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Sweeping Shapes: Optimal Crumb Cleanup. Yonit Bousany, Mary Leah Karker, Joseph O’Rourke, Leona Sparaco. What does it mean to sweep a shape?.

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sweeping shapes optimal crumb cleanup

Sweeping Shapes: Optimal Crumb Cleanup

Yonit Bousany, Mary Leah Karker, Joseph O’Rourke, Leona Sparaco

slide3

Restricting our attention to two-sweeps and triangles, the minimum sweeping cost is always achieved by enclosing the triangle in a minimum perimeter parallelogram.

one flush lemma
One-flush Lemma

The minimal perimeter enclosing parallelogram is always flush against at least one edge of the convex hull.

[Mitchell and Polishchuk 2006]

conjecture
Conjecture

However.... Conjecture: Minimal cost sweeping can be achieved with two sweeps for any convex shape.

theorem
Theorem

Normalize triangle so that the longest edge=1. Let θ be the ab-apex.

  • If θ ≥ 90, the min cost sweep is determined by the parallelogram 2-flush against a and b.
  • If θ ≤ 90, the min cost sweep is determined by the rectangle 1-flush against the shortest side.
proof of one subcase
Proof of One Subcase

hb < 1

hb (1-b) < (1-b)

hb- b hb < 1-b

b·hb = 1·h1

hb - h1 < 1 - b

hb + b < 1 + h1

the best way of sweeping a shape is not necessarily achieved with two sweeps
The best way of sweeping a shape is not necessarily achieved with two sweeps:

An example requiring three sweeps.

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