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Joined 2026-09-21 Message
Incoming links: Half-open interval
Canonical equality of coloured interval rows 2026-10-31
Represent a row by sorted disjoint half-open intervals, each carrying a colour. Merge touching intervals precisely when their colours agree. The resulting ordered maximal runs form a canonical representation of occupancy and colour: equality of those runs is equivalent to pointwise equality, while ignoring arbitrary subdivisions inside one colour run. For two rows containing and intervals, constructing and comparing runs takes time complexity, independently of coordinate magnitude or interval lengths.
Half-open interval 2026-10-31
Let brick have positive integer width . Regard its horizontal occupied region as and its vertical occupied region as . These half-open interval conventions allow adjoining bricks to touch without overlapping. Since the vertical coordinates are integers, two bricks can overlap only when they are on the same row. A suitable predicate isPositive widths and nonnegative integer heights are the underlying domain conditions for a wall built on the baseplate. Distinct members of the set are compared even if their
Brick references happen to be identical: one brick specification can describe several placements. Testing object identity alone would miss two overlapping placements of the same specification.Represent a wall by a list of rows indexed by height. Within each row keep brick placements sorted by their left coordinate, with disjoint occupied half-open intervals. The public
Brick in the paper is a mutable input specification; internal placements snapshot its width and colour, so later client edits cannot invalidate an existing wall. Empty intermediate rows cannot occur in a supportive wall with occupied rows above them.The following Java uses
import java.util.*; and import java.awt.Color;. Keep the source's Brick input software class. All other representation types are private nested software classes of Wall, which prevents clients from inserting directly into a row. The row(int y) query returns a detached row for the equality operation in part (e).final class Wall {
private static final class Placed {
final long x, end;
final Color colour;
Placed(long x, int width, Color colour) {
this.x = x;
this.end = x + width;
this.colour = colour;
}
}
private static final class Row {
private final List<Placed> bricks = new ArrayList<Placed>();
boolean contains(long x) {
for (Placed b : bricks) {
if (x < b.x) return false;
if (x < b.end) return true;
}
return false;
}
boolean canInsert(Placed b) {
for (Placed old : bricks)
if (b.x < old.end && old.x < b.end) return false;
return true;
}
void insert(Placed b) {
int i = 0;
while (i < bricks.size() && bricks.get(i).x < b.x) ++i;
bricks.add(i, b);
}
Row copy() {
Row r = new Row();
r.bricks.addAll(bricks);
return r;
}
// Insert the equality members from part (e) here.
}
private final List<Row> rows = new ArrayList<Row>();
public void addBrick(Brick b, int x, int y) throws Exception {
if (b == null || b.col == null || b.width <= 0 || y < 0)
throw new Exception("invalid brick or height");
// Snapshot mutable input. Long arithmetic protects x + width.
int width = b.width;
Color colour = new Color(b.col.getRGB(), true);
Placed placed = new Placed((long) x, width, colour);
if (y > rows.size()) throw new Exception("unsupported height");
Row target = y == rows.size() ? new Row() : rows.get(y);
if (!target.canInsert(placed)) throw new Exception("overlap");
if (y > 0) {
Row below = rows.get(y - 1);
if (!below.contains(placed.x) || !below.contains(placed.end - 1))
throw new Exception("unsupported end");
}
// All checks precede the first modification of this wall.
target.insert(placed);
if (y == rows.size()) rows.add(target);
}
public boolean contains(int x, int y) {
return y >= 0 && y < rows.size() && rows.get(y).contains((long) x);
}
public Object row(int y) {
if (y < 0) throw new IllegalArgumentException("negative row");
return y < rows.size() ? rows.get(y).copy() : new Row();
}
}Here
Placed instances and their captured Color values are immutable objects, so copying a row's outer list suffices to create a detached snapshot. Returning its static type as Object exposes just the equality operation needed below, without exposing a mutable row API. A fuller application could instead return an immutable row-view object interface.The empty list represents a valid supportive wall. A rejected insertion performs no wall update. On success, the overlap test establishes validity in the target row; all other rows are unchanged. The support checks establish support for the new brick; adding material cannot remove support from old bricks. Appending a row only at the current height count keeps the row representation contiguous. Integer overflow in a right endpoint is prevented by converting the horizontal coordinate to
long before adding the positive int width. For instance, a brick at Integer.MAX_VALUE can still have a mathematically meaningful right endpoint.