Recent work on constrained graph layout has involved projection of simple two-variable linear equality and inequality constraints in the context of majorization or gradient-projection based optimization. While useful classes of containment, alignment and rectangular non-overlap constraints could be built using this framework, a severe limitation was that the layout used an axis-separation approach such that all constraints had to be axis aligned. In this paper we use techniques from Procrustes Analysis to extend the gradient-projection approach to useful types of non-linear constraints. The constraints require subgraphs to be locally fixed into various geometries—such as circular cycles or local layout obtained by a combinatorial algorithm (e.g. orthogonal or layered-directed)—but then allow these sub-graph geometries to be integrated into a larger layout through translation, rotation and scaling.\ud
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机译:关于约束图布局的最新工作涉及在基于最大化或基于梯度投影的优化情况下,简单的两变量线性等式和不等式约束的投影。尽管可以使用此框架构建有用的包含,对齐和矩形非重叠约束类别,但一个严重的限制是,布局使用了轴分离方法,因此所有约束都必须进行轴对齐。在本文中,我们使用Procrustes Analysis中的技术将梯度投影方法扩展到有用的非线性约束类型。约束要求将子图局部固定到各种几何形状中,例如圆形循环或通过组合算法获得的局部布局(例如正交或分层定向),但随后允许这些子图几何形状通过平移集成到更大的布局中,旋转和缩放。\ ud
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