);
-    if w0 == 0 || h0 == 0 || img.rgba.len() != w0 * h0 * 4 {
+    if w0 == 0 || h0 == 0 || w0.checked_mul(h0).and_then(|n| n.checked_mul(4)) != Some(img.rgba.len()) {
         return Ok(TraceResult::default());
     }
     let (mut pix, mut w, mut h, mut factor) = working_copy(img, 1);
diff --git a/crates/trace/src/mosaic.rs b/crates/trace/src/mosaic.rs
index ac2ae3051..b8bd46a37 100644
--- a/crates/trace/src/mosaic.rs
+++ b/crates/trace/src/mosaic.rs
@@ -2,38 +2,72 @@
 
 use crate::Raster;
 
+/// The most tiles along a side.
+const MAX_SIDE: u32 = 4096;
+
 /// Average colour (straight RGBA) of each of `cols` × `rows` tiles covering `img`, row by row.
-/// Transparent pixels count by their alpha, so a tile over transparency fades out.
+/// Transparent pixels count by their alpha, so a tile over transparency fades out. Tiles over the
+/// same pixels (more tiles than pixels) are averaged once, and a raster whose bytes are not
+/// `width * height * 4` gives transparent tiles.
 pub fn mosaic(img: &Raster, cols: u32, rows: u32) -> Vec<[u8; 4]> {
-    let (cols, rows) = (cols.max(1), rows.max(1));
-    let mut out = Vec::with_capacity((cols * rows) as usize);
+    let (cols, rows) = (cols.clamp(1, MAX_SIDE), rows.clamp(1, MAX_SIDE));
+    let tiles = cols as usize * rows as usize;
+    let (w, h) = (img.width as usize, img.height as usize);
+    if w.checked_mul(h).and_then(|n| n.checked_mul(4)) != Some(img.rgba.len()) {
+        return vec![[0; 4]; tiles];
+    }
+    let mut out = Vec::with_capacity(tiles);
+    // The pixel rows of the last row of tiles, and where its tiles start in `out`.
+    let mut last_row: Option<((u32, u32), usize)> = None;
     for r in 0..rows {
         let y0 = (r as u64 * img.height as u64 / rows as u64) as u32;
         let y1 = (((r + 1) as u64 * img.height as u64 / rows as u64) as u32).max(y0 + 1).min(img.height);
+        if let Some((range, start)) = last_row
+            && range == (y0, y1)
+        {
+            out.extend_from_within(start..start + cols as usize);
+            continue;
+        }
+        let start = out.len();
+        let mut last: Option<((u32, u32), [u8; 4])> = None;
         for c in 0..cols {
             let x0 = (c as u64 * img.width as u64 / cols as u64) as u32;
             let x1 = (((c + 1) as u64 * img.width as u64 / cols as u64) as u32).max(x0 + 1).min(img.width);
-            let (mut sr, mut sg, mut sb, mut sa, mut n) = (0u64, 0u64, 0u64, 0u64, 0u64);
-            for y in y0..y1.max(y0 + 1).min(img.height) {
-                for x in x0..x1.max(x0 + 1).min(img.width) {
-                    let [pr, pg, pb, pa] = img.pixel(x, y);
-                    let a = pa as u64;
-                    sr += pr as u64 * a;
-                    sg += pg as u64 * a;
-                    sb += pb as u64 * a;
-                    sa += a;
-                    n += 1;
-                }
+            if let Some((range, tile)) = last
+                && range == (x0, x1)
+            {
+                out.push(tile);
+                continue;
             }
-            out.push(match (sr.checked_div(sa), sg.checked_div(sa), sb.checked_div(sa)) {
-                (Some(r), Some(g), Some(b)) => [r as u8, g as u8, b as u8, (sa / n.max(1)) as u8],
-                _ => [0, 0, 0, 0],
-            });
+            let tile = average(img, (x0, x1), (y0, y1));
+            out.push(tile);
+            last = Some(((x0, x1), tile));
         }
+        last_row = Some(((y0, y1), start));
     }
     out
 }
 
+/// The average colour of pixels `xs` × `ys` (half-open).
+fn average(img: &Raster, (x0, x1): (u32, u32), (y0, y1): (u32, u32)) -> [u8; 4] {
+    let (mut sr, mut sg, mut sb, mut sa, mut n) = (0u64, 0u64, 0u64, 0u64, 0u64);
+    for y in y0..y1.max(y0 + 1).min(img.height) {
+        for x in x0..x1.max(x0 + 1).min(img.width) {
+            let [pr, pg, pb, pa] = img.pixel(x, y);
+            let a = pa as u64;
+            sr += pr as u64 * a;
+            sg += pg as u64 * a;
+            sb += pb as u64 * a;
+            sa += a;
+            n += 1;
+        }
+    }
+    match (sr.checked_div(sa), sg.checked_div(sa), sb.checked_div(sa)) {
+        (Some(r), Some(g), Some(b)) => [r as u8, g as u8, b as u8, (sa / n.max(1)) as u8],
+        _ => [0, 0, 0, 0],
+    }
+}
+
 #[cfg(test)]
 mod tests {
     use super::*;
@@ -49,6 +83,12 @@ mod tests {
         assert!((m[0][0] as i32 - 127).abs() <= 1 && (m[0][2] as i32 - 127).abs() <= 1);
         // More tiles than pixels still works.
         assert_eq!(mosaic(&img, 20, 8).len(), 160);
+        // Tiles over the same pixels average the same.
+        let m = mosaic(&img, 20, 8);
+        assert!(m.chunks(20).all(|row| row == &m[..20]));
+        // Bytes that are not the raster's size: transparent.
+        let short = Raster { width: 4, height: 4, rgba: vec![] };
+        assert_eq!(mosaic(&short, 2, 2), vec![[0; 4]; 4]);
         // Transparency.
         let clear = Raster::from_fn(2, 2, |_, _| [9, 9, 9, 0]);
         assert_eq!(mosaic(&clear, 1, 1), vec![[0, 0, 0, 0]]);