The equivalence of the subregion representation and the wall representation for a certain class of rectangular dissections

A rectangular dissection is a partition of a rectangular space R info n ≱ 1 disjoint rectangles {r1, r2, . . ., rn. Two classes of dissections that are of particular interest in floor-space design and very-large-scale integration (VLSI) space partitioning are called T-plans and T*-plans, where the T*-plans form a subclass of the T-plans. We consider here the subregion tree representation t(D) of a T*-plan D, which describes the successive partitioning operations by which the dissection D is derived. There are four types of partitioning operations in a T*-plan: (1) the horizontal partitioning; (2) the vertical partitioning; (3) the left-spiral partitioning; and (4) the right-spiral partitioning. We show that for a T*-plan the subregion representation and the wall representation are equivalent in the sense that one can be obtained from the other in a unique fashion. The importance of this equivalence property lies in that while the two representations allow different types of design constraints to be represented in a more natural way, these constraints may be converted to the same representation for a more efficient solution of the design problem.

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