3. Representation of the placement problem

Placement is an ordered combination of shapes. All combinations can be represented by a tree-structured state graph. The optimal placement problem then becomes a problem of finding the optimal path between the root of the tree, where no shape is placed, and a leaf of the tree, whose corresponding placement occupies the minimum area. To solve this optimization problem numerically, we need to define a state representation for traversing the search space and adopt a state exploration algorithm.