用于平面图中矩形形状模块的算法设计与基于图的分类
Algorithmic Design and Graph-Based Classification for Rectilinear-Shaped Modules in Floor Plans
AI总结：
本文提出了一种基于图论的算法框架，用于在平面图中实现L形和T形模块，通过引入形状保持约束解决复杂模块形状的构造问题。
摘要：
我们提出了一种图论框架，用于构建支持非矩形模块的平面图，特别强调L形和T形几何结构。与传统方法主要关注矩形模块和外边界约束不同，我们的方法明确纳入了在矩形平面图表示中实现更复杂模块形状时出现的结构限制。该框架基于三角形图，并探讨了如何利用算法图论技术嵌入L形和T形模块，同时保持规定的邻接关系。我们证明并非所有三角形图都能实现这样的实现，并识别出阻止所需模块几何存在的结构限制。为捕捉这些限制，我们引入了一种形状保持约束，确保模块几何不能通过边界变形改变，因为此类改变会增加相邻模块的组合复杂性或违反邻接关系。我们提出了一种基于优先级规范顺序的线性时间构造算法，该算法在包含至少一个内部K4或满足特定存在条件的两个内部K4子图的图中，能够实现L形和T形模块。该算法简单、构造性强且可以直接实施，使其适用于实际平面图生成工作流程。我们最后讨论了扩展到其他模块形状和更广泛支持图结构的可能性。
英文摘要：
We present a graph-theoretic framework for constructing floor plans that support non-rectangular modules, with particular emphasis on L-shaped and T-shaped geometries. Unlike traditional approaches that primarily focus on rectangular modules and outer boundary constraints, our method explicitly incorporates structural restrictions that arise when realizing more complex module shapes within rectangular floor-plan representations. The framework is based on triangulated graphs and investigates how algorithmic graph theory techniques can be used to embed L and T-shaped modules while preserving prescribed adjacencies. We show that not every triangulated graph admits such realizations and identify structural limitations that prevent the existence of the desired module geometries. To capture these limitations, we introduce a shape-preservation constraint that ensures module geometries cannot be altered through boundary deformation, as such changes would either increase the combinatorial complexity of neighboring modules or violate adjacency relationships. We propose a linear-time construction algorithm based on a prioritized canonical ordering that realizes L and T-shaped modules in graphs containing at least one internal K4, or two internal K4 subgraphs satisfying specific existence conditions. The algorithm is simple, constructive, and directly implementable, making it suitable for practical floor-plan generation workflows. We conclude by discussing extensions to additional module shapes and broader classes of supporting graph structures.
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