## 5.85. cumulative_two_d

Origin
Constraint

Arguments
 $\mathrm{\pi \pi ΄\pi ²\pi \pi °\pi ½\pi Ά\pi »\pi ΄\pi }$ $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\begin{array}{c}\mathrm{\pi \pi \pi \pi \pi }\mathtt{1}-\mathrm{\pi \pi \pi \pi },\hfill \\ \mathrm{\pi \pi \pi £\pi }\mathtt{1}-\mathrm{\pi \pi \pi \pi },\hfill \\ \mathrm{\pi \pi \pi \pi }\mathtt{1}-\mathrm{\pi \pi \pi \pi },\hfill \\ \mathrm{\pi \pi \pi \pi \pi }\mathtt{2}-\mathrm{\pi \pi \pi \pi },\hfill \\ \mathrm{\pi \pi \pi £\pi }\mathtt{2}-\mathrm{\pi \pi \pi \pi },\hfill \\ \mathrm{\pi \pi \pi \pi }\mathtt{2}-\mathrm{\pi \pi \pi \pi },\hfill \\ \mathrm{\pi \pi \pi \pi \pi \pi }-\mathrm{\pi \pi \pi \pi }\hfill \end{array}\right)$ $\mathrm{\pi »\pi Έ\pi Ό\pi Έ\pi }$ $\mathrm{\pi \pi \pi }$
Restrictions
 $\mathrm{\pi \pi \pi \pi \pi \pi \pi }_\mathrm{\pi \pi }_\mathrm{\pi \pi \pi \pi \pi }$$\left(2,\mathrm{\pi \pi ΄\pi ²\pi \pi °\pi ½\pi Ά\pi »\pi ΄\pi },\left[\mathrm{\pi \pi \pi \pi \pi }\mathtt{1},\mathrm{\pi \pi \pi £\pi }\mathtt{1},\mathrm{\pi \pi \pi \pi }\mathtt{1}\right]\right)$ $\mathrm{\pi \pi \pi \pi \pi \pi \pi }_\mathrm{\pi \pi }_\mathrm{\pi \pi \pi \pi \pi }$$\left(2,\mathrm{\pi \pi ΄\pi ²\pi \pi °\pi ½\pi Ά\pi »\pi ΄\pi },\left[\mathrm{\pi \pi \pi \pi \pi }\mathtt{2},\mathrm{\pi \pi \pi £\pi }\mathtt{2},\mathrm{\pi \pi \pi \pi }\mathtt{2}\right]\right)$ $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }$$\left(\mathrm{\pi \pi ΄\pi ²\pi \pi °\pi ½\pi Ά\pi »\pi ΄\pi },\mathrm{\pi \pi \pi \pi \pi \pi }\right)$ $\mathrm{\pi \pi ΄\pi ²\pi \pi °\pi ½\pi Ά\pi »\pi ΄\pi }.\mathrm{\pi \pi \pi £\pi }\mathtt{1}\beta ₯0$ $\mathrm{\pi \pi ΄\pi ²\pi \pi °\pi ½\pi Ά\pi »\pi ΄\pi }.\mathrm{\pi \pi \pi £\pi }\mathtt{2}\beta ₯0$ $\mathrm{\pi \pi ΄\pi ²\pi \pi °\pi ½\pi Ά\pi »\pi ΄\pi }.\mathrm{\pi \pi \pi \pi \pi \pi }\beta ₯0$ $\mathrm{\pi »\pi Έ\pi Ό\pi Έ\pi }\beta ₯0$
Purpose

Consider a set $\mathrm{\beta }$ of rectangles described by the $\mathrm{\pi \pi ΄\pi ²\pi \pi °\pi ½\pi Ά\pi »\pi ΄\pi }$ collection. Enforces that at each point of the plane, the cumulated height of the set of rectangles that overlap that point, does not exceed a given limit.

Example
$\left(\begin{array}{c}β©\begin{array}{ccccccc}\mathrm{\pi \pi \pi \pi \pi }\mathtt{1}-1\hfill & \mathrm{\pi \pi \pi £\pi }\mathtt{1}-4\hfill & \mathrm{\pi \pi \pi \pi }\mathtt{1}-4\hfill & \mathrm{\pi \pi \pi \pi \pi }\mathtt{2}-3\hfill & \mathrm{\pi \pi \pi £\pi }\mathtt{2}-3\hfill & \mathrm{\pi \pi \pi \pi }\mathtt{2}-5\hfill & \mathrm{\pi \pi \pi \pi \pi \pi }-4,\hfill \\ \mathrm{\pi \pi \pi \pi \pi }\mathtt{1}-3\hfill & \mathrm{\pi \pi \pi £\pi }\mathtt{1}-2\hfill & \mathrm{\pi \pi \pi \pi }\mathtt{1}-4\hfill & \mathrm{\pi \pi \pi \pi \pi }\mathtt{2}-1\hfill & \mathrm{\pi \pi \pi £\pi }\mathtt{2}-2\hfill & \mathrm{\pi \pi \pi \pi }\mathtt{2}-2\hfill & \mathrm{\pi \pi \pi \pi \pi \pi }-2,\hfill \\ \mathrm{\pi \pi \pi \pi \pi }\mathtt{1}-1\hfill & \mathrm{\pi \pi \pi £\pi }\mathtt{1}-2\hfill & \mathrm{\pi \pi \pi \pi }\mathtt{1}-2\hfill & \mathrm{\pi \pi \pi \pi \pi }\mathtt{2}-1\hfill & \mathrm{\pi \pi \pi £\pi }\mathtt{2}-2\hfill & \mathrm{\pi \pi \pi \pi }\mathtt{2}-2\hfill & \mathrm{\pi \pi \pi \pi \pi \pi }-3,\hfill \\ \mathrm{\pi \pi \pi \pi \pi }\mathtt{1}-4\hfill & \mathrm{\pi \pi \pi £\pi }\mathtt{1}-1\hfill & \mathrm{\pi \pi \pi \pi }\mathtt{1}-4\hfill & \mathrm{\pi \pi \pi \pi \pi }\mathtt{2}-1\hfill & \mathrm{\pi \pi \pi £\pi }\mathtt{2}-1\hfill & \mathrm{\pi \pi \pi \pi }\mathtt{2}-1\hfill & \mathrm{\pi \pi \pi \pi \pi \pi }-1\hfill \end{array}βͺ,4\hfill \end{array}\right)$

PartΒ (A) of FigureΒ 5.85.1 shows the 4 parallelepipeds of height 4, 2, 3 and 1 associated with the items of the $\mathrm{\pi \pi ΄\pi ²\pi \pi °\pi ½\pi Ά\pi »\pi ΄\pi }$ collection (parallelepipeds since each rectangle also has a height). PartΒ (B) gives the corresponding cumulated 2-dimensional profile, where each number is the cumulated height of all the rectangles that contain the corresponding region. The constraint holds since the highest peak of the cumulated 2-dimensional profile does not exceed the upper limit 4 imposed by the last argument of the constraint.

Typical
 $|\mathrm{\pi \pi ΄\pi ²\pi \pi °\pi ½\pi Ά\pi »\pi ΄\pi }|>1$ $\mathrm{\pi \pi ΄\pi ²\pi \pi °\pi ½\pi Ά\pi »\pi ΄\pi }.\mathrm{\pi \pi \pi £\pi }\mathtt{1}>0$ $\mathrm{\pi \pi ΄\pi ²\pi \pi °\pi ½\pi Ά\pi »\pi ΄\pi }.\mathrm{\pi \pi \pi £\pi }\mathtt{2}>0$ $\mathrm{\pi \pi ΄\pi ²\pi \pi °\pi ½\pi Ά\pi »\pi ΄\pi }.\mathrm{\pi \pi \pi \pi \pi \pi }>0$ $\mathrm{\pi »\pi Έ\pi Ό\pi Έ\pi }<$$\mathrm{\pi \pi \pi }$$\left(\mathrm{\pi \pi ΄\pi ²\pi \pi °\pi ½\pi Ά\pi »\pi ΄\pi }.\mathrm{\pi \pi \pi \pi \pi \pi }\right)$
Symmetries
• Items of $\mathrm{\pi \pi ΄\pi ²\pi \pi °\pi ½\pi Ά\pi »\pi ΄\pi }$ are permutable.

• Attributes of $\mathrm{\pi \pi ΄\pi ²\pi \pi °\pi ½\pi Ά\pi »\pi ΄\pi }$ are permutable w.r.t. permutation $\left(\mathrm{\pi \pi \pi \pi \pi }\mathtt{1},\mathrm{\pi \pi \pi \pi \pi }\mathtt{2}\right)$ $\left(\mathrm{\pi \pi \pi £\pi }\mathtt{1},\mathrm{\pi \pi \pi £\pi }\mathtt{2}\right)$ $\left(\mathrm{\pi \pi \pi \pi }\mathtt{1},\mathrm{\pi \pi \pi \pi }\mathtt{2}\right)$ $\left(\mathrm{\pi \pi \pi \pi \pi \pi }\right)$ (permutation applied to all items).

• $\mathrm{\pi \pi ΄\pi ²\pi \pi °\pi ½\pi Ά\pi »\pi ΄\pi }.\mathrm{\pi \pi \pi \pi \pi \pi }$ can be decreased to any value $\beta ₯0$.

• One and the same constant can be added to the $\mathrm{\pi \pi \pi \pi \pi }\mathtt{1}$ and $\mathrm{\pi \pi \pi \pi }\mathtt{1}$ attributes of all items of $\mathrm{\pi \pi ΄\pi ²\pi \pi °\pi ½\pi Ά\pi »\pi ΄\pi }$.

• One and the same constant can be added to the $\mathrm{\pi \pi \pi \pi \pi }\mathtt{2}$ and $\mathrm{\pi \pi \pi \pi }\mathtt{2}$ attributes of all items of $\mathrm{\pi \pi ΄\pi ²\pi \pi °\pi ½\pi Ά\pi »\pi ΄\pi }$.

• $\mathrm{\pi »\pi Έ\pi Ό\pi Έ\pi }$ can be increased.

Usage

The constraint is a necessary condition for the $\mathrm{\pi \pi \pi \pi \pi }$ constraint in 3 dimensions (i.e.,Β the placement of parallelepipeds in such a way that they do not pairwise overlap and that each parallelepiped has his sides parallel to the sides of the placement space).

Algorithm

A first natural way to handle this constraint would be to accumulate the compulsory partΒ [Lahrichi82] of the different rectangles in a quadtreeΒ [Samet89]. To each leave of the quadtree we associate the cumulated height of the rectangles containing the corresponding region.

Systems

geost in Choco.

related: $\mathrm{\pi \pi \pi \pi \pi }$Β ( is a necessary condition for $\mathrm{\pi \pi \pi \pi \pi }$: forget one dimension when the number of dimensions is equal to 3).

specialisation: $\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi \pi }$Β ($\mathrm{\pi \pi \pi \pi \pi \pi }$ of size 1 with a $\mathrm{\pi \pi \pi \pi \pi \pi }$ replaced by $\mathrm{\pi \pi \pi \pi }$ of $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }$ 1), $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }$Β ($\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }$ with a $\mathrm{\pi \pi \pi \pi \pi \pi }$ replaced by $\mathrm{\pi \pi \pi \pi }$ with same $\mathrm{\pi \pi \pi \pi \pi \pi }$).

Keywords