Algorithmic complexity of shape grammar implementation

Abstract

Computer-based shape grammar implementations aim to support creative design exploration by automating rule-application. This paper reviews existing shape grammar implementations in terms of their algorithmic complexity, extends the definition of shape grammars with sets of transformations for rule application, categorizes (parametric and non-parametric) sets of transformations, and analyses these categories in terms of the resulting algorithmic complexity. Specifically, it describes how different sets of transformations admit different numbers of targets (i.e., potential inputs) for rule application. In the non-parametric case, this number is quadratic or cubic, while in the parametric case, it can be non-polynomial, depending on the size of the target shape. The analysis thus yields lower bounds for the algorithmic complexity of shape grammar implementations that hold independently of the employed algorithm or data structure. Based on these bounds, we propose novel matching algorithms for non-parametric and parametric shape grammar implementation and analyze their complexity. The results provide guidance for future, general-purpose shape grammar implementations for design exploration.

Animation as a computational framework for architectural design composition

“…In a design process executed by hand, the designer is free to create and chose shapes and give them meaning in a complete ontological manner: a rectangle can be a Kitchen counter. This is a feat that is not yet possible through a computer, even though shape recognition and meaningful understanding can come from selected sets in A. I research (Wortmann and Stuffs 2018). Certainly, this freedom is crucial for the designer to explore what is possible within the design constraints.…”

Supporting Creativity : with Emergent Shapes in Shape Grammars

“…In very brief words, the research has been focused in conceptual and theoretical aspects, as the ones exposed in [4], in analysis, i.e., the development of specific shape grammars of languages of design extracted from corpuses of designs in architecture, product design or painting, see [4] [5] [6], for instance, and in synthesis, i.e., building specific shape grammars to define original languages of designs, as in [7] [8] [9] [10]. More research include the development of algorithms and data structures for shape manipulation and rule matching and application processes, which are very interesting for us in the present paper, as in [11] [12] [13] [14] [15] [16] [17] [18] [19], where some of this papers focus also on appropriate interfaces and generic and reusable shape grammar interpreters, including for didactical purposes.…”

“…In [18] an approach is proposed that uses graphs, more specifically, graph grammars, to represent shapes and shape grammars, relegating the problem to another of graph representation and manipulation. The problem of sub-shape detection is a computationally hard one and is equivalent in hardness to the subgraph isomorphism problem, see [19], where the computational complexity of algorithms used in different kinds of shape grammars, including in sub-shape detection, is analyzed.…”

Shapes, structures and shape grammar implementation

“…But, despite some promising results, these implementations have not yet satisfactorily met the potential suggested by theoretical developments of the shape grammar formalism [1]. The visual nature of shape grammars means they are particularly suited to the early stages of design, where ambiguity and emergence play an important role [5,6], but implementing the visual shape computations that result from applying shape rules has proven to be difficult within the logics of symbolic computation that underlie CAD and CACD systems [7,8].…”

“…via a general interpreter, should be able to accommodate the changes in structure that occur during shape computations. However, the problem of shape recognition is difficult, and for parametric shapes is NP-hard [7]. Instead, shape rule application is often made tractable by assuming a fixed part structure which focusses recognition on known parts [5,17], and this paper investigates the consequences of such an assumption.…”