REPRESENTATION AND GENERATION OF RECTANGULAR DISSECTIONS Ulrich Flemming School of Architecture Yale University New Haven, Connecticut 06520 Summary. The stepwise development of a data struc- ture for representing dissections of rectangles into rectangular components is outlined. The data structure is used to define a finite set of "principal options" for a class of space allocation problems, and a proce- dure is described which generates dissections repre- senting these options. i. Introduction Figure 1.2 shows 21 floor plans each of which re- presents a feasible solution for the space allocation problem described in figure i.I. The elements of this problem are: A (rectangular) design area surrounded by four ex- ternal spaces N,E~S and W. - Four internal spaces which have to be "densely packed" within the design area, i.e. allocated such that no two spaces overlap and no area remains which does not belong to one of the given spaces. - Dimensional constraints restricting the shapes of the internal spaces and the design area. - Topological constraints requiring direct adjacen- cies between certain pairs of spaces. The internal spaces represent the the rooms of an effi- ciency apartment which is accessible from the east, e.g. via a public corridor, and receives natural light from the west. The problems dealt with in this paper con- sist, like the sample problem~ of a number of internal spaces which have to be densely packed within a design area subject to dimensional and topological constraints. The solutions of figure 1.2 exhibit distinct geo- metric or topological properties, within the limita- tions imposed by these properties, the internal spaces were dimensioned such that in each solution, the peri- meter of the design area becomes minimal (the minimum perimeters are listed in brackets above each plan). Figure 1.2 is intended to demonstrate two points: Space allocation problems are often undercon- strained and can have a large number of feasible solutions. By inspecting just one of these solu- tions, a designer hardly learns enough about the options available to him. Even the generation of a solution that is optimal according to a certain criterion can be insuffi- cient. It might prohibit a designer from looking for other solutions which are hardly worse with respect to the used criterion, but have advantages according to other criteria. Observations of this kind are confirmed if problems containing a larger number of internal spaces are studied. 2 These results raise questions about the prac- tical usefulness of computer programs which generate single feasible or (sub)optimal solutions to space allocation problems. More generally, they raise ques- tions about the nature of such problems, and in order to study these questions in depth, a procedure is needed which structures the (possibly infinite) sets of solutions such that, in each case, a finite set of "principal options" can be generated and analyzed. A meaningful definition of such a set depends crucial- ly upon the data structure selected in order to repre- sent the objects to be generated. Following an ap- proach suggested by Tompa II, the development of such a data structure is traced in the following sections. For reasons given below, the discussion is restricted to a certain class of floor plans known a "rectangular dissections." The work presented here was first docu- mented in Flemming(1977a). 2 2. Data Reality It is known that the spaces in a floor plan can be restricted by constraints which depend upon particular geometric properties of that plan and can vary with these properties. I Constraints of this kind are called dependent constraints in the following; independent constraints are not influenced by the geometry of a plan. It appears difficult~ if not impossible, to deter- mine both the basic geometry of a solution and the dimensions of the spaces contained in it in one inte- grated step. All approaches known to me rely, either implicitly or explicitly, on a procedure consisting of two steps. Step i defines certain geometric properties of the solution, and step 2 determines the dimensions of the spaces in response to these characteristics. 1,9 This observation provides the starting point for the following discussion. It is also known that the formulation of the de- pendent constraints becomes largely facilitated if the shapes of the internal spaces are predetermined. I It can then be possible to express both the dependent and independent constraints through a system of simultane- ous equations or inequalities and execute step 2 by means of well-established numerical techniques. The formulation of the dependent Constraints becomes par- ticularly easy if all spaces, including the design area, are rectangular. In this case, each solution consists of a rectangle dissected into rectangular components each of which is occupied by exactly one of the internal spaces that have to be allocated. Such a plan is called a dissection in the following, and the procedure outlined below is restricted to the genera- tion of dissections, without such limitations, a deeper analysis of the questions raised in the previ- ous section appears extremely difficult at the present time. 138